Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems

TL;DR

This paper visualizes hidden chaotic attractors in three Lorenz-like systems by numerically connecting them through a specific parameter path, enhancing understanding of complex dynamics.

Contribution

It introduces a method to connect hidden attractors across different systems using numerical continuation along a parameter path.

Findings

Hidden attractors can be connected via parameter continuation.

The method reveals the structure of chaotic sets in these systems.

Enhanced visualization of complex chaotic dynamics.

Abstract

In this report, by the numerical continuation method we visualize and connect hidden chaotic sets in the Glukhovsky-Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system.

Tables

Table 1. Table 1: Initial point ( x 0 , y 0 , z 0 ) subscript 𝑥 0 subscript 𝑦 0 subscript 𝑧 0 (x_{0},\,y_{0},\,z_{0}) and time interval [ 0 , T ] 0 𝑇 [0,\leavevmode\nobreak\ T] of numerical integration for each part of the path.

Path	$(x_0,y_0,z_0)$	$T$
$\mathrm{GD}\to L$	$(10.64,\mathrm{\hspace{0.17em}60.78},\mathrm{\hspace{0.17em}390})$	${10}^4$
$L\to R$	$(0.2,\mathrm{\hspace{0.17em}0.2},\mathrm{\hspace{0.17em}0.35})$	$1.1\cdot {10}^4$

Equations

$$\begin{cases}\dot{x}$ = $-\sigma(x-y)-ayz\\ \dot{y}$ = $rx-y-xz\\ \dot{z}$ = $-bz+xy,\end{cases}$$

$$a=0$$

$$b=1,a>0,\sigma >ar.$$

$$(x,y,z)\rightarrow\bigg{(}x,\frac{\zeta}{\sigma-ar},r-\frac{\zeta}{\sigma-ar}\,y\bigg{)},$$

$$\begin{cases}\dot{x}$ = $-\sigma x+\zeta z+\alpha yz\\ \dot{y}$ = $\rho-y-xz\\ \dot{z}$ = $-z+xy,\end{cases}$$

$$\zeta >0,\rho =\frac{r(\sigma -ar)}{\zeta }>0,\alpha =\frac{{\zeta }^{2}a}{(\sigma -ar{)}^2}>0.$$

$$a<0,\sigma =-ar,$$

$$(x,y,z)\rightarrow\bigg{(}\nu_{1}^{-1}\,y,\nu_{1}^{-1}\nu_{2}^{-1}h\,x,\nu_{1}^{-1}\nu_{2}^{-1}h\,z)\bigg{)},\quad t\to\nu_{1}\,t$$

$$\begin{cases}\dot{x}$ = $hy-\nu_{1}x-yz,\\ \dot{y}$ = $hx-\nu_{2}y+xz,\\ \dot{z}$ = $-z+xy,\end{cases}$$

$$\sigma ={\nu }_1^{-1}{\nu }_2,b={\nu }_1^{-1},a=-{\nu }_2^2{h}^{-2},r={\nu }_1^{-1}{\nu }_2^{-1}{h}^2.$$

$${\mathrm{S}}_{\pm }=(\pm x_1,\pm y_1,z_1).$$

$$x_1=\frac{\sigma b\sqrt{\xi }}{\sigma b+a\xi },y_1=\sqrt{\xi },z_1=\frac{\sigma \xi }{\sigma b+a\xi },$$

$$\xi =\frac{\sigma b}{2a^2}\left[a(r-2)-\sigma +\sqrt{(\sigma -ar{)}^2+4a\sigma }\right].$$

$$a^2{\sigma }^2(\sigma -2)r^3$$

$$+{\sigma }^2\left({\sigma }^3+2(3a-1){\sigma }^2-8a\sigma +8a\right)r-$$

$$-{\sigma }^3\left({\sigma }^3+4{\sigma }^2-16a\right)<0.$$

$$\dot{u}=f(u),$$

$$\mathcal{B}=\left\{(x,y,z)\in {\mathbb{R}}^3\text{leavevmode}|\text{leavevmode}V(x,y,z)\le \frac{b(\sigma +\delta r{)}^2}{2c(a+\delta )}\right\},$$

