Second-order delay ordinary differential equations, their symmetries and application to a traffic problem

TL;DR

This paper applies Lie group theory to second-order delay ordinary differential equations, revealing their symmetry structures and deriving exact solutions, including a traffic flow model example.

Contribution

It extends symmetry analysis to second-order delay differential equations, identifying their symmetry algebra dimensions and providing a method to find exact solutions.

Findings

Symmetry algebra dimension ranges from 0 to 6 for nonlinear DODSs.

Linear or linearizable DODSs have infinite-dimensional symmetry algebras.

Exact solutions for a traffic flow DODS model are obtained.

Abstract

This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_-$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_-$. Two previous articles were devoted to {\it first}-order DODSs, here we concentrate on a large class of {\it second}-order ones. We show that within this class the symmetry algebra can be of dimension $n$ with $0 \leq n \leq 6$ for nonlinear DODSs and must be $n=\infty$ for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.

Equations

$$\dot{y}=f(x,y,y_{-}),\frac{\partial f}{\partial y_{-}}\not\equiv 0,x\in I,$$

$$x_{-}=g(x,y,y_{-}),x_{-}<x,g(x,y,y_{-})\not\equiv \text{mbox}const.$$

$$X=\eta (x,y)\frac{\partial }{\partial y}.$$

$$\ddot{y}=f(x,y,y_{-},\dot{y},{\dot{y}}_{-}),{\left(\frac{\partial f}{\partial y_{-}}\right)}^2+{\left(\frac{\partial f}{\partial {\dot{y}}_{-}}\right)}^2\not\equiv 0,x\in I,$$

$$x_{-}=g(x,y,y_{-},\dot{y},{\dot{y}}_{-}),x_{-}<x,g(x,y,y_{-},\dot{y},{\dot{y}}_{-})\not\equiv \text{mbox}const.$$

$$X_{\alpha }={\xi }_{\alpha }(x,y)\frac{\partial }{\partial x}+{\eta }_{\alpha }(x,y)\frac{\partial }{\partial y}.$$

$$\text{mbox}\mathbf{pr}{X}_{\alpha }={\xi }_{\alpha }\frac{\partial }{\partial x}+{\eta }_{\alpha }\frac{\partial }{\partial y}+{\xi }_{\alpha }^{-}\frac{\partial }{\partial x_{-}}+{\eta }_{\alpha }^{-}\frac{\partial }{\partial y_{-}}+{\zeta }_{\alpha }^1\frac{\partial }{\partial \dot{y}}+{\zeta }_{\alpha }^{1-}\frac{\partial }{\partial {\dot{y}}_{-}}+{\zeta }_{\alpha }^2\frac{\partial }{\partial \ddot{y}}$$

$${\xi }_{\alpha }={\xi }_{\alpha }(x,y),{\eta }_{\alpha }={\eta }_{\alpha }(x,y),{\xi }_{\alpha }^{-}={\xi }_{\alpha }(x_{-},y_{-}),{\eta }_{\alpha }^{-}={\eta }_{\alpha }(x_{-},y_{-}),$$

$${\zeta }_{\alpha }^1(x,y,\dot{y})=D({\eta }_{\alpha })-\dot{y}D({\xi }_{\alpha }),{\zeta }_{\alpha }^{1-}(x_{-},y_{-},{\dot{y}}_{-})=D_{-}({\eta }_{\alpha }^{-})-{\dot{y}}_{-}{D}_{-}({\xi }_{\alpha }^{-}),$$

$${\zeta }_{\alpha }^2(x,y,\dot{y},\ddot{y})=D({\zeta }_{\alpha }^1)-\ddot{y}D({\xi }_{\alpha }),$$

$$\text{mbox}\mathbf{pr}{X}_{\alpha }\Phi (x,y,x_{-},y_{-},\dot{y},{\dot{y}}_{-},\ddot{y})=0,\alpha =1,...,n.$$

$$k=\text{mbox}dimM-(\text{mbox}dimG-\text{mbox}dim{G}_0)$$

$$F(I_1,...,I_k)=0,G(I_1,...,I_k)=0,\text{mbox}det\left(\frac{\partial (F,G)}{\partial (\ddot{y},x_{-})}\right)\ne 0$$

$$Z=\left(\begin{array}[]{ccccccc}\xi_{1}&\eta_{1}&\xi_{1}^{-}&\eta_{1}^{-}&\zeta_{1}^{1}&\zeta_{1}^{1-}&{\zeta}_{1}^{2}\\ \vdots&&&&\\ \xi_{n}&\eta_{n}&\xi_{n}^{-}&\eta_{n}^{-}&\zeta_{n}^{1}&\zeta_{n}^{1-}&{\zeta}_{n}^{2}\\ \end{array}\right).$$

$$k=\text{mbox}dimM-\text{mbox}rankZ$$

$${\text{mbox}\mathbf{pr}{X}_{\alpha }\Phi (x,y,x_{-},y_{-},\dot{y},{\dot{y}}_{-},\ddot{y})|}_{\Phi =0}=0,\alpha =1,...,n.$$

$$X_{\alpha }={\eta }_{\alpha }(x,y)\frac{\partial }{\partial y}.$$

$$S_m=\{{\partial }_y,x{\partial }_y,{\chi }_2(x){\partial }_y,...,{\chi }_m(x){\partial }_y\},0\le m<\infty ,$$

$$H_m=\{{\partial }_y,x{\partial }_y,{\chi }_2(x){\partial }_y,...,{\chi }_m(x){\partial }_y,y{\partial }_y\},0\le m<\infty ,$$

$$\ddot{y}=a_1(x)\dot{y}+a_2(x){\dot{y}}_{-}+a_3(x)y+a_4(x)y_{-}+b(x),x_{-}=g(x),$$

$$a_2^2(x)+a_4^2(x)\not\equiv 0,g(x)<x,g(x)\not\equiv \text{mbox}const.$$

$$\ddot{y}=a_1(x)\dot{y}+a_2(x){\dot{y}}_{-}+a_3(x)y+a_4(x)y_{-},x_{-}=g(x).$$

$$X_1=\frac{\partial }{\partial y},X_2=x\frac{\partial }{\partial y},X_3=\chi (x)\frac{\partial }{\partial y},X_4=y\frac{\partial }{\partial y},\ddot{\chi }(x)\not\equiv 0.$$

$$I_{1}=x,\qquad I_{2}=x_{-},\qquad I_{3}={\left|\begin{array}[]{cccc}\ddot{y}&\dot{y}&y&y_{-}\\ 0&0&1&1\\ 0&1&x&x_{-}\\ \ddot{\chi}(x)&\dot{\chi}(x)&{\chi}(x)&{\chi}(x_{-})\\ \end{array}\right|\over\left|\begin{array}[]{cccc}\dot{y}&\dot{y}_{-}&y&y_{-}\\ 0&0&1&1\\ 1&1&x&x_{-}\\ \dot{\chi}(x)&\dot{\chi}(x_{-})&{\chi}(x)&{\chi}(x_{-})\\ \end{array}\right|}.$$

$$\left|\begin{array}[]{cccc}\ddot{y}&\dot{y}&y&y_{-}\\ 0&0&1&1\\ 0&1&x&x_{-}\\ \ddot{\chi}(x)&\dot{\chi}(x)&{\chi}(x)&{\chi}(x_{-})\\ \end{array}\right|=f(x)\left|\begin{array}[]{cccc}\dot{y}&\dot{y}_{-}&y&y_{-}\\ 0&0&1&1\\ 1&1&x&x_{-}\\ \dot{\chi}(x)&\dot{\chi}(x_{-})&{\chi}(x)&{\chi}(x_{-})\\ \end{array}\right|$$

$$x_{-}=g(x).$$

$$\left|\begin{array}[]{ccc}0&1&1\\ 1&x&x_{-}\\ \dot{\chi}(x)&{\chi}(x)&{\chi}(x_{-})\\ \end{array}\right|_{x_{-}=g(x)}={\chi}(x)-{\chi}(g(x))-\dot{\chi}(x)(x-g(x)){\not\equiv}0$$

$$X_1=\frac{\partial }{\partial y},X_2=x\frac{\partial }{\partial y},X_3={\chi }_2(x)\frac{\partial }{\partial y},X_4={\chi }_3(x)\frac{\partial }{\partial y},$$

$$I_{1}=x,\qquad I_{2}=x_{-},\qquad I_{3}=\left|\begin{array}[]{ccccc}\ddot{y}&\dot{y}&\dot{y}_{-}&y&y_{-}\\ 0&0&0&1&1\\ 0&1&1&x&x_{-}\\ \ddot{\chi}_{2}(x)&\dot{\chi}_{2}(x)&\dot{\chi}_{2}(x_{-})&{\chi}_{2}(x)&{\chi}_{2}(x_{-})\\ \ddot{\chi}_{3}(x)&\dot{\chi}_{3}(x)&\dot{\chi}_{3}(x_{-})&{\chi}_{3}(x)&{\chi}_{3}(x_{-})\\ \end{array}\right|.$$

$$\left|\begin{array}[]{ccccc}\ddot{y}&\dot{y}&\dot{y}_{-}&y&y_{-}\\ 0&0&0&1&1\\ 0&1&1&x&x_{-}\\ \ddot{\chi}_{1}(x)&\dot{\chi}_{1}(x)&\dot{\chi}_{1}(x_{-})&{\chi}_{1}(x)&{\chi}_{1}(x_{-})\\ \ddot{\chi}_{2}(x)&\dot{\chi}_{2}(x)&\dot{\chi}_{2}(x_{-})&{\chi}_{2}(x)&{\chi}_{2}(x_{-})\\ \end{array}\right|=f(x)$$

$$\left|\begin{array}[]{cccc}0&0&1&1\\ 1&1&x&x_{-}\\ \dot{\chi}_{1}(x)&\dot{\chi}_{1}(x_{-})&{\chi}_{1}(x)&{\chi}_{1}(x_{-})\\ \dot{\chi}_{2}(x)&\dot{\chi}_{2}(x_{-})&{\chi}_{2}(x)&{\chi}_{2}(x_{-})\\ \end{array}\right|_{x_{-}=g(x)}{\not\equiv}0$$

Full text

Second-order delay ordinary differential equations, their symmetries and application to a traffic problem

Vladimir A. Dorodnitsyna, Roman Kozlovb,

Sergey V. Meleshkoc and Pavel Winternitzd

a Keldysh Institute of Applied Mathematics, Russian Academy of Science,

Miusskaya Pl. 4, Moscow, 125047, Russia;

e-mail: Dorodnitsyn@Keldysh.ru

b Department of Business and Management Science, Norwegian School of Economics, Helleveien 30, 5045, Bergen, Norway;

e-mail: Roman.Kozlov@nhh.no

c School of Mathematics, Institute of Science, Suranaree University of Technology, 30000, Thailand;

e-mail: sergey@math.sut.ac.th

d Centre de Recherches Mathématiques and Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, H3C 3J7, Canada;

e-mail: wintern@crm.umontreal.ca

Abstract

This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_{-}$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_{-}$. Two previous articles were devoted to first-order DODSs, here we concentrate on a large class of second-order ones. We show that within this class the symmetry algebra can be of dimension $n$ with $0\leq n\leq 6$ for nonlinear DODSs and must be $n=\infty$ for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.

