The infinitely many zeros of stochastic coupled oscillators driven by random forces

TL;DR

This paper extends the study of zeros in stochastic oscillators to coupled systems, analyzing harmonic and nonlinear cases, and evaluates numerical methods for capturing this oscillatory behavior.

Contribution

It introduces the analysis of infinitely many zeros in coupled stochastic oscillators, including harmonic and nonlinear types, and assesses numerical integrators for these dynamics.

Findings

Coupled harmonic oscillators exhibit infinitely many zeros under random forces.

Certain classes of nonlinear oscillators also show this oscillatory behavior.

Numerical integrators can effectively reproduce the zeros in these stochastic systems.

Abstract

In this work, previous results concerning the infinitely many zeros of single stochastic oscillators driven by random forces are extended to the general class of coupled stochastic oscillators. We focus on three main subjects: 1) the analysis of this oscillatory behavior for the case of coupled harmonic oscillators; 2) the identification of some classes of coupled nonlinear oscillators showing this oscillatory dynamics and 3) the capability of some numerical integrators - thought as discrete dynamical systems - for reproducing the infinitely many zeros of coupled harmonic oscillators driven by random forces.

Equations

$$d\mathbf{x}\left(t\right)=\mathbf{Ax}\left(t\right)dt+\mathbf{B}d{\mathbf{w}}_t,$$

$$\mathbf{x}(t)=\left[\begin{array}{c}x(t)\\ y(t)\end{array}\right],\mathbf{A}=\left[\begin{array}{cc}0& \mathbf{I}\\ -{\Lambda }^2& 0\end{array}\right],\text{ and}\mathbf{B}=\left[\begin{array}{c}0\\ \Pi \end{array}\right],$$

$$\mathcal{P}\left(\underset{n\to \infty }{\mathrm{limsup}}\frac{S_n}{\sqrt{2s_n^2\log \log s_n^2}}\ge 1\right)=1,$$

$$\mathcal{P}\left(\underset{n\to \infty }{\mathrm{liminf}}\frac{S_n}{\sqrt{2s_n^2\log \log s_n^2}}\le -1\right)=1.$$

$$\Lambda =Pdiag[{\lambda }_1,\dots ,{\lambda }_d]P^{⊺},$$

$$f(\Lambda )=Pdiag[f({\lambda }_1),\dots ,f({\lambda }_d)]P^{⊺}.$$

$$\displaystyle\left[\begin{array}[c]{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]$$

$$+\underset{t_0}{\overset{t}{\int }}\left[\begin{array}{c}{\Lambda }^{-1}\sin (\Lambda \left(t-s\right))\\ \cos (\Lambda \left(t-s\right))\end{array}\right]\Pi d{\mathbf{w}}_s,$$

$$x^1(t)=D(t)+V(t),$$

$$D(t)=\sum _{k=1}^d\left(P_{1k}\cos ({\lambda }_k(t-t_0))⟨P_k,x_0⟩+P_{1k}{\lambda }_k^{-1}\sin ({\lambda }_k(t-t_0))⟨P_k,y_0⟩\right),$$

$$V(t)=\sum _{l=1}^m\left(\underset{t_0}{\overset{t}{\int }}\left(\sum _{k=1}^d{c}_k^l\sin \left({\lambda }_k\left(t-s\right)\right)\right)d{\mathbf{w}}_s^l\right),$$

$$V(t)=\sum _{l=1}^m\left(\underset{t_0}{\overset{t}{\int }}\left(\sum _{j=1}^{d^{\ast }}{e}_j^l\sin \left({\lambda }_j^{\ast }\left(t-s\right)\right)\right)d{\mathbf{w}}_s^l\right),$$

$$S_n:=V(t_n)=\sum _{r=1}^n{V}_{nr},$$

$$V_{nr}=\sum _{l=1}^m\left(\underset{t_0+(r-1)\Delta }{\overset{t_0+r\Delta }{\int }}\left(\sum _{k=1}^d{c}_k^l\sin \left({\lambda }_k(t_n-s)\right)\right)d{\mathbf{w}}_s^l\right),$$