Full text

I Introduction

In 1963, meteorologist Edward Lorenz suggested an approximate mathematical model (the Lorenz system) for the Rayleigh-Bénard convection and discovered numerically a chaotic attractor in this model Lorenz-1963 . This discovery stimulated rapid development of the chaos theory, numerical methods for attractor investigation, and till now has received a great deal of attention from different fields Celikovsky-1994 ; Li-2015-PhysLetA ; Doedel-2015 ; LeonovK-2015-AMC ; LeonovKKK-2016-CNSCS ; Cermak-2017 . The Lorenz system gave rise to various generalizations, e.g. Lorenz-like systems, some of which are also simplified mathematical models of physical phenomena. In this paper, we consider the following Lorenz-like system

[TABLE]

where parameters $r$, $\sigma$, $b$ are positive and $a$ is real. System (1) with

[TABLE]

coincides with the classical Lorenz system.

Consider

[TABLE]

Then by the following linear transformation (see, e.g., LeonovB-1992 ):

[TABLE]

system (1) is transformed to the Glukhovsky-Dolzhansky system GlukhovskyD-1980 :

[TABLE]

where

[TABLE]

The Glukhovsky-Dolzhansky system describes the convective fluid motion inside a rotating ellipsoidal cavity.

If we set

[TABLE]

then after the linear transformation (see, e.g., LeonovB-1992 ):

[TABLE]

with positive $\nu_{1},\nu_{2},h$, we obtain the Rabinovich system Rabinovich-1978 ; PikovskiRT-1978 , describing the interaction of three resonantly coupled waves, two of which being parametrically excited:

[TABLE]

where

[TABLE]

Hereinafter, the Lorenz, Glukhovsky-Dolzhansky, and Rabinovich systems are studied in the framework of system (1) under the corresponding assumptions on parameters ((2), (3), or (7)), respectively. For the considered assumptions on parameters, if $r<1$, then (1) has a unique111 In general, system (1) can possess up to five equilibria LeonovB-1992 . equilibrium ${\bf\rm S_{0}}=(0,0,0)$, which is globally asymptotically Lyapunov stable LeonovB-1992 ; BoichenkoLR-2005 . If $r>1$, then system (1) has three equilibria: ${\bf\rm S_{0}}=(0,0,0)$ and

[TABLE]

Here,

[TABLE]

and

[TABLE]

The stability of equilibria $S_{\pm}$ of system (1) depends on the parameters $r$, $\sigma$, $a$ and $b$.

Lemma 1* (see, e.g. LeonovKM-2015-EPJST ).*

For a certain $\sigma>2$, the equilibria $S_{\pm}$ of system (1) with (3) (and, thus, of Glukhovsky-Dolzhansky system (5)) are stable if and only if the following condition holds:

[TABLE]

Lemma 2* (see, e.g. KuznetsovLMS-2016-INCAAM ).*

The equilibria $S_{\pm}$ of system (1) with (7) (and, thus, of the Rabinovich system (8)) are stable if and only if one of the following conditions holds:

1. (i)

$0\leq\,ar+1\,<\frac{2r}{r-\sqrt{r(r-1)}}$, 2. (ii)

$ar+1<0$,   $b>b_{\rm cr}=\frac{4a(r-1)(ar+1)\sqrt{r(r-1)}+(ar-1)^{3}}{(ar+1)^{2}-4ar^{2}}.$

The particular interest in the considered Lorenz-like systems is due to the existence of chaotic attractors in their phase spaces. In the next section, we will present the definition of attractor from analytical and numerical perspectives.

II Attractors of dynamical systems

II.1 Attractors of dynamical systems

Consider system (1) as an autonomous differential equation in a general form:

[TABLE]

where $u=(x,y,z)\in\mathbb{R}^{3}$, and the continuously differentiable vector-function $f:\mathbb{R}^{3}\to\mathbb{R}^{3}$ may represent the right-hand side of system (1). Define by ${u}(t,{u}_{0})$ a solution of (12) such that ${u}(0,{u}_{0})={u}_{0}$. For system (12), a bounded closed invariant set K is

1. (i)

a (local) attractor if it is a minimal locally attractive set (i.e., $\lim_{t\to+\infty}{\rm dist}(K,{u}(t,{u}_{0}))=0$ for all ${u_{0}}\in K(\varepsilon)$, where $K(\varepsilon)$ is a certain $\varepsilon$-neighborhood of set $K$), 2. (ii)

a global attractor if it is a minimal globally attractive set (i.e., $\lim_{t\to+\infty}{\rm dist}(K,{u}(t,{u}_{0}))=0$ for all ${u_{0}}\in\mathbb{R}^{n}$),

where ${\rm dist}(K,{u})=\inf_{{v}\in K}||{v}-{u}||$ is the distance from the point ${u}\in\mathbb{R}^{3}$ to the set $K\subset\mathbb{R}^{3}$ (see, e.g. LeonovKM-2015-EPJST ).