1 Introduction and formulation of the problem

Two previous articles were devoted to the adaptation of Lie group and Lie algebra theory to the study of delay ordinary differential equations [6, 7]. In these articles we restricted ourselves to the case of first-order DODEs, supplemented by a general delay equation. Thus we considered first-order delay ordinary differential systems (DODSs) of the form

[TABLE]

Here $I$ is a finite or semifinite interval and $f$ and $g$ are arbitrary smooth functions. For details and motivation we refer to our articles [6, 7].

Our main results were the following:

We classified DODSs of the form (1.1) into conjugacy classes under arbitrary Lie point transformations and found that their Lie point symmetry groups can have dimension $n$ with $0\leq n\leq 3$, or infinity. If $n=\infty$ the DODE is linear or linearizable by a point transformation. In general the Lie algebra of the infinite-dimensional symmetry group of a linear DODS is a solvable Lie algebra with an infinite-dimensional nilradical and is realized by vector fields of the form

[TABLE] 2. 2.

If the symmetry algebra of a DODS contains a 2-dimensional subalgebra realized by linearly connected vector fields, then this DODS is linearizable (or already linear) with $g=g(x)$. 3. 3.

The symmetry algebra for genuinely nonlinear DODEs has dimension $n\leq 3$. For algebras with $n=2$ or $3$ we presented a method for obtaining exact particular solutions.

Here we provide a similar analysis of SECOND-ORDER delay ordinary differential systems

[TABLE]

Thus a DODS consists of a delay ordinary differential equation (DODE (1.2a)) and a delay equation (1.2b)

In most of the existing literature the delay $\Delta x=x-x_{-}=\tau>0$ is constant. An alternative is to impose specific conditions characterizing the physical problem under consideration. We impose the conditions given in (1.2b) and leave the function $g$ free and to be determined by symmetry considerations.

As in the case or ordinary differential equations (ODEs) we will be working with vector fields of the form

[TABLE]

Integrating them, we obtain (local) Lie point transformations acting on the independent variable $x$, the dependent one $y$ and thus also on functions $y(x)$. They must be prolonged to act on DODSs viewed as functions of $x$, $y$, $\dot{y}$ and $\ddot{y}$ evaluated at two points, the reference point $x$ and the delay point $x_{-}$. The transformations will leave the DODS (1.2) invariant on its solution set, i.e. the prolongation of the vector field (1.3) must annihilate the DODS on its solution set. From this point of view the symmetry group of DODS are reminiscent of those of ordinary difference systems that leave invariant the solution set of two equations: the difference equation itself and an equation determining the lattice [3, 4, 5, 15, 16, 17, 18, 19].

Delay differential equations have many properties that distinguish them from differential equations without delay [8, 9, 13, 20, 43]. For earlier work on solutions of delay equations be they numerical, or exact, using group theory and other approaches we refer to [12, 22, 25, 26, 30, 33, 34, 35, 36, 38, 40]. We recall that DODEs are standardly solved numerically or otherwise using the method of steps as described e.g. in [23] and adapted to nonconstant delay in our previous articles [6, 7].

In Section 2 we describe our general classification procedure. All nonlinear DODSs of the type (1.2) are classified into symmetry classes in Section 3. In Section 4 we investigate linear DODSs. An algorithm for calculating exact analytic solutions of DODSs is given in Section 5. Applications to a ”follow the leader” model of traffic flow are given in Section 6. Finally, the concluding remarks are presented in Section 7.

2 Second-order DODSs invariant under local Lie point transformations groups

The method of constructing invariant first-order DODSs in Refs. [6, 7] can be generalized to DODSs of any order and any number of delay points. For the DODS (1.2) we prolong the vector field (1.3) to

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

where $D$ and $D_{-}$ are the total derivative operators in points $x$ and $x_{-}$, respectively. Simply put, the coefficients $\zeta_{\alpha}^{1}$ and ${\zeta}_{\alpha}^{2}$ are calculated as for ODEs [28, 29] and the coefficients $\xi_{\alpha}^{-}$, $\eta_{\alpha}^{-}$ and $\zeta_{\alpha}^{1-}$ are obtained by shifting $x$ and $y$ to $x_{-}$ and $y_{-}$ in $\xi_{\alpha}$, $\eta_{\alpha}$ and $\zeta_{\alpha}^{1}$.

We construct, classify and represent all symmetry classes of DODSs of the type (1.2) using essentially the same method as we used in previous articles [6, 7] for first-order DODSs:

For each algebra of vector fields in the list given in [11] we construct the prolongations (2.1) of the chosen basis vectors $\{X_{\alpha},\alpha=1,...,n\}$. 2. 2.

Running through all the algebras of the list, we construct the ”strongly invariant” DODSs out of universal point invariants satisfying

[TABLE]

The method of characteristics gives us a set of independent elementary invariants of the corresponding Lie group action. We label them $I_{1}$, … $I_{k}$, where $k$ satisfies

[TABLE]

with $M\sim(x,y,x_{-},y_{-},\dot{y},\dot{y}_{-},\ddot{y})$. In (2.3) $G$ is the local Lie point symmetry group corresponding to the considered Lie algebra and $G_{0}\subset G$ is the stabilizer of a generic point on the manifold $M$.

The strongly invariant DODSs can all be written as

[TABLE]

($F$ and $G$ are otherwise arbitrary smooth functions).

It is convenient to introduce a matrix $Z$ for each algebra of prolonged vector fields, namely

[TABLE]

The number of strong invariants in then expressed as

[TABLE]

where $\mbox{rank}\ Z$ is evaluated at a generic point. 3. 3.

Complement the strongly invariant DODSs found in Step 2 above by weakly invariant ones. These are found using ”weak invariants”. They satisfy a weaker equation than (2.2), namely

[TABLE]

The weak invariants $J_{a}$ are then invariant for a Lie group action on a submanifold of $M$: $G(J_{a})\neq J_{a}$, but $\left.G(J_{a})\right|_{J_{a}=0}=0$.

3 Representative list of second-order nonlinear DODSs

We proceed as outlined in Section 2 and as we did in our previous article on first-order DODSs [6]. Thus we run through the list of Lie algebras of vector fields presented in [11], proceeding by dimension. For each algebra in the list we construct a basis for all strong and weak invariants in the space with coordinates $(x,y,x_{-},y_{-},\dot{y},\dot{y}_{-},\ddot{y})$. We then construct the most general DODS of the form (1.2).

We shall not present any of the standard calculations here. The final results are summed up in Tables 1, 2, 3 and 4. The algebras have dimensions $\mbox{dim}\ L$ satisfying $1\leq\mbox{dim}\ L\leq 6$. The reason for this cut-off is that only linear and linearizable second-order DODSs (of the form (1.2)) have symmetry algebras of dimensions larger than $\mbox{dim}\ L=6$.

Linear DODSs are studied below in Section 4. Contrary to second-order linear ODEs (with no delay) linear DODSs of any order allow infinite-dimensional Abelian Lie point symmetry algebras, realized by linearly connected vector fields. These algebras are just expressions of the linear superposition principle for linear DODSs.

Let us now discuss the individual tables. Table 1 corresponds to algebras of dimension 1 and 2. In general the delay equation as well as the DODE is nonlinear. For special choices of the functions $f$ and $g$ they may be linear and then the symmetry algebra is infinite-dimensional. Notice that for $\mbox{\bf A}_{2,1}$ and $\mbox{\bf A}_{2,3}$ the vector fields are linearly connected. Contrary to the case of first-order DODSs this does not imply that the second-order DODSs are linear.

From Table 2 and Ref. [39] we see that each isomorphy class of three-dimensional real Lie algebras allows at least one class of invariant DODSs and the DODEs and delay equations involve arbitrary functions $f$ and $g$ of two variables. We mention that the particular case $a=1$ of algebra $\mbox{\bf A}_{3,3}^{a}$, one of the algebras ${\mathfrak{sl}}(2,\mathbb{R})$ (namely $\mbox{\bf A}_{3,11}$), and the algebra $3{\mathfrak{n}}_{1,1}$ are realized entirely by linearly connected vector fields since all three vector fields involve only $y$-derivatives. This does not force the DODSs to be linear.

The classification results for $\mbox{dim}\ L=4$ are summed up in Table 3. Alltogether 13 classes of four-dimensional indecomposable real Lie algebra exist [39]. Two of them ${\mathfrak{s}}_{4,7}$ and ${\mathfrak{s}}_{4,9}$ are missing in Table 3 because they cannot be represented by real vector fields in two dimensions [11, 37]. The decomposable algebras ${\mathfrak{n}}_{1,1}\oplus{\mathfrak{n}}_{3,1}$ and ${\mathfrak{n}}_{1,1}\oplus{\mathfrak{o}}(3)$ are missing for the same reason. Two other Lie algebras are omitted in Table 3 for a different reason. They are realized by linearly connected vector fields and we shall see in Section 4 that they lead to linear (or linearizable) DODSs. They correspond to the solvable Lie algebra ${\mathfrak{s}}_{4,3}$ with basis $\{\partial_{y},x\partial_{y},\chi(x)\partial_{y},y\partial_{y}\}$, $\ddot{\chi}(x){\not\equiv}0$ and the Abelian algebra $4{\mathfrak{n}}_{1,1}$ with basis $\{\partial_{y},x\partial_{y},\chi_{1}(x)\partial_{y},\chi_{2}(x)\partial_{y}\}$, where the functions $\{1,x,\chi_{1}(x),\chi_{2}(x)\}$ are linearly independent.

All algebras in Table 3 correspond to nonlinear DODEs and delay relations that both involve arbitrary functions $f$ and $g$ of one variable each. For $f$ and $g$ generic the equations are nonlinear. None of the symmetry algebras consists entirely of linearly connected vector fields, though many of them have three-dimensional subalgebras of linearly connected vector fields.