$${\sigma }_{nr}^2$$

$$=\sum _{l=1}^m\underset{t_0+(r-1)\Delta }{\overset{t_0+r\Delta }{\int }}{\left(\sum _{k=1}^d{c}_k^l\sin \left({\lambda }_k\left(t_n-s\right)\right)\right)}^{2}ds.$$

$$S_n=\sum _{r=1}^n{\sigma }_{nr}{\eta }_r,$$

$$s_n^2=\sum _{r=1}^n{\sigma }_{nr}^2.$$

$$s_n^2$$

$$-\exp (\text{i}({\lambda }_j-{\lambda }_k)t_n)\underset{t_0}{\overset{t_n}{\int }}\exp (-\text{i}({\lambda }_j-{\lambda }_k)s)ds\},$$

$${\textstyle\int\limits_{t_{0}}^{t_{n}}}\exp(-\text{$i$}\theta s)ds=\left\{\begin{array}[c]{cc}n\Delta&\text{if }\theta=0\text{ mod }2\pi\\ \left(\exp(-\text{$i$}\theta t_{0})-\exp(-\text{$i$}\theta t_{n})\right)/(\text{$i$}\theta)&\text{otherwise}\end{array}\right.,$$

$$s_n^2=\frac{1}{2}\sum _{l=1}^m\sum _{k=1}^d{\left(c_k^l\right)}^{2}n\Delta +C_n,$$

$$\underset{n\to \infty }{\lim }\frac{s_n^2}{n}=\frac{1}{2}\sum _{l=1}^m\sum _{k=1}^d{\left(c_k^l\right)}^2\Delta >0.$$

$${\sigma }_{nr}^2$$

$$\le \Delta d\sum _{l=1}^m\sum _{k=1}^d{\left|c_k^l\right|}^2,$$

$$S_n>\left(1-\epsilon \right)\sqrt{2s_n^2\left(\log \log s_n^2\right)}\text{ for infinitely many values of }n\text{ (almost surely).}$$

$$\left|D(t_n)\right|\le {\left|P\right|}^2(\left|x_0\right|+\left|y_0\right|\underset{k}{\max }\left\{{\lambda }_k^{-1}\right\})$$

$$x^1(t_n)>0\text{ infinitely often as }n\to \infty \text{ (almost surely).}$$

$$S_n<\left(-1+\epsilon \right)\sqrt{2s_n^2\left(\log \log s_n^2\right)}\text{ for infinitely many values of }n\text{ (almost surely)}$$

$$x^1(t_n)<0\text{ infinitely often as }n\to \infty \text{ (almost surely).}$$

$$\begin{array}[c]{l}dx(t)=y(t)dt,\\ dy(t)=-f(x(t),y(t))dt+\Pi d\widetilde{\mathbf{w}}_{t},\end{array}$$

$$\left|f(x,y)\right|\le K_1(1+\left|x\right|+\left|y\right|),$$

Full text

1 Introduction

Motivated by their capability to describe the time evolution of complex random phenomena, models of nonlinear oscillators driven by random forces have become a focus of intensive studies (see, e.g., [6], [1], [23], [12], [13]). Naturally, the added noise modifies the dynamics of the deterministic oscillators and so new distinctive dynamical features arise in these random systems. Since the complexity of the random dynamics depends on the type of nonlinearity and the level of noise, many of the results on this matter have been achieved for specific classes of stochastic oscillators. In particular, a number of properties have been studied for the simple harmonic oscillator such as the stationary probability distribution, the linear growth of energy along the paths, the oscillation of the solution, and the symplectic structure of Hamiltonian oscillators, among others (see, e.g., [14], [3], [15], [21], [20]). Some of these properties have been also analyzed for coupled harmonic oscillators. However, to the best of our knowledge, there are no studies concerning the oscillatory behavior around the origin of stochastic coupled oscillators.

On the other hand, demanded by an increasing number of practical applications (see e.g., [7], [1], [8], and references therein), the numerical simulation of stochastic oscillators has also a high interest. In particular, it is required to use specialized numerical integrators that preserve the dynamics of the oscillators since general multipurpose integrators fail to achieve this target. This is so because, in general, the dynamics of discrete dynamical systems is far richer than that of the continuous ones. Consequently, specific oriented integrators for stochastic oscillators have also been proposed, for instance, in [5], [17], [4], [18], [22]. Distinctively, in [5], the family of the Locally Linearized methods have been proved to simultaneously reproduce various dynamical properties of the stochastic harmonic oscillators including the oscillatory behavior around [math] of the single oscillators.