Note that system (1) is dissipative in the sense that it possesses a bounded convex absorbing set LeonovB-1992 ; LeonovKM-2015-EPJST

[TABLE]

where $V(u)=V(x,y,z)=x^{2}+\delta y^{2}+(a+\delta)\left(z-\frac{\sigma+\delta r}{a+\delta}\right)^{2}$, $\delta$ is an arbitrary positive number such that $a+\delta>0$ and $c=\min(\sigma,1,\frac{b}{2})$. Thus, solutions of (12) exist for $t\in[0,+\infty)$ and system (1) possesses a global attractor Chueshov-2002-book ; LeonovKM-2015-EPJST , which contains the set of equilibria and can be constructed as $\cap_{\tau>0}\overline{\cup_{t\geq\tau}\varphi^{t}\left(\mathcal{B}\right)}$.

Computational errors (caused by finite precision arithmetic and numerical integration of differential equations) and sensitivity to initial conditions allow one to get a reliable visualization of a chaotic attractor by only one pseudo-trajectory computed on a sufficiently large time interval. For that, one needs to choose an initial point in attractor’s basin of attraction and observe how the trajectory starting from this initial point after a transient process visualizes the attractor. Thus, from a computational point of view, it is natural to suggest the following classification of attractors, based on the simplicity of finding the basin of attraction in the phase space.

Definition**.**

KuznetsovLV-2010-IFAC ; LeonovKV-2011-PLA ; LeonovK-2013-IJBC ; LeonovKM-2015-EPJST An attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of a stationary state (an equilibrium); otherwise, it is called a hidden attractor.

Remark*.*

Sustained chaos is often (almost) indistinguishable numerically from transient chaos (transient chaotic set in the phase space), which can nevertheless persist for a long time. Similar to the above definition, in general, a chaotic set can be called hidden if it does not involve and attract trajectories from a small vicinities of stationary states; otherwise, it is called self-excited.

For a self-excited attractor, its basin of attraction is connected with an unstable equilibrium and, therefore, self-excited attractors can be localized numerically by the standard computational procedure in which after a transient process a trajectory, started in a neighborhood of an unstable equilibrium (e.g., from a point of its unstable manifold), is attracted to the state of oscillation and then traces it. Thus, self-excited attractors can be easily visualized (see, e.g. the classical Lorenz, Rossler, and Hennon attractors can be visualized by a trajectory from a vicinity of unstable zero equilibrium).

For a hidden attractor, its basin of attraction is not connected with equilibria, and, thus, the search and visualization of hidden attractors in the phase space may be a challenging task. Hidden attractors are attractors in the systems without equilibria (see, e.g. rotating electromechanical systems with Sommerfeld effect described in 1902 Sommerfeld-1902 ; KiselevaKL-2016-IFAC ), and in the systems with only one stable equilibrium (see, e.g. counterexamples LeonovK-2011-DAN ; LeonovK-2013-IJBC to the Aizerman’s (1949) and Kalman’s (1957) conjectures on the monostability of nonlinear control systems Aizerman-1949 ; Kalman-1957 ). One of the first related problems is the second part of Hilbert’s 16th problem (1900) Hilbert-1901 on the number and mutual disposition of limit cycles in two-dimensional polynomial systems where nested limit cycles (a special case of multistability and coexistence of attractors) exhibit hidden periodic oscillations (see, e.g., Bautin-1939 ; KuznetsovKL-2013-DEDS ; LeonovK-2013-IJBC ). The classification of attractors as being hidden or self-excited was introduced by G. Leonov and N. Kuznetsov in connection with the discovery of the first hidden Chua attractor LeonovK-2009-PhysCon ; KuznetsovLV-2010-IFAC ; LeonovKV-2011-PLA ; LeonovKV-2012-PhysD ; KuznetsovKLV-2013 and has captured much attention of scientists from around the world (see, e.g. LiZY-2014-HA ; BurkinK-2014-HA ; LiSprott-2014-HA ; Chen-2015-IFAC-HA ; ZhusubaliyevMCM-2015-HA ; KuznetsovKMS-2015-HA ; ChenLYBXW-2015-HA ; Semenov20151553 ; PhamRFF-2014-HA ; BorahR-2017-HA ; MenacerLC-2016-HA ; MessiasR-2017 ; Zelinka-2016-HA ; DancaKC-2016 ; Danca-2016-HA ; WeiPKW-2016-HA ; PhamVJVK-2016-HA ; JafariPGMK-2016-HA ; DudkowskiJKKLP-2016 ).