Finally, let us discuss Table 4 devoted to nonlinear DODSs with symmetry algebras of dimension 5 and 6. We present a representative list of all five- and six-dimensional Lie algebras that can be realized by vector fields in two dimensions, do not contain a four-dimensional subalgebra realized by linearly connected vector fields and do leave a DODS invariant. For $\mbox{dim}\ L=5$ there are 8 such algebras. Each class of DODSs depends on two arbitrary constants but no arbitrary functions appear. The Levi decomposable Lie algebra ${\mathfrak{sl}}(2,\mathbb{R})\ltimes 2{\mathfrak{n}}_{1,1}$ appears in two inequivalent ways. The algebras $\mbox{\bf A}_{5,1}$ – $\mbox{\bf A}_{5,4}$ are solvable with nilradical ${\mathfrak{n}}_{4,1}$, i.e. $\{X_{1},...,X_{4}\}$; $\mbox{\bf A}_{5,5}$ is also solvable with nilradical ${\mathfrak{n}}_{3,1}$, i.e. $\{X_{1},X_{2},X_{3}\}$; $\mbox{\bf A}_{5,6}$ and $\mbox{\bf A}_{5,7}$ are isomorphic but not conjugate. The radical $\{X_{4},X_{5}\}$ is linearly connected for $\mbox{\bf A}_{5,6}$ but not for $\mbox{\bf A}_{5,7}$.

For $\mbox{dim}\ L=6$ there exist only three Lie algebras that fulfil all conditions. The DODSs depend on one constant, no arbitrary functions. We mention that in cases $\mbox{\bf A}_{5,1}-\mbox{\bf A}_{5,5}$ and $\mbox{\bf A}_{6,2}$ the DODE is actually linear, but the delay relation depends on the solution $y$ and this induces nonlinearity.

For $\mbox{dim}\ L\geq 7$ the only Lie algebra realized by the considered vector fields that does not contain a four-dimensional subalgebra of linearly connected vector fields is ${\mathfrak{sl}}(3,\mathbb{R})$ (see [11]). There is no second-order DODS invariant under the Lie group ${{SL}}(3,\mathbb{R})$. All other Lie algebras with $\mbox{dim}\ L\geq 7$ are contained in the series 20, …, 28 of [11] and lead to linear DODSs.

4 Symmetry algebras of linear second-order DODSs

In our previous articles [6, 7] on DODSs we considered in particular the symmetries of linear and linearizable first-order DODSs. It was shown that a special role is played by symmetry algebras of linearly connected vector fields. These are vector fields that in the case of a two-dimensional real manifold can all be simultaneously reduced to the form

[TABLE]

The representative list of algebras of vector fields in a real plane ([11], Table 1) contains 3 individual algebras of this type, namely algebras 9, 10 and 11 in the table (here we use notations of [11]), and two infinite families, 20 and 21 in the same table.

We adapt these results to our needs. We include algebras 9 and 10 in the families 20 and 21, respectively. The two families now are

[TABLE]

[TABLE]

where we put $\chi_{0}(x)=1$, $\chi_{1}(x)=x$ and the functions $\chi_{0}(x)$, …, $\chi_{m}(x)$ are linearly independent. We remark that algebra 11, namely ${\mathfrak{sl}}(2,\mathbb{R})\sim\{\partial_{y},y\partial_{y},y^{2}\partial_{y}\}$, is not included in the infinite families.

It was shown in Ref. [6] that if a symmetry algebra of a first-order DODE contains a two-dimensional subalgebra of linearly connected vector fields, it is linearizable. That is the case of $S_{1}=\{\partial_{y},x\partial_{y}\}$ for linear inhomogeneous DODSs or $H_{0}=\{\partial_{y},y\partial_{y}\}$ for homogeneous ones. Here we show that for second-order DODSs (with one delay point) a four-dimensional subalgebra of linearly connected vector fields has the same effect.

Thus, we now turn to the general second-order DODS with one delay point $x_{-}$

[TABLE]

where $a_{i}(x)$, $b(x)$ and $g(x)$ are arbitrary real functions, smooth in some interval $x\in I$. The functions satisfy

[TABLE]

We will mainly use homogeneous DODS, i.e. (4.3) with $b(x)=0$

[TABLE]

Theorem 4.1

Let the DODS (1.2) admit a four-dimensional symmetry algebra realized by linearly connected vector fields. Then the DODS can be transformed into the linear DODS (4.3).

Proof. Precisely two different nonisomorphic four-dimensional Lie algebras of linearly connected vector fields in two dimensions exist [11, 6]. Both were listed in Table 1 of Ref. [11] in the infinite series of algebras 20 and 21. Any four-dimensional Lie algebra realized by linearly connected vector fields can hence be transformed into one of them.

Solvable algebra ${\mathfrak{s}}_{4,3}$ with operators $\mbox{\bf A}_{4,5}$ (also $H_{3}$ in (4.2))

[TABLE]

In this case there are three invariants in the space $(x,y,x_{-},y_{-},\dot{y},\dot{y}_{-},\ddot{y})$:

[TABLE]

We obtain the invariant linear homogeneous DODE

[TABLE]

with the invariant delay relation

[TABLE]

Note that the DODE is a second order one if and only if the condition

[TABLE]

holds. 2. 2.

Abelian algebra $4{\mathfrak{n}}_{1,1}$ with operators $\mbox{\bf A}_{4,22}$ (also $S_{3}$ in (4.1))

[TABLE]

where $\{1,x,\chi_{2}(x),\chi_{3}(x)\}$ are linearly independent functions.

In the space $(x,x_{-},y,y_{-},\dot{y},\dot{y}_{-},\ddot{y})$ there are three invariants

[TABLE]

We obtain the invariant linear inhomogeneous DODE

[TABLE]

with the same invariant delay relation (4.8). The obtained DODE is of order 2 if and only if

[TABLE]

holds.

In both cases we obtain the statement of the theorem provided that we have a second-order DODE. $\Box$

Thus, DODSs which admit four linearly connected symmetries are linearizable. Theorem 4.1 tells us that any Lie point symmetry algebra containing a four-dimensional subalgebra realized by linearly connected vector fields will correspond to an invariant DODS that can be transformed into the linear DODE with a solution-independent delay relation. The larger Lie algebra will at most put constraints on the functions involved in the DODS.

In the rest of this section we consider the symmetry properties of the linear DODS (4.3). First of all we mention a property that inhomogeneous DODSs share with ordinary inhomogeneous ODSs (without delay). Namely, if we know at least one solution $\sigma(x)$ of the inhomogeneous equation we can reduce the equation to a homogeneous one by the transformation

[TABLE]

For linear DODSs this does not affect the delay equation. We can formulate an important theorem.

Theorem 4.2

The linear homogeneous DODS (4.5) admits an infinite dimensional Lie point symmetry algebra represented by the vector fields

[TABLE]

where $\rho(x)$ is the general solution of the homogeneous linear DODS (4.5).

Proof. Application of the prolongation (2.1) of the vector field (1.3) to the delay equation in (4.5) gives

[TABLE]

Since $x$, $y$, $y_{-}$, $\dot{y}$ and $\dot{y}_{-}$ can be considered as independent while $x_{-}$ and $\ddot{y}$ are related to them via equations (4.5) we get

[TABLE]

Now we apply the prolongation to the DODE in (4.5) on the solutions of equation (4.5):

[TABLE]

Splitting equation (4.15) for terms with $\dot{y}^{2}$, $\dot{y}$, $\dot{y}_{-}$ and terms without derivatives, we obtain the equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From equation (4.16) we obtain

[TABLE]

Substituting this result into equations (4.17)-(4.19), we get the system

[TABLE]

[TABLE]

[TABLE]

The last equation splits for terms with $y$, $y_{-}$ and the remaining terms as

[TABLE]

[TABLE]

[TABLE]

The equation (4.21) can be integrated to obtain

[TABLE]

Substituting (4.27) into the equations (4.22), (4.24) and (4.25), we obtain three equations for $\xi(x)$:

[TABLE]

[TABLE]

[TABLE]

Here in addition to Eq. (4.27) we used Eq. (4.14), differential consequences of both these equations as well as the delay equation $x_{-}=g(x)$.

Three differential equations (4.28),(4.29),(4.30) together with the discrete equation (4.14) form an overdetermined differential-discrete system. For general $a_{i}(x)$ and $g(x)$ it has only one solution, namely $\xi(x)\equiv 0$.

Thus we obtain symmetries with coefficients

[TABLE]

where $B(x)$ solves the considered homogeneous linear DODS (4.5). Finally, we can rewrite the admitted symmetries as given in (4.12). $\Box$

Theorem 4.3

For the inhomogeneous linear DODS (4.3) the theorem is virtually the same. The particular solution $\sigma(x)$ of the inhomogeneous DODS (4.3) enters into the definition of $Y$ and we have

[TABLE]

instead of (4.12)

The proof is obvious. Moreover, this is a property shared by linear DODSs and linear ODEs (without delay).

The four-dimensional Lie algebras (4.6) and (4.9) of Theorem 4.1 are subalgebras of the infinite-dimensional Lie algebras (4.12) of Theorem 4.2 or algebras (4.32) of Theorem 4.3. Theorems 4.2 and 4.3 are very useful for recognizing linearizable DODSs. They can however not be used for symmetry reduction so they do not provide any explicit analytic solutions. Indeed, knowing all symmetries specified in (4.12) in the case of homogenous DODSs (in (4.32) in the case of inhomogenous DODSs) is equivalent to knowing all solutions.

The important question that remains is does the linear homogeneous DODS (4.5) allow any further symmetries for special cases of the functions $a_{i}(x)$ and $g(x)$? The answer is given by Theorem 4.5 below.

Now we turn to the examination of the equations (4.14), (4.28), (4.29) and (4.30) which $\xi(x)$ must satisfy. We will make use of the following result.

Lemma 4.4

Let us consider an overdetermined differential-discrete system

[TABLE]

Nontrivial solutions exist if and only if the functions $g(x)$ and $K(x)$ satisfy the compatibility condition

[TABLE]

These solutions can be presented as

[TABLE]

Proof. Differentiating (4.33a) with respect to $x$ we get

[TABLE]

Using both equations of the considered system (4.33), we obtain an identity

[TABLE]

We conclude that either $\xi(x)\equiv 0$ or the condition (4.34) must hold. The function $\xi(x)$ must still satisfy the first order ODE (4.33b). We can write the general solution as given in (4.35). $\Box$

Using this lemma, we can establish the following result.