In this work, we are interested in the study of the oscillatory behavior of the stochastic coupled oscillators driven by random forces. We focus on three main aspects: 1) the analysis of this oscillatory behavior for the case of coupled harmonic oscillators, a property that has only been demonstrated for simple oscillators ([14],[13]); 2) the identification of some classes of coupled nonlinear oscillators that display this dynamics; and 3) the capability of the Locally Linearized integrators - as discrete dynamical systems - of reproducing the infinitely many zeros of the coupled harmonic oscillators driven by random forces, which complements known results of these integrators for simple harmonic oscillators [5].

2 The infinitely many zeros of the coupled harmonic oscillators

Let us first consider the undamped harmonic oscillator, defined by the $2d$-dimensional Stochastic Differential Equation (SDE) with additive noise

[TABLE]

for $t\geq t_{0}\geq 0$, with initial condition $\mathbf{x}(t_{0})=(x_{0},y_{0})^{\top},$ $x_{0},y_{0}\in\mathbb{R}^{d}$ and $d>1$. Here,

[TABLE]

being $\Lambda\in\mathbb{R}^{d\times d}$ a nonsingular symmetric matrix, $\Pi\in\mathbb{R}^{d\times m}$ a matrix, $\mathbf{I}$ the $d-$dimensional identity matrix, and $\mathbf{w}_{t}$ an $m$-dimensional standard Wiener process on the filtered complete probability space $\left(\Omega,\mathfrak{F,}\left(\mathfrak{F}_{t}\right)_{t\geq t_{0}},\mathbb{P}\right)$.

In what follows, the symbol $\left\langle\cdot,\cdot\right\rangle$ denotes the Euclidean scalar product associated to the Euclidean vector norm $\left|\cdot\right|$. For matrices, $\left|\cdot\right|$ denotes the Frobenious matrix norm. In addition, the following lemma will be useful.

Lemma 1

Let $\eta_{1},...,\eta_{n},...$ be independent $\mathcal{N}(0,1)$ distributed random variables. Let $\{\sigma_{nr}\}$ be a bounded triangular array of real numbers. Set $S_{n}=$ ${\textstyle\sum\limits_{r=1}^{n}}\sigma_{nr}\eta_{r}$ and $s_{n}^{2}={\textstyle\sum\limits_{r=1}^{n}}\sigma_{nr}^{2}.$ If $\underset{n\rightarrow\infty}{\lim\inf}\frac{s_{n}^{2}}{n}>0,$ then

[TABLE]

and

[TABLE]

Proof. This is a direct consequence of Corollary $1$ of Theorem $2$ in [19].

The following theorem shows the infinitely many oscillations of the paths of coupled harmonic oscillators (1), which extends the Theorem $4.1$ in [13] (Section $8.4$) that refers to the paths of simple harmonic oscillators (i.e., those defined by (1) with $d=1$).

Theorem 2

Consider the coupled harmonic oscillator (1). Then, almost surely, each component of the solution $\mathbf{x}(t)$ has infinitely many zeros on $[t_{0}$ $\infty)$ for every $t_{0}\geq 0.$

Proof. Let us start considering the first component $x^{1}$ of the solution of (1). By the spectral theorem for the real symmetric matrix $\Lambda$ we have the factorization

[TABLE]

where $\lambda_{1},\ldots,\lambda_{d}$ are the eigenvalues of $\Lambda$, and $P$ is a real orthogonal matrix with entries $[P_{k,j}]$ for $k,j=1,\ldots,d$. Then, for $f(\Lambda)=\sin(\Lambda)$ and for $f(\Lambda)=\cos(\Lambda)$, we have (see, e.g., [9])

[TABLE]

Since the solution of (1) satisfies (see, e.g., [13])

[TABLE]

then

[TABLE]

where

[TABLE]

and

[TABLE]

being $c_{k}^{l}=P_{1k}\lambda_{k}^{-1}\left\langle P_{k},\Pi_{l}\right\rangle$, and $P_{k},$ $\Pi_{l}$ the column vectors of $P$ and $\Pi$, respectively.