II.2 Hidden attractor localization via numerical continuation method

One of the effective methods for numerical localization of hidden attractors in multidimensional dynamical systems is based on the homotopy and numerical continuation method (NCM). The idea is to construct a sequence of similar systems such that for the first (starting) system the initial point for numerical computation of oscillating solution (starting oscillation) can be obtained analytically. Thus, it is often possible to consider the starting system with self-excited starting oscillation; then the transformation of this starting oscillation in the phase space is tracked numerically while passing from one system to another; the last system corresponds to the system in which a hidden attractor is searched.

For studying the scenario of transition to chaos, we consider system (12) with $f(u)=f(u,\lambda)$, where $\lambda\in\Lambda\subset\mathbb{R}^{d}$ is a vector of parameters, whose variation in the parameter space $\Lambda$ determines the scenario. Let $\lambda_{\rm end}\in\Lambda$ define a point corresponding to the system, where a hidden attractor is searched. Choose a point $\lambda_{\rm begin}\in\Lambda$ such that we can analytically or numerically localize a certain nontrivial (oscillating) attractor $\mathcal{A}^{1}$ in system (12) with $\lambda=\lambda_{\rm begin}$ (e.g., one can consider an initial self-excited attractor defined by a trajectory ${u}^{1}(t)$ numerically integrated on a sufficiently large time interval $t\in[0,T]$ with initial point ${u}^{1}(0)$ in the vicinity of an unstable equilibrium). Consider a *path222 In the simplest case, when $d=1$, the path is a line segment.

• in the parameter space $\Lambda$ , i.e. a continuous function $\gamma\leavevmode\nobreak\ :\leavevmode\nobreak\ [0,\,1]\to\Lambda$, for which $\gamma(0)=\lambda_{\rm begin}$ and $\gamma(1)=\lambda_{\rm end}$, and a sequence of points $\{\lambda^{j}\}_{j=1}^{k}$ on the path, where $\lambda^{1}=\lambda_{\rm begin}$, $\lambda^{k}=\lambda_{\rm end}$, such that the distance between $\lambda^{j}$ and $\lambda^{j+1}$ is sufficiently small. On each next step of the procedure, the initial point for a trajectory to be integrated is chosen as the last point of the trajectory integrated on the previous step: ${u}^{j+1}(0)={u}^{j}(T)$. Following this procedure and sequentially increasing $j$, two alternatives are possible: the points of $\mathcal{A}^{j}$ are in the basin of attraction of attractor $\mathcal{A}^{j+1}$, or while passing from system (12) with $\lambda=\lambda^{j}$ to system (12) with $\lambda=\lambda^{j+1}$, a loss of stability bifurcation is observed and attractor $\mathcal{A}^{j}$ vanishes. If, while changing $\lambda$ from $\lambda_{\rm begin}$ to $\lambda_{\rm end}$, there is no loss of stability bifurcation of the considered attractors, then a hidden attractor for $\lambda^{k}=\lambda_{\rm end}$ (at the end of the procedure) is localized.

Classical attractors obtained in the Lorenz, Rabinovich, and Glukhovsky-Dolzhansky systems are self-excited, each can be visualized easily by a trajectory from a small vicinity of one of the unstable equilibria (see Lorenz-1963 , Rabinovich-1978 , GlukhovskyD-1980 , respectively).

Recently, hidden attractors were discovered in the Glukhovsky-Dolzhansky system (5) for $\sigma=4$ (see LeonovKM-2015-CNSNS ; LeonovKM-2015-EPJST ) and in the Rabinovich system (see KuznetsovLMS-2016-INCAAM ) by numerical continuation method. For $\sigma=10$, $b=8/3$ and $r=24$ in the Lorenz system, there exists a hidden bounded chaotic set (similar to the classical Lorenz attractor), which is numerically indistinguishable from sustained chaos since it persists for a very long time (see corresponding discussions in YorkeY-1979 ; YuanYW-2017-HA ). Our aim here is to find a continuous path in the parameter space of system (1) that connects the above hidden chaotic set in the Lorenz system with the hidden Glukhovsky-Dolzhansky and Rabinovich attractors.