Theorem 4.5

Let the linear homogeneous DODS (4.5) have functions $a_{i}(x)$ and $g(x)$ such that the equations (4.14), (4.28), (4.29) and (4.30) have a nontrivial solution $\xi(x)$, then the symmetry algebra is larger. In addition to the symmetries (4.12) it contains one additional symmetry of the form

[TABLE]

Proof. We have to consider two cases

$a_{2}(x){\not\equiv}0$

In this case we rewrite the equation (4.28) as

[TABLE]

The equations (4.14) and (4.37) can have a nontrivial solution $\xi(x)$ if the compatibility condition (4.34) holds. If in addition this solution $\xi(x)$ and functions $a_{i}(x)$ and $g(x)$ satisfy the equations (4.29) and (4.30), then there is an additional symmetry of the form (4.36) with

[TABLE] 2. 2.

$a_{2}(x)\equiv 0$, $a_{4}(x){\not\equiv}0$

Now the equation (4.28) holds identically. The equation (4.30) can be rewritten as

[TABLE]

The overdetermined differential–discrete system (4.14) and (4.39) has a nontrivial solution if functions $g(x)$ and $K_{2}(x)$ satisfy the compatibility condition (4.34). If in addition this solution $\xi(x)$ and functions $a_{1}(x)$ and $a_{3}(x)$ satisfy the equation (4.29), then there is an additional symmetry (4.36) with

[TABLE]

$\Box$

The additional symmetry (4.36) can be used to simplify the DODSs. For convenience we formulate this result only for the linear homogenous DODSs. However, the same result can be established for the linear inhomogenous DODSs.

Theorem 4.6

Let the linear homogeneous DODS (4.5) allow the additional symmetry (4.36). Then the DODS can be transformed to

[TABLE]

with

[TABLE]

where $\alpha$, $\beta$, $\gamma$ and $C$ are constants. This DODS admits the symmetry algebra

[TABLE]

Proof. Let us consider a linear homogenous DODS (4.5) which admits a symmetry of the form (4.36) with $\xi(x){\not\equiv}0$.

Let us find a change of variables $(x,y)\rightarrow(\bar{x},\bar{y})$ that straightens out the vector field $Z$ of (4.36) into $\bar{Z}=\partial_{\bar{x}}$ and preserves the linearity of the DODS. To preserve the linearity the transformation must have the form

[TABLE]

In the new variables we have

[TABLE]

and we impose

[TABLE]

The solution of this system is

[TABLE]

where $A_{0}$, $B_{0}$ and $C_{0}$ are inessential constants. Choosing $A_{0}=0$, $B_{0}=1$ and $C_{0}=0$, we obtain the new variables

[TABLE]

The DODS that we obtain is linear and invariant under transformation of $\bar{x}$. Dropping the bars on $\bar{x}$ and $\bar{y}$, we rewrite the DODS as

[TABLE]

where $a_{1}$ and $C$ are constants. The translational invariance corresponding to $Z=\partial_{x}$ would allow $a_{i}$ to be functions of $\Delta x=x-x_{-}$, however the delay condition imposes $\Delta x=\mbox{const}$.

A further transformation

[TABLE]

takes (4.48) into (4.41) with symmetry algebra (4.42). $\Box$

5 Group invariant solutions of DODSs

The main reason for applying Lie group and Lie algebra theory to equations of any type is to obtain exact solutions. For PDEs, once a symmetry group is known, one looks for solutions invariant under a subgroup $G_{0}\subseteq G$, i.e. solutions annihilated by the Lie algebra $L_{0}$ of $G_{0}$. This amounts to adding one, or more linear first-order PDEs to the one which we are solving. By first solving these additional equations using the method of characteristics we obtain the general form of the solution $y({\bf x})$, ${\bf x}\in\mathbb{R}^{p}$ in terms of the invariants $I_{k}$, $k<p$ of the subgroup $G_{0}$. The reduced solution depends on fewer variables since we have $k<p$. Solving the reduced equation, we obtain a subset of solutions satisfying specific boundary or initial conditions (see e.g. in Olver’s book [28]).

For ODEs it is also possible to obtain group invariant solutions by imposing invariance under a subgroup of the symmetry group. This again provides very specific solutions for a restricted type of initial conditions. A more efficient procedure for ODEs is to take an element of the symmetry algebra and to ”straighten out” the corresponding vector field by a change of variables

[TABLE]

The same transformation will take the original ODE into one not containing the variable $u$

[TABLE]

Putting $\dot{u}=v(t)$, we obtain an ODE for $v(t)$ of order $n-1$. If we can integrate it to obtain $v=v(t,C_{1},...,C_{n-1})$, then we can integrate $v$ to obtain $u=u(t,C_{1},...,C_{n})$, i.e. the general solution. If $\tilde{E}=0$ allows a symmetry group, we can repeat the procedure and reduce the order to $n-2$.

From this point of view DODSs are more similar to PDEs than to ODEs. In our earlier article [6] we used a one-dimensional subalgebra $L_{0}\subseteq L$ of the symmetry algebra of the given DODS to obtain group invariant solutions. These were particular solutions satisfying very specific initial conditions. The same method works for DODSs of any order (with one delay).

Thus, we are given a second-order DODS (1.2), i.e. the functions $f$ and $g$ are given. We calculate the symmetry algebra of the DODS, i.e. solve the equations

[TABLE]

for the coefficients $\xi(x,y)$ and $\eta(x,y)$ in (1.3). Thus, the symmetry algebra is known and we can classify its one-dimensional subalgebras into conjugacy classes under the action of the group $G$ leaving the DODS (1.2) invariant. This group is also known: $G\sim G_{0}\triangleright\mbox{exp}\ L$, where $G_{0}$ is a discrete group leaving the DODS invariant ($\mbox{exp}\ L$ is the connected component of $G$). For methods of classifying subalgebras of Lie algebras see [31, 32] and references therein.

Each subalgebra is represented by one vector field of the form (1.3). We must find the invariants of the corresponding one-parameter subgroup acting in the $(x,y)$ plane. All invariants will be functions of three basic invariants $I_{k}(x,y,x_{-},y_{-})$, $k=1,2,3$ determined from the equation

[TABLE]

For the method to provide explicit solutions there must exist a set of basic invariants satisfying

[TABLE]

together with the condition

[TABLE]

This allows us to set the invariants equal to constants

[TABLE]

and to solve for $y$ and $x_{-}$ obtaining the ”reduction formulas”

[TABLE]

According to the implicit function theorem the functions $h(x,A)$ and $k(x,A,B)$ are determined uniquely up to the indicated constants, at least in some neighbourhood of $x\in I$. Substituting into the DODS, we obtain equations for the constants $A$ and $B$ and (5.8) then gives the exact invariant solution. Unfortunately, this method for DODSs give only particular solutions. Thus, the initial data on the first step $x\in[x_{-1},x_{0}]$ must satisfy $y=h(x,A)$.

Examples will be given in Section 6.

Let us mention that another approach to finding exact solutions of delay differential equations, which makes use of functional separation of variables, was proposed in [33, 34, 35, 36]

6 Exact analytical solutions of a DODS describing traffic flow

Traffic flow is one of the areas in which time delay plays an important role. The time delay $\tau$ corresponds to the time it takes a driver of a car to react to changes of velocity of a preceding vehicle in a row of cars. One of the standard nonlinear ”follow-the-leader” models is given by the system of delay ordinary differential equations [10, 24]

[TABLE]

where $\alpha$, $n_{1}$, $n_{2}$ are adjustable constant parameters and $\tau$ is the constant time delay $t-t_{-}=\tau>0$. The function $x_{n}(t)$ is the position of the $n$-th vehicle at time $t$, the dots denote time derivatives. Equations (6.1) can be viewed as a finite ($0\leq n\leq N<\infty$) or infinite ($0\leq n<\infty$) set of DODEs, or as a delay ordinary differential-difference equation (since $x_{n}(t)$ is a function of a continuous variable $t$ and a discrete one $n$). The leading car’s position is $x_{0}(t)$ and must be given as part of the input in the model.

In order to apply the Lie theory formalism developed in this article we restrict to the case of two cars: the leading one with position $x_{0}(t)$ and a following one with position $x_{1}(t)\equiv x(t)$. The system (6.1) then simplifies to the DODE

[TABLE]

where we replaced $(x,y)$ by $(t,x)$ (as usual in traffic flow studies) and allowed $t_{-}$ to be a function of the indicated variables, instead of imposing $t_{-}=t-\tau$ with constant $\tau$. We have put $x_{-}=x(t_{-})$, $x_{0,-}=x_{0}(t_{-})$. The function $x_{0}(t)$ must be given for all $t$, $t_{-1}=t_{0}-\tau<t<\infty$.

A group analysis of equations (6.2) and also (6.1) is in progress. Here we will just present several examples showing how group analysis can lead to exact solutions invariant under some subgroup of the symmetry group.

Example 6.1

Let us start with the simplest and the most studied case. The leading car is moving with a constant velocity $v$, $x_{0}(t)=vt$, and the time delay is constant. We then have

[TABLE]

This DODS is invariant under transformations corresponding to

[TABLE]

Invariants of $X$ in the space $(t,t_{-},x,x_{-})$ are

[TABLE]

We substitute

[TABLE]

into (6.3) and obtain $B=\tau$. Thus, we have a particular solution

[TABLE]

While the above considerations are very simple and obvious, they do indicate that the model (6.1) makes sense. If the leading car moves with constant velocity, the following car should move with the same velocity. If we have $A=0$ the two cars would be in the same position (i.e. on top of each other if we have a single lane). Hence we impose $A\neq 0$, or $A<0$ since $x(t)$ is the position of the second car. $\Diamond$

Example 6.2

Consider the equation (6.2) with

[TABLE]

where $n=1-n_{1}^{-1}$. The DODS

[TABLE]

admits a two-dimensional symmetry algebra with basis

[TABLE]

A solution invariant under the subgroup generated by $X=X_{1}-n\beta X_{2}$ has the form

[TABLE]

$A$ and $B$ are integration constants, $\beta$ is the result of an $x$ translation.