Without loss of generality, let us assume that $\lambda_{k}>0$ and $\lambda_{k}\neq$ $\lambda_{r}$ for all $k\neq r$ with $k,r=1,\ldots,d$. Indeed, when there are only $d^{\ast}<d$ different values $\lambda_{j}^{\ast}$ of $\left|\lambda_{k}\right|$, $k=1,\ldots,d$ and $j=1,\ldots,d^{\ast}$, the expression (5) can be rewritten as

[TABLE]

where $e_{j}^{l}={\displaystyle\sum\limits_{k=1}^{d}}c_{k}^{l}\delta_{\lambda_{j}^{\ast}}^{\left|\lambda_{k}\right|}(1_{\lambda_{k}>0}-1_{\lambda_{k}<0})$, and $\delta$ is the Kronecker delta. For this expression of $V(t)$ the analysis below would be the same as that for (5) with the above mentioned assumptions on $\lambda_{k}$.

Consider an arbitrary $\Delta>0$ and the time instants $t_{n}=t_{0}+n\Delta$, with $n=1,2,\ldots$. In addition, for all $n$, define

[TABLE]

where $V(t_{n})$ is defined in (5), and

[TABLE]

for all $n,r=1,2,\ldots$. Because the independence of $\mathbf{w}_{s}^{1},...,\mathbf{w}_{s}^{m}$ and the independence of the increments of $\mathbf{w}_{s}^{l}$ on disjoint intervals, $\{V_{nr}\}_{r\geq 1}$ defines a double sequence of $i.i.d.$ Gaussian random variables with zero mean and variance

[TABLE]

In this way, (6) can be written as

[TABLE]

where $\eta_{1},...,\eta_{n}$ are $i.i.d.$ $\mathcal{N}(0,1)$ random variables. Thus, the variance $s_{n}^{2}$ of $S_{n}$ satisfies

[TABLE]

The expression (7) and the identity $sin(\theta)=\left(\exp(i\theta)-\exp(-i\theta)\right)/(2i)$ (where $i=\sqrt{-1}$) imply that

[TABLE]

where $\operatorname{Re}$ denotes the real part of a complex number. Since

[TABLE]

we have

[TABLE]

where $C_{n}$ is uniformly bounded for all $n$. Thus,

[TABLE]

In addition, since

[TABLE]

for all $n$ and $r$, the Law of the Iterated Logarithms of Lemma 1 holds for $S_{n}$. Thus, for $0<\varepsilon<1$, (2) implies that

[TABLE]

In addition, since

[TABLE]

for all $n$, for the fist component (4) of the solution of (1) we have that

[TABLE]

Similarly, (3) implies that

[TABLE]

for $0<\varepsilon<1$, and so

[TABLE]

Thus, since the sample path of the solution to (1) is continuous, $x^{1}(t)$ must have, almost surely, infinitely many zeros on $[t_{0}$ $\infty)$. For the remainder of the components of the solution of (1) we can proceed in a similar manner. This concludes the proof.

3 The infinitely many zeros of coupled nonlinear oscillators

Let us consider the coupled nonlinear oscillator defined by the $2d$-dimensional ($d>1$) SDE with additive noise

[TABLE]

where $\Pi\in\mathbb{R}^{d\times m}$ is a matrix, $\widetilde{\mathbf{w}}_{t}$ is a $m$-dimensional standard Wiener process on a filtered complete probability space different of that of the equation (1), and $f:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a smooth function satisfying the linear growth condition

[TABLE]

for some positive constant $K_{1}$.

For analysis of the oscillatory behavior of (8), next Lemma will be useful.

Lemma 3

Let $\left(x(t),y(t)\right)^{\top}\in\mathbb{R}^{2d}$ be the unique solution of the harmonic oscillator equation (1) on $[0,T]$ for any $T>0$. Suppose that $\Phi_{t}:=\phi(x(t),y(t)):\mathbb{R}\rightarrow\mathbb{R}^{m}$ is a function satisfying the linear growth condition

[TABLE]

Then, there is a probabilistic measure $\widetilde{\mathbb{P}}$ on $\left(\Omega,\mathfrak{F}\right)$ absolutely continuous with respect to $\mathbb{P}$ and an $m$-dimensional standard Wiener process $\widetilde{\mathbf{w}}_{t}$ on $\left(\Omega,\mathfrak{F,}\left(\mathfrak{F}_{t}\right)_{t\geq t_{0}},\widetilde{\mathbb{P}}\right)$ such that $\left(x(t),y(t)\right)^{\intercal}$ is also the unique solution of the nonlinear equation

[TABLE]

on $[0,T]$.