III

Localization of hidden attractors on one path

In this experiment for system (1), we consider three sets of parameters: $P_{\rm GD}\,\left(r=346,\,a=0.01,\,\sigma=4,\,b=1\right)$ (for the Glukhovsky-Dolzhansky system — GD), $P_{\rm L}\,\left(r=24,\,a=0,\,\sigma=10,\,b=8/3\right)$ (for the Lorenz system — L), and $P_{\rm R}\,\left(r=24,\,a=-1/r-0.01,\,\sigma=-ar,\,b=b_{\rm cr}+0.14\right)$ (for the Rabinovich system — R). Here, we change the parameters in such a way that hidden Glukhovsky-Dolzhansky and Rabinovich attractors are located not too close to the unstable zero equilibrium so as to avoid a situation that numerically integrated trajectory persists for a long time and then falls on an unstable manifold of the unstable zero equilibrium, then leaves the transient chaotic set, and finally tends to one of the stable equilibria (see e.g. the corresponding discussion on the Lorenz system in YorkeY-1979 ).

Hidden chaotic attractor in the Glukhovsky-Dolzhansky system with $P_{\rm GD}\,\left(r=346,\,a=0.01,\,\sigma=4,\,b=1\right)$ can be obtained from a self-excited attractor by numerical continuation method LeonovKM-2015-CNSNS ; LeonovKM-2015-EPJST . See the corresponding path in the space of parameters in Fig. 1 and the localization procedure in Fig. 2.

These sets of parameters define three points, $P_{\rm GD}$, $P_{\rm L}$ and $P_{\rm R}$, in the 4D parameter space $(r,\,a,\,\sigma,\,b)$. Consider two line segments, $P_{\rm GD}\rightarrow P_{\rm L}$ and $P_{\rm L}\rightarrow P_{\rm R}$, defining two parts of the path in the continuation procedure. Choose the partition of the line segments into $N_{\rm st}=10$ parts and define intermediate values of parameters as follows: $P_{\rm GD\to L}^{i}=P_{\rm GD}+\frac{i}{N_{\rm st}}(P_{\rm L}-P_{\rm GD})$ and $P_{\rm L\to R}^{i}=P_{\rm L}+\frac{i}{N_{\rm st}}(P_{\rm R}-P_{\rm L})$, where $i=1,\ldots,N_{\rm st}$. Initial points for trajectories of system (1) that define hidden chaotic sets are presented in Table 1. At each iteration of the procedure, a chaotic attractor (defined by the trajectory in the phase space of system (1)) is computed. The last computed point of the trajectory at the previous step is used as the initial point for computation at the next step.

By this procedure, starting from the hidden Glukhovsky-Dolzhansky attractor it is possible to localize numerically hidden chaotic sets in the Lorenz and Rabinovich systems. For the considered parameters, the trajectories, starting in small neighborhoods of unstable zero equilibrium, are not attracted by the computed chaotic set, and the outgoing separatrices of unstable zero equilibrium tend to two symmetric stable equilibria. Thus, the computed chaotic sets are hidden according to the above classification.

Remark*.*

The path and its partition are chosen such that during the procedure the obtained intermediate attractors are self-excited (equilibria $S_{\pm}$ are unstable) and the basin of attraction of the attractor at the current step intersects with the attractor obtained on the previous step.

Hereinafter, it is reasonable to try to increase the length of the step (i.e. decrease the number of the steps) in the continuation procedure, but we may face the situation where the basin of attraction of the current attractor does not intersect the previous attractor, or intersects it only partially. In this case, the result of the procedure depends on the time interval of the numerical integration of the trajectory.

All numerical experiments were performed in MATLAB R2016b using standard procedures for numerical ODE integration.

IV Conclusion

In this report, by means of the numerical continuation method we localize hidden chaotic sets on one path: from the Glukhovsky-Dolzhansky system through the Lorenz system to the Rabinovich system. This helps better understanding of hidden chaotic attractors and their relationships.

Acknowledgement

This work was supported by the Russian Science Foundation (project 14-21-00041).