Substituting (6.9) into the DODS (6.7), we get restrictions for the constants

[TABLE]

Again the model and the solution (6.9) make sense. This time the leading car moves with acceleration $\ddot{x}_{0}=kn(n-1)t^{n-2}$. The second car tries to adapt with solution (6.9). The distance would be constant for $A=k$. This is however not possible since $A=k$ contradicts the constraint (6.10). To avoid a collision we must have $A<k$. For $\alpha>0$, $n_{1}>1$ a collision is inevitable. $\Diamond$

Example 6.3

The equation (6.2) with

[TABLE]

leads to the DODS

[TABLE]

which admits the symmetry

[TABLE]

A solution invariant with respect to $X$ has the form

[TABLE]

Substituting it into the DODS, we get a restriction for the constants. They should satisfy the equation

[TABLE]

In this case the acceleration of the first car is exponential (by assumption). The second car adapts with the same type of exponential acceleration. A constant distance between the 2 cars requires the identity $A=k$ but this is ruled out by the constraint (6.14). To avoid a collision we must have $A<k$. This is only possible for $\alpha>0$. $\Diamond$

The last two examples show the usefulness of simple exact solutions of DODS. They help identify useful limits on the parameters of the model like $\alpha$, $n_{1}$ and $n_{2}$ in (6.2). For other discussions of traffic models and their solutions (with no use of group theory) see [42, 21, 41, 27, 2].

7 Conclusion

Let us sum up the main results of this article devoted to second-order DODSs with one delay point $x_{-}$ as presented in equations (1.2). We shall also put the results into a more general context.

A second-order genuinely nonlinear DODS of the type (1.2) can have a Lie point symmetry algebra $L$ of dimension $\mbox{dim}\ L\leq 6$. Representatives of all such DODSs and their symmetry algebras are listed in Tables $1$–$4$. If we have $\mbox{dim}\ L>6$, then the DODS is linearisable by an invertible transformation. 2. 2.

All linear or linearisable DODSs of this type have infinite-dimensional Lie point symmetry algebras. These algebras are solvable and their nilradicals are Abelian and of codimension 1 or 2. The nilradical $NR(L)$ is infinite-dimensional because it represents the linear superposition principle. The fact that the initial data depend on a function, $y(x)=\varphi(x)$, $x\in[x_{-},x_{0}]$, where $\varphi(x)$ is an arbitrary function, implies that an infinite set of solutions must exist. The factor algebra $L/NR(L)$ always contains one nonnilpotent element that reduces to $X=y\partial_{y}$ once the DODS is transformed into a linear homogeneous DODS. 3. 3.

In special cases, when the overdetermined differential-discrete system of equations (4.14), (4.28), (4.29) and (4.30) has nontrivial solutions $\xi(x){\not\equiv}0$, the factor algebra $L/NR(L)$ contains a second nonnilpotent element $Z=\xi(x)\partial_{x}+\eta(x,y)\partial_{y}$, $\xi(x){\not\equiv 0}$. A further transformation of variables takes the DODS into a linear homogeneous DODS with constant coefficients

[TABLE]

and its symmetry algebra into

[TABLE]

Let us apply the standard method of solving linear ODEs to the DODSs. We put

[TABLE]

into (7.1) and obtain a ”modified characteristic equation” for $\lambda$

[TABLE]

Any solution $\lambda$ of the transcendental equation (7.4) gives a solution of (7.1). It is however difficult to obtain possible constraints on the parameters $\alpha$, $\beta$, $\gamma$ and $C$ for which an infinite set of solutions (7.4) exists and the class of initial data for which this set of solutions is complete. For further information on delay ordinary differential equations, in particular linear once with constant coefficients, we refer to Driver’s book [8]. 4. 4.

In order to show that a DODS (1.2) is linearisable it is sufficient to show that it allows a four-dimensional symmetry algebra realized by linearly connected vector fields (see Theorem 4.1). 5. 5.

A one-dimensional subalgebra of the symmetry algebra of a DODS allows us to obtain particular invariant solutions of the DODS if the vector fields (1.3) representing this algebra satisfies the condition

[TABLE]

Let the one parameter group $\mbox{exp}(\epsilon X)$ acting on the space have three invariants $J_{1}=J_{1}(x,y)$, $J_{2}=J_{2}(x,y,x_{-},y_{-})$, $J_{3}=J_{1}(x_{-}.y_{-})$ satisfying (locally) condition (5.6). Than it is possible to express $y$, $y_{-}$ and $x_{-}$ as functions of $x$ and some constants as in (5.8). Ultimately we can also express $dy_{-}/dx_{-}$, $d^{2}y_{-}/dx_{-}^{2}$ and all higher derivatives $d^{k}y_{-}/dx_{-}^{k}$ in terms of $x$. Substituting $y=h(x,A)$, $x_{-}=k(x,A,B)$, $y_{-}=h(x_{-},A)$ as well as all derivatives $\dot{y}(x)$, $\dot{y}(x_{-})$ and $\ddot{y}(x)$ into the DODS, we obtain functional equations from which we determine the constants $A$ and $B$ compatible with the DODS.

Some of the above results can be extended to DODSs of arbitrary order $n$.

Results 1. An $n$-th order nonlinearisable DODS with one delay point $x_{-}$ will have a finite-dimensional symmetry algebra. The reason is that all individual algebras that can be realized by vector fields in two dimensions (algebras 1-19 in Table 1 of [11]) have dimensions $\mbox{dim}\ L\leq 8$. Those in the series 20-28 in the same table have arbitrary large dimensions $\dim\ L=r+k$ with $0\leq k\leq 3$ and $r=\mbox{dim}\ L_{0}$, where $L_{0}$ is a maximal Abelian subalgebra of $L$ and is realized by linearly connected vector fields. The actual value of $r$ is related to the order of the DODS (we have $r=1$ for $n=1$ [6, 7] and $r=3$ for $n=2$ as shown above). The maximal dimensions of the symmetry algebra are $\mbox{dim}\ L=3$ and $\mbox{dim}\ L=6$, respectively.

Results 2. All linear and linearisable DODSs of any order will have infinite-dimensional symmetry algebras with the same structure as those for $n=1$ and $n=2$ and for the same reason.

Results 3. In order to show that a DODS of order $n$ is linearisable it is sufficient to show that it has $k$-dimensional Abelian subalgebra with $k<\infty$. We conjecture that $k$ satisfies $k=2n$.

It is interesting to compare results concerning Lie point symmetries known for DODSs with those for ODEs without delay. First-order ODEs have infinite-dimensional symmetry algebras because of the existence of an integrating multiplier. For higher order ODEs the symmetry algebras are all finite-dimensional with $\mbox{dim}\ L=8$ for $n=2$ and $\mbox{dim}\ L=n+4$ for $n\geq 3$. A solvable subalgebra $L_{0}\subseteq L$ of dimension $n_{s}$ can be used to reduce the order of the equation to $n-n_{s}$. For $n_{s}\geq n$ this means that we can reduce the order to zero and thus obtain the general solution (at least in implicit form). All of these properties are well known and can be found e.g. in Olver’s book [28].

For DODSs the situation is different. The concept of integrating multiplier has not been introduced. The symmetry algebra is infinite-dimensional if and only if the DODS (of any order) is linear or linearisable. This is similar to the case of linearisable partial differential equations [14, 1].

In both cases this algebra is useful for establishing linearizability, finding the ”target” linear equations and finding the linearizing transformation.

A further difference between DODSs and ODEs is that for DODSs invariant solutions are very particular and satisfy very special initial conditions.

Acknowledgments

The research of VD was partly supported by research grant No. 18-01-00890 of the Russian Foundation for Basic Research. The research of PW was partially supported by a research grant from NSERC of Canada.

Appendix

In Tables 1-4 we provide a classification of nonlinear invariant DODSs by dimension of the Lie algebras of the admitted symmetries. In each table the first three columns give Lie algebras and their realizations. Column 1 provides the Lie algebra isomorphism class using the notations of [39]. For example, ${\mathfrak{n}}_{i,k}$ denotes the $k$-th nilpotent Lie algebra of dimension $i$. The only nilpotent algebras in the table are ${\mathfrak{n}}_{1,1}$, ${\mathfrak{n}}_{3,1}$ and ${\mathfrak{n}}_{4,1}$. Similarly, ${\mathfrak{s}}_{i,k}$ denotes the $k$-th solvable (but not nilpotent) Lie algebra of dimension $i$. Simple Lie algebras are presented by their standard notations (${\mathfrak{sl}}(2,\mathbb{R})$, ${\mathfrak{o}}(3,\mathbb{R})$). Column 2 gives notations $\mbox{\bf A}_{i,k}$ for algebras in the list of subalgebras of $\mbox{diff}(2,\mathbb{R})$. These notations were introduced in [6] for algebras of dimension $1$–$4$. As before $i$ is the dimension of the algebra. The numbers in brackets give the algebra notations used in Table 1 of Ref. [11]. Column 3 contains vector fields spanning each representative algebra.

The two last columns give invariant DODEs and invariant delay relations. For dimensions 1, 2 and 3 the tables contain all Lie algebras which can be realized as subalgebras of $\mbox{diff}(2,\mathbb{R})$. For dimensions 4, 5 and 6 only algebras which provide invariant DODSs are given.

Table 1. Classification of nonlinear invariant DODSs.

Symmetry algebra dimensions 1 and 2.

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{1,1}&\mbox{\bf A}_{1,1}(9)&{\displaystyle X_{1}={\partial_{y}}}&{\displaystyle\ddot{y}=f(x,\Delta y,\dot{y},\dot{y}_{-})}&{\displaystyle x_{-}=g(x,\Delta y,\dot{y},\dot{y}_{-})}\\ &&&&\\ \hline\cr\hline\cr&&&&\\ {\mathfrak{s}}_{2,1}&\mbox{\bf A}_{2,1}(10)&{\displaystyle X_{1}={\partial_{y}};\ X_{2}=y{\partial_{y}}}&{\displaystyle\ddot{y}=\dot{y}f\left(x,{\dot{y}\over\Delta y},{\dot{y}_{-}\over\Delta y}\right)}&{\displaystyle x_{-}=g\left(x,{\dot{y}\over\Delta y},{\dot{y}_{-}\over\Delta y}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{2,2}(22)&{\displaystyle X_{1}={\partial_{y}};\ X_{2}=x{\partial_{x}}+y{\partial_{y}}}&{\displaystyle\ddot{y}={1\over x}f\left({y_{x}},\dot{y},\dot{y}_{-}\right)}&{\displaystyle x_{-}=xg\left({y_{x}},\dot{y},\dot{y}_{-}\right)}\\ &&&&\\ \hline\cr&&&&\\ 2{\mathfrak{n}}_{1,1}&\mbox{\bf A}_{2,3}(20)&{\displaystyle\left\{X_{1}={\partial_{y}}\right\},\ \left\{X_{2}=x{\partial_{y}}\right\}}&{\displaystyle\ddot{y}=f\left(x,\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)}&{\displaystyle x_{-}=g\left(x,\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{2,4}(22)&{\displaystyle\left\{X_{1}={\partial_{x}}\right\},\ \left\{X_{2}={\partial_{y}}\right\}}&{\displaystyle\ddot{y}=f(\Delta y,\dot{y},\dot{y}_{-})}&{\displaystyle\Delta x=g(\Delta y,\dot{y},\dot{y}_{-})}\\ &&&&\\ \hline\cr\end{array}

Table 2. Classification of nonlinear invariant DODSs. Symmetry algebra dimension

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{3,1}&\mbox{\bf A}_{3,1}(22)&{\displaystyle X_{1}={\partial_{y}};\ X_{2}=x{\partial_{y}},\ X_{3}={\partial_{x}}}&{\displaystyle\ddot{y}=f\left(\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)}&{\displaystyle\Delta x=g\left(\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right).}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{3,1}&\mbox{\bf A}_{3,2}^{a}(12)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{x}},\ X_{2}={\partial_{y}};}\\ {\displaystyle X_{3}=x{\partial_{x}}+ay{\partial_{y}},\ 0<|a|\leq 1}\\ \end{array}&{\displaystyle\ddot{y}=|\Delta x|^{a-2}f\left({\dot{y}\over|\Delta x|^{a-1}},{\dot{y}_{-}\over|\Delta x|^{a-1}}\right)}&{\displaystyle\Delta x=|\Delta y|^{1\over a}g\left({\dot{y}\over|\Delta x|^{a-1}},{\dot{y}_{-}\over|\Delta x|^{a-1}}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,3}^{a}(21,22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}};}\\ {\displaystyle X_{3}=(1-a)x{\partial_{x}}+y{\partial_{y}},}\\ {\displaystyle 0<|a|\leq 1}\\ \end{array}\qquad\mbox{i)}\ a\neq 1&\ddot{y}=|x|^{2a-1\over 1-a}f\left({{\displaystyle\dot{y}-{y_{x}}}\over{\displaystyle|x|^{a\over 1-a}}},{{\displaystyle\dot{y}-\dot{y}_{-}}\over{\displaystyle|x|^{a\over 1-a}}}\right)&x_{-}=xg\left({{\displaystyle\dot{y}-{y_{x}}}\over{\displaystyle|x|^{a\over 1-a}}},{{\displaystyle\dot{y}-\dot{y}_{-}}\over{\displaystyle|x|^{a\over 1-a}}}\right)\\ &&&&\\ \cline{4-5}\cr&&&&\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\quad\mbox{ii)}\ a=1&\ddot{y}=\left({\dot{y}-{y_{x}}}\right)f\left(x,{{\displaystyle\dot{y}-\dot{y}_{-}}\over{\displaystyle\dot{y}-{y_{x}}}}\right)&x_{-}=g\left(x,{{\displaystyle\dot{y}-\dot{y}_{-}}\over{\displaystyle\dot{y}-{y_{x}}}}\right)\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{3,2}&\mbox{\bf A}_{3,4}(25)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{x}},\ X_{2}={\partial_{y}};}\\ {\displaystyle X_{3}=x{\partial_{x}}+(x+y){\partial_{y}}}\\ \end{array}&\ddot{y}={\displaystyle{1\over\Delta x}}f\left(\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)&\Delta x=e^{y_{x}}g\left(\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,5}(22)&{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}};\ X_{3}={\partial_{x}}+y{\partial_{y}}}&\ddot{y}=e^{x}f\left(e^{-x}\left(\dot{y}-{y_{x}}\right),e^{-x}\left(\dot{y}-\dot{y}_{-}\right)\right)&\Delta x=g\left(e^{-x}\left(\dot{y}-{y_{x}}\right),e^{-x}\left(\dot{y}-\dot{y}_{-}\right)\right)\\ &&&&\\ \hline\cr\end{array}

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{3,3}&\mbox{\bf A}_{3,6}^{b}(1)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{x}},\ X_{2}={\partial_{y}};}\\ {\displaystyle X_{3}=(bx+y){\partial_{x}}+(by-x){\partial_{y}},}\\ {\displaystyle b\geq 0}\\ \end{array}&\begin{array}[]{c}{\displaystyle\ddot{y}=(1+\dot{y}^{2})^{3/2}e^{b\arctan\dot{y}}}\\ \\ \times{\displaystyle f\left({\dot{y}-{y_{x}}\over 1+\dot{y}{y_{x}}},{\dot{y}-\dot{y}_{-}\over 1+\dot{y}\dot{y}_{-}}\right)}\\ \end{array}&\begin{array}[]{c}{\displaystyle\Delta xe^{b\arctan{y_{x}}}\sqrt{1+y_{x}^{2}}}\\ \\ ={\displaystyle g\left({\dot{y}-{y_{x}}\over 1+\dot{y}{y_{x}}},{\dot{y}-\dot{y}_{-}\over 1+\dot{y}\dot{y}_{-}}\right)}\\ \end{array}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,7}^{b}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}};}\\ {\displaystyle X_{3}=(1+x^{2}){\partial_{x}}+(x+b)y{\partial_{y}},}\\ {\displaystyle b\geq 0}\\ \end{array}&\ddot{y}={\displaystyle{e^{b\arctan x}\over(1+x^{2})^{3/2}}}{\displaystyle f(u_{7},v_{7})}&{\displaystyle x_{-}={x-g(u_{7},v_{7})\over 1+xg(u_{7},v_{7})}}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{sl}}(2,\mathbb{R})&\mbox{\bf A}_{3,8}(18)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{x}}+y{\partial_{y}},}\\ {\displaystyle X_{3}=2xy{\partial_{x}}+y^{2}{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}=-{\dot{y}\over 2x}+{\dot{y}^{3}\over x}f\left({1\over\dot{y}}-{2x\over\Delta y},{1\over\dot{y}_{-}}+{2x_{-}\over\Delta y}\right)}&{\displaystyle x_{-}={(\Delta y)^{2}\over x}g\left({1\over\dot{y}}-{2x\over\Delta y},{1\over\dot{y}_{-}}+{2x_{-}\over\Delta y}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,9}(2)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{x}}+y{\partial_{y}},}\\ {\displaystyle X_{3}=2xy{\partial_{x}}+(y^{2}-x^{2}){\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={(1+\dot{y}^{2})\dot{y}\over x}+{(1+\dot{y}^{2})^{3/2}\over x}f(u_{9},v_{9})}&{(x-x_{-})^{2}+(\Delta y)^{2}}=xx_{-}g(u_{9},v_{9})\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,10}(17)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{x}}+y{\partial_{y}},}\\ {\displaystyle X_{3}=2xy{\partial_{x}}+(y^{2}+x^{2}){\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={|1-\dot{y}^{2}|\dot{y}\over x}+{|1-\dot{y}^{2}|^{3/2}\over x}f(u_{10},v_{10})}&{(x-x_{-})^{2}-(\Delta y)^{2}}=xx_{-}g(u_{10},v_{10})\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,11}(11)&{\displaystyle X_{1}={\partial_{y}},\ X_{2}=y{\partial_{y}},\ X_{3}=y^{2}{\partial_{y}}}&{\displaystyle\ddot{y}=2{\dot{y}^{2}\over\Delta y}+\dot{y}f\left(x,{(\Delta y)^{2}\over\dot{y}\dot{y}_{-}}\right)}&{\displaystyle x_{-}=g\left(x,{(\Delta y)^{2}\over\dot{y}\dot{y}_{-}}\right)}\\ &&&&\\ \hline\cr\end{array}

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{o}}(3,\mathbb{R})&\mbox{\bf A}_{3,12}(3)&\begin{array}[]{l}{\displaystyle X_{1}=(1+x^{2}){\partial_{x}}+xy{\partial_{y}},}\\ {\displaystyle X_{2}=xy{\partial_{x}}+(1+y^{2}){\partial_{y}},}\\ {\displaystyle X_{3}=y{\partial_{x}}-x{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}=\left({1+\dot{y}^{2}+(y-x\dot{y})^{2}}\over 1+x^{2}+y^{2}\right)^{3/2}f(u_{12},v_{12})}&{\displaystyle{(x-x_{-})^{2}\left(1+y_{x}^{2}+\left(y-x{y_{x}}\right)^{2}\right)\over(1+x^{2}+y^{2})(1+x_{-}^{2}+y_{-}^{2})}=g(u_{12},v_{12})}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{1,1}\oplus{\mathfrak{s}}_{2,1}&\mbox{\bf A}_{3,13}(23)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{y}};\ X_{3}=y{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}=\dot{y}f\left({\dot{y}\over\Delta y},{\dot{y}_{-}\over\Delta y}\right)}&{\displaystyle\Delta x=g\left({\dot{y}\over\Delta y},{\dot{y}_{-}\over\Delta y}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{3,14}(22)&\begin{array}[]{l}{\displaystyle\left\{X_{1}=x{\partial_{y}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{y}};\ X_{3}=x{\partial_{x}}+y{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}={1\over x}f\left(\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)}&x_{-}=xg\left(\dot{y}-{y_{x}},\dot{y}-\dot{y}_{-}\right)\\ &&&&\\ \hline\cr&&&&\\ 3{\mathfrak{n}}_{1,1}&\mbox{\bf A}_{3,15}(20)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{y}}\right\},\ \left\{X_{2}=x{\partial_{y}}\right\},}\\ {\displaystyle\left\{X_{3}=\chi(x){\partial_{y}}\right\},\ \ddot{\chi}(x){\not\equiv}0}\\ \end{array}&\begin{array}[]{c}{\displaystyle\ddot{y}={\ddot{\chi}(x)\over\dot{\chi}(x)-{{\chi}_{x}}}(\dot{y}-{y_{x}})}\\ \\ {\displaystyle+\ddot{\chi}(x)f\left(x,{{\displaystyle\dot{y}-{y_{x}}}\over\dot{\chi}(x)-{{\chi}_{x}}}-{{\displaystyle\dot{y}-\dot{y}_{-}}\over\dot{\chi}(x)-\dot{\chi}(x_{-})}\right)}\end{array}&{\displaystyle x_{-}=g\left(x,{{\displaystyle\dot{y}-{y_{x}}}\over\dot{\chi}(x)-{{\chi}_{x}}}-{{\displaystyle\dot{y}-\dot{y}_{-}}\over\dot{\chi}(x)-\dot{\chi}(x_{-})}\right)}\\ &&&&\\ \hline\cr\end{array}

In Table 2 there were used the following notations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Table 3. Classification of nonlinear invariant DODSs. Symmetry algebra dimension

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{4,1}&\mbox{\bf A}_{4,1}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}};}\\ {\displaystyle X_{3}=x^{2}{\partial_{y}},\ X_{4}={\partial_{x}}}\\ \end{array}&{\displaystyle\ddot{y}={\dot{y}-\dot{y}_{-}\over\Delta x}+f\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)}&{\displaystyle{\Delta x}=g\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,1}&\mbox{\bf A}_{4,2}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=e^{x}{\partial_{y}};}\\ {\displaystyle X_{4}={\partial_{x}}}\\ \end{array}&{\displaystyle\ddot{y}={e^{\Delta x}\over e^{\Delta x}-1}(\dot{y}-\dot{y}_{-})+f(z_{2})}&{\displaystyle{\Delta x}=g(z_{2})}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,2}&\mbox{\bf A}_{4,3}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=x^{2}{\partial_{y}};}\\ {\displaystyle X_{4}={\partial_{x}}+y{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={\dot{y}-\dot{y}_{-}\over\Delta x}+e^{x}f\left(e^{-x}\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)\right)}&{\displaystyle{\Delta x}=g\left(e^{-x}\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)\right)}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,3}&\mbox{\bf A}_{4,4}^{a,\alpha}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=|x|^{\alpha}{\partial_{y}};}\\ {\displaystyle X_{4}=(1-a)x{\partial_{x}}+y{\partial_{y}},}\\ {\displaystyle a\in[-1,0)\cup(0,1),\ \alpha\neq\{0,{1\over 1-a},1\}}\\ {\displaystyle\mbox{(see~{}\cite[cite]{[\@@bibref{}{SWbook}{}{}]} for additional restriction on a$and$\alpha)}}\\ \end{array}&\begin{array}[]{c}{\displaystyle\ddot{y}={(\alpha-1)|x|^{\alpha-2}(\dot{y}-\dot{y}_{-})\over|x|^{\alpha-1}\mbox{sgn}(x)-|x_{-}|^{\alpha-1}\mbox{sgn}(x_{-})}}\\ \\ {\displaystyle+x^{{2a-1\over 1-a}}f(z_{4})}\\ \end{array}&{\displaystyle x_{-}=xg(z_{4})}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,4}&\mbox{\bf A}_{4,6}^{a}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=e^{ax}{\partial_{y}};}\\ {\displaystyle X_{4}={\partial_{x}}+y{\partial_{y}},\ a\neq 0,1}\\ \end{array}&{\displaystyle\ddot{y}={ae^{a\Delta x}\over e^{a\Delta x}-1}(\dot{y}-\dot{y}_{-})+e^{x}f(z_{6})}&{\displaystyle{\Delta x}=g(z_{6})}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,5}&\mbox{\bf A}_{4,7}^{\alpha,\beta}(22)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},}\\ {\displaystyle X_{2}=e^{\alpha x}\cos(\beta x){\partial_{y}},\ X_{3}=e^{\alpha x}\sin(\beta x){\partial_{y}};}\\ {\displaystyle X_{4}={\partial_{x}}+y{\partial_{y}},}\\ {\displaystyle\beta\neq 0}\\ \end{array}&\begin{array}[]{c}{\displaystyle\ddot{y}=\left({\beta\cos(\beta\Delta x)\over\sin(\beta\Delta x)}+\alpha\right)\dot{y}-{\beta e^{\alpha\Delta x}\over\sin(\beta\Delta x)}\dot{y}_{-}}\\ \\ +e^{x}f(z_{7})\\ \end{array}&{\displaystyle{\Delta x}=g(z_{7})}\\ &&&&\\ \hline\cr\end{array}

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,6}&\mbox{\bf A}_{4,8}(24)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}={\partial_{x}},\ X_{3}=x{\partial_{y}};}\\ {\displaystyle X_{4}=x{\partial_{x}}}\\ \end{array}&{\displaystyle\ddot{y}={1\over(\Delta x)^{2}}f\left(\Delta x\left(\dot{y}_{-}-{y_{x}}\right)\right)}&{\displaystyle\Delta x={g\left(\Delta x\left(\dot{y}_{-}-{y_{x}}\right)\right)\over\dot{y}-{y_{x}}}}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,8}&\mbox{\bf A}_{4,9}^{a}(24)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}={\partial_{x}},\ X_{3}=x{\partial_{y}};}\\ {\displaystyle X_{4}=x{\partial_{x}}+ay{\partial_{y}},\ a\neq 0,1}\\ \end{array}&{\displaystyle\ddot{y}=|\Delta x|^{\alpha-2}f\left(|\Delta x|^{1-\alpha}\left(\dot{y}_{-}-{y_{x}}\right)\right)}&{\displaystyle|\Delta x|^{1-\alpha}={g\left(|\Delta x|^{1-\alpha}\left(\dot{y}_{-}-{y_{x}}\right)\right)\over\dot{y}-{y_{x}}}}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,10}&\mbox{\bf A}_{4,10}(25)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}={\partial_{x}},\ X_{3}=x{\partial_{y}};}\\ {\displaystyle X_{4}=x{\partial_{x}}+(2y+x^{2}){\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={\dot{y}-\dot{y}_{-}\over\Delta x}+f\left((\Delta x)\mbox{exp}\left[{\displaystyle\dot{y}_{-}-y_{x}\over\Delta x}\right]\right)}&{\displaystyle{\Delta x\over\mbox{exp}\left[{\displaystyle{\dot{y}-{y_{x}}\over\Delta x}}\right]}=g\left((\Delta x)\mbox{exp}\left[{\displaystyle\dot{y}_{-}-{y_{x}}\over\Delta x}\right]\right)}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,11}&\mbox{\bf A}_{4,11}(24)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}={\partial_{x}},\ X_{3}=x{\partial_{y}};}\\ {\displaystyle X_{4}=x{\partial_{x}}+y{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={1\over\Delta x}f\left(\dot{y}_{-}-{y_{x}}\right)}&{\displaystyle\dot{y}-{y_{x}}=g\left(\dot{y}_{-}-{y_{x}}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{4,12}(23)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}={\partial_{x}};}\\ {\displaystyle X_{4}=y{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}=\left(\dot{y}-{y_{x}}\right)f\left({\displaystyle\dot{y}_{-}-{y_{x}}\over\displaystyle\dot{y}-{y_{x}}}\right)}&{\displaystyle\Delta x=g\left({\displaystyle\dot{y}_{-}-{y_{x}}\over\displaystyle\dot{y}-{y_{x}}}\right)}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{4,12}&\mbox{\bf A}_{4,13}(4)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{x}},\ X_{2}={\partial_{y}};}\\ {\displaystyle X_{3}=x{\partial_{x}}+y{\partial_{y}},\ X_{4}=y{\partial_{x}}-x{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={(1+\dot{y}^{2})^{2}\over\Delta x\left(\dot{y}-{\displaystyle{y_{x}}}\right)}f\left({\dot{y}-\dot{y}_{-}\over 1+\dot{y}\dot{y}_{-}}\right)}&{\displaystyle{\dot{y}-{\displaystyle{y_{x}}}\over 1+\dot{y}{\displaystyle{y_{x}}}}=g\left({\dot{y}-\dot{y}_{-}\over 1+\dot{y}\dot{y}_{-}}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{4,14}(23)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}};}\\ {\displaystyle X_{3}=y{\partial_{y}},\ X_{4}=(1+x^{2}){\partial_{x}}+xy{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={\dot{y}-{\displaystyle{y_{x}}}\over(x^{2}+1)}f(z),\ z=\sqrt{{x_{-}^{2}+1\over x^{2}+1}}{\displaystyle\dot{y}_{-}-{y_{x}}\over\displaystyle\dot{y}-{y_{x}}}}&{\displaystyle x_{-}={x-g(z)\over 1+xg(z)}}\\ &&&&\\ \hline\cr\end{array}

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{1,1}\oplus{\mathfrak{s}}_{3,1}&\mbox{\bf A}_{4,15}^{a}(22)&\begin{array}[]{l}{\displaystyle\left\{X_{1}=|x|^{1\over 1-a}{\partial_{y}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{y}},\ X_{3}=x{\partial_{y}};\ X_{4}=(1-a)x{\partial_{x}}+y{\partial_{y}}\right\}},\\ {\displaystyle a\in[-1,0)\cup(0,1)}\\ \end{array}&\begin{array}[]{c}{\displaystyle\ddot{y}={{a\over 1-a}|x|^{2a-1\over 1-a}(\dot{y}-\dot{y}_{-})\over|x|^{a\over 1-a}\mbox{sgn}(x)-|x_{-}|^{a\over 1-a}\mbox{sgn}(x_{-})}}\\ \\ {\displaystyle+x^{2a-1\over 1-a}f(z_{15})}\\ \end{array}&{\displaystyle x_{-}=xg(z_{15})}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{1,1}\oplus{\mathfrak{s}}_{3,2}&\mbox{\bf A}_{4,16}(22)&\begin{array}[]{l}{\displaystyle\left\{X_{1}=e^{x}{\partial_{y}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{y}},\ X_{3}=x{\partial_{y}};\ X_{4}={\partial_{x}}+y{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}={e^{\Delta x}\over e^{\Delta x}-1}(\dot{y}-\dot{y}_{-})+e^{x}f(z_{16})}&{\displaystyle{\Delta x}=g(z_{16})}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{1,1}\oplus{\mathfrak{s}}_{3,3}&\mbox{\bf A}_{4,17}^{\alpha}(22)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{y}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{x}},\ X_{3}=e^{\alpha x}\cos x{\partial_{y}},\ X_{4}=e^{\alpha x}\sin x{\partial_{y}}\right\},}\\ {\displaystyle{\alpha\geq 0}}\\ \end{array}&{\displaystyle\ddot{y}=\left({\cos(\Delta x)\over\sin(\Delta x)}+\alpha\right)\dot{y}-{e^{\alpha\Delta x}\over\sin(\Delta x)}\dot{y}_{-}+f(z_{17})}&{\displaystyle{\Delta x}=g(z_{17})}\\ &&&&\\ \hline\cr\end{array}

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{n}}_{1,1}\oplus{\mathfrak{sl}}(2,\mathbb{R})&\mbox{\bf A}_{4,18}(14)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{y}},\ X_{3}=y{\partial_{y}},\ X_{4}=y^{2}{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}=2{\dot{y}^{2}\over\Delta y}+\dot{y}f\left({(\Delta y)^{2}\over\dot{y}\dot{y}_{-}}\right)}&{\displaystyle\Delta x=g\left({(\Delta y)^{2}\over\dot{y}\dot{y}_{-}}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{4,19}(19)&\begin{array}[]{l}{\displaystyle\left\{X_{1}=x{\partial_{x}}\right\},}\\ {\displaystyle\left\{X_{2}={\partial_{y}},\ X_{3}=x{\partial_{x}}+y{\partial_{y}},\right.}\\ {\displaystyle\left.X_{4}=2xy{\partial_{x}}+y^{2}{\partial_{y}}\right\}}\\ \end{array}&\begin{array}[]{c}{\displaystyle\ddot{y}=-{\dot{y}\over 2x}+{x_{-}\dot{y}^{3}\over(\Delta y)^{2}}}\\ \\ {\displaystyle\times f\left({(\Delta y)^{2}\over xx_{-}}\left({1\over\dot{y}_{-}}+{2x_{-}\over\Delta y}\right)^{2}\right)}\\ \end{array}&\begin{array}[]{c}{\displaystyle{(\Delta y)^{2}\over xx_{-}}\left({1\over\dot{y}}-{2x\over\Delta y}\right)^{2}}\\ \\ {\displaystyle=g\left({(\Delta y)^{2}\over xx_{-}}\left({1\over\dot{y}_{-}}+{2x_{-}\over\Delta y}\right)^{2}\right)}\\ \end{array}\\ &&&&\\ \hline\cr&&&&\\ 2{\mathfrak{s}}_{2,1}&\mbox{\bf A}_{4,20}(13)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}};\ X_{2}=x{\partial_{x}}\right\},}\\ {\displaystyle\left\{X_{3}={\partial_{y}};\ X_{4}=y{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}={\dot{y}\over\Delta x}f\left({\dot{y}_{-}\over\dot{y}}\right)}&{\displaystyle\Delta x={\Delta y\over\dot{y}}g\left({\dot{y}_{-}\over\dot{y}}\right)}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{4,21}(23)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{y}};\ X_{2}=x{\partial_{x}}+y{\partial_{y}}\right\},}\\ {\displaystyle\left\{X_{3}=x{\partial_{y}};\ X_{4}=x{\partial_{x}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}={\displaystyle\dot{y}-{y_{x}}\over x}f\left({\displaystyle\dot{y}_{-}-{y_{x}}\over\displaystyle\dot{y}-{y_{x}}}\right)}&{\displaystyle x_{-}=xg\left({\displaystyle\dot{y}_{-}-{y_{x}}\over\displaystyle\dot{y}-{y_{x}}}\right)}\\ &&&&\\ \hline\cr\end{array}

In Table 3 there were used the notations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Table 4. Classification of nonlinear invariant DODSs.

Symmetry algebra dimensions 5 and 6.

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{5,33}&\mbox{\bf A}_{5,1}(24)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=x^{2}{\partial_{y}},}\\ {\displaystyle X_{4}={\partial_{x}};\ X_{5}=x{\partial_{x}}}\\ \end{array}&{\displaystyle\ddot{y}=2{\dot{y}-{y_{x}}\over\Delta x}+{C_{1}\over(\Delta x)^{2}}}&{\displaystyle\Delta x={C_{2}\over\dot{y}+\dot{y}_{-}-2{y_{x}}}}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{5,34}&\mbox{\bf A}_{5,2}(25)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=x^{2}{\partial_{y}},}\\ {\displaystyle X_{4}={\partial_{x}};}\ {\displaystyle X_{5}=x{\partial_{x}}+(3y+x^{3}){\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}={4\dot{y}+2\dot{y}_{-}-6{y_{x}}\over\Delta x}+C_{1}\Delta x}&{\displaystyle{1\over(\Delta x)^{2}}\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)-\ln|\Delta x|=C_{2}}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{5,35}&\mbox{\bf A}_{5,3}(24)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=x^{2}{\partial_{y}},}\\ {\displaystyle X_{4}={\partial_{x}};}\ {\displaystyle X_{5}=x{\partial_{x}}+\alpha y{\partial_{y}},\ \alpha\neq\{0,2\}}\\ \end{array}&{\displaystyle\ddot{y}=2{\dot{y}-{y_{x}}\over\Delta x}+C_{1}|\Delta x|^{\alpha-2}}&{\displaystyle|\Delta x|^{\alpha-1}=C_{2}\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{5,36}&\mbox{\bf A}_{5,4}(24)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}=x^{2}{\partial_{y}},}\\ {\displaystyle X_{4}={\partial_{x}};}\ {\displaystyle X_{5}=x{\partial_{x}}+2y{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}=2{\dot{y}-{y_{x}}\over\Delta x}+C_{1}}&{\displaystyle\Delta x=C_{2}\left(\dot{y}+\dot{y}_{-}-2{y_{x}}\right)}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{5,44}&\mbox{\bf A}_{5,5}(26)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{y}},\ X_{2}=x{\partial_{y}},\ X_{3}={\partial_{x}};}\\ {\displaystyle X_{4}=x{\partial_{x}},\ X_{5}=y{\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}=C_{1}{\dot{y}-{y_{x}}\over\Delta x}}&{\displaystyle\Delta x={(1-C_{2})\Delta y\over\dot{y}-C_{2}\dot{y}_{-}}}\\ &&&&\\ \hline\cr\end{array}

\begin{array}[]{|c|c|l|c|c|}\hline\cr&&&&\\ \mbox{Lie algebra}&\mbox{Case}&\mbox{Operators}&\mbox{DODE}&\mbox{Delay relation}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{sl}}(2,\mathbb{R})\ltimes 2{\mathfrak{n}}_{1,1}&\mbox{\bf A}_{5,6}(27)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}},\ X_{2}=2x{\partial_{x}}+y{\partial_{y}},\ X_{3}=x^{2}{\partial_{x}}+xy{\partial_{y}}\right\}}\\ {\displaystyle\ltimes\left(\left\{X_{4}={\partial_{y}}\right\},\ \left\{X_{5}=x{\partial_{y}}\right\}\right)}\\ \end{array}&{\displaystyle\ddot{y}=C_{1}\displaystyle\left(\dot{y}-{y_{x}}\right)^{3}}&{\displaystyle\Delta x={C_{2}\over\left(\dot{y}-{y_{x}}\right)\left(\dot{y}_{-}-{y_{x}}\right)}}\\ &&&&\\ \cline{2-5}\cr&&&&\\ &\mbox{\bf A}_{5,7}(5)&\begin{array}[]{l}{\displaystyle\left\{X_{1}=x{\partial_{x}}-y{\partial_{y}},\ X_{2}=y{\partial_{x}},\ X_{3}=x{\partial_{y}}\right\}}\\ {\displaystyle\ltimes\left(\left\{X_{4}={\partial_{x}}\right\},\ \left\{X_{5}={\partial_{y}}\right\}\right)}\\ \end{array}&{\displaystyle\ddot{y}=C_{1}(\Delta x)^{3}\left(\dot{y}-{y_{x}}\right)^{3}}&{\displaystyle(\Delta x)^{2}={C_{2}(\dot{y}-\dot{y}_{-})\over\left(\dot{y}-{y_{x}}\right)\left(\dot{y}_{-}-{y_{x}}\right)}}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{s}}_{2,1}\oplus{\mathfrak{sl}}(2,\mathbb{R})&\mbox{\bf A}_{5,8}(15)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}},\ X_{2}=x{\partial_{x}}\right\},}\\ {\displaystyle\left\{X_{3}={\partial_{y}},\ X_{4}=y{\partial_{y}},\ X_{5}=y^{2}{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}=2{\dot{y}^{2}\over\Delta y}+C_{1}{\dot{y}\over\Delta x}}&{\displaystyle(\Delta x)^{2}=C_{2}{(\Delta y)^{2}\over\dot{y}\dot{y}_{-}}}\\ &&&&\\ \hline\cr\hline\cr&&&&\\ {\mathfrak{s0}}(3,1)&\mbox{\bf A}_{6,1}(7)&\begin{array}[]{l}{\displaystyle X_{1}={\partial_{x}},\ X_{2}={\partial_{y}},\ X_{3}=x{\partial_{x}}+y{\partial_{y}},\ X_{4}=y{\partial_{x}}-x{\partial_{y}},}\\ {\displaystyle X_{5}=(x^{2}-y^{2}){\partial_{x}}+2xy{\partial_{y}},\ X_{6}=2xy{\partial_{x}}+(y^{2}-x^{2}){\partial_{y}}}\\ \end{array}&{\displaystyle\ddot{y}-2{\dot{y}^{2}+1\over y_{x}^{2}+1}{\dot{y}-y_{x}\over\Delta x}=0}&\begin{array}[]{c}{\displaystyle\arctan(\dot{y})+\arctan(\dot{y}_{-})}\\ {\displaystyle-2\arctan(y_{x})=C}\\ \end{array}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{sl}}(2,\mathbb{R})\ltimes 3{\mathfrak{n}}_{1,1}&\mbox{\bf A}_{6,2}(27)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}},\ X_{2}=x{\partial_{x}}+y{\partial_{y}},\ X_{3}=x^{2}{\partial_{x}}+2xy{\partial_{y}}\right\}}\\ {\displaystyle\ltimes\left(\left\{X_{4}={\partial_{y}}\right\},\ \left\{X_{5}=x{\partial_{y}}\right\},\ \left\{X_{6}=x^{2}{\partial_{y}}\right\}\right)}\\ \end{array}&{\displaystyle\ddot{y}-2{\dot{y}-{y_{x}}\over\Delta x}=0}&{\displaystyle\dot{y}+\dot{y}_{-}-2{y_{x}}=C}\\ &&&&\\ \hline\cr&&&&\\ {\mathfrak{sl}}(2,\mathbb{R})\oplus{\mathfrak{sl}}(2,\mathbb{R})&\mbox{\bf A}_{6,3}(16)&\begin{array}[]{l}{\displaystyle\left\{X_{1}={\partial_{x}},\ X_{2}=x{\partial_{x}},\ X_{3}=x^{2}{\partial_{x}}\right\},}\\ {\displaystyle\left\{X_{4}={\partial_{y}},\ X_{5}=y{\partial_{y}},\ X_{6}=y^{2}{\partial_{y}}\right\}}\\ \end{array}&{\displaystyle\ddot{y}-2{\dot{y}^{2}\over\Delta y}+{2\dot{y}\over\Delta x}=0}&{\displaystyle(\Delta x)^{2}=C{(\Delta y)^{2}\over\dot{y}\dot{y}_{-}}}\\ &&&&\\ \hline\cr\end{array}