Proof. Let $\mathbf{x}_{t}=\left(x(t),y(t)\right)^{\intercal}$ be the solution of the equation (1) on $[0,T]$. From the condition (10) it follows that

[TABLE]

where $C=3K^{2}$.

Since $\mathbf{x}_{t}$ is the solution of the linear SDE with additive noise (1), $\mathbf{x}_{t}\sim\mathcal{N}_{2d}(\mu_{t},\Sigma_{t})$ for all $t\in[0,T]$, where the mean $\mu_{t}$ and the variance $\Sigma_{t}$ of $\mathbf{x}_{t}$ are continuous functions on $[0,T]$ (see, e.g., [2]). Here, $\mathcal{N}_{2d}$ denotes $2d-$variate normal distribution. The random vector $\mathbf{x}_{t}$ can be written as $\mathbf{x}_{t}=\mu_{t}+\Sigma_{t}^{1/2}Z_{t}$, where $\Sigma_{t}^{1/2}$ is the symmetric square root of $\Sigma_{t}$, and $Z_{t}\sim\mathcal{N}_{2d}(\mathbf{0},\mathbf{I})$. Therefore,

[TABLE]

Since $\left|Z_{t}\right|^{2}$ is a random variable that has chi-squared distribution with $2d$ degrees of freedom, $E\left(\exp\left(\alpha\left|Z_{t}\right|^{2}\right)\right)\leq 1/\left(1-2\alpha\right)^{d}$ for $\alpha<1/2$ ([11], pp. 420). Therefore, for all $a<1/\left(8C\max_{t\in[0,T]}\left|\Sigma_{t}\right|\right)$, it holds that

[TABLE]

where $D=$ $\exp\left(aC+2aC\max_{t\in[0,T]}\left|\mu_{t}\right|^{2}\right)$. The proof is then completed as a direct consequence of the Cameron-Martin-Girsanov theorem (see, e.g., [13], pp. 274).

Next theorem provides conditions that guarantee a link between the solutions of the harmonic and nonlinear oscillator equations.

Theorem 4

Let $\left(x(t),y(t)\right)^{\intercal}\in\mathbb{R}^{2d}$ be the unique solution of the harmonic oscillator equation (1) on $[0,T]$ for $T>0$. Let $\Phi_{t}:=\phi(x(t),y(t)):\mathbb{R}\rightarrow\mathbb{R}^{m}$ be a function such that

[TABLE]

where the function $f$ satisfies the linear growth condition (9). Then, there is a probabilistic measure $\widetilde{\mathbb{P}}$ on $\left(\Omega,\mathfrak{F}\right)$ absolutely continuous with respect to $\mathbb{P}$ and an $m$-dimensional standard Wiener process $\widetilde{\mathbf{w}}_{t}$ on $\left(\Omega,\mathfrak{F,}\left(\mathfrak{F}_{t}\right)_{t\geq t_{0}},\widetilde{\mathbb{P}}\right)$ such that $\left(x(t),y(t)\right)^{\intercal}$ is also the unique solution of the nonlinear oscillator equation (8) on $[0,T]$.

Proof. Since $\Phi_{t}$ solves the equation (12), $\Phi_{t}=\Pi^{-}\left(\Lambda^{2}x(t)-f(x(t),y(t))\right)$, where the matrix $\Pi^{-}$ is a generalized inverse of $\Pi$. This and condition (9) imply that $\Phi_{t}$ satisfies the linear growth condition (10). Then, the assumptions of Lemma 3 are fulfilled, which completes the proof.

Notice that the assumptions of Theorem 4 are directly satisfied in the case, for instance, that $\Pi$ in (1) is a nonsingular $d\times d$ matrix.

Next theorem deals with the infinite oscillations of the paths of the coupled nonlinear oscillator (8).

Theorem 5

Each component of the solution of the coupled nonlinear oscillator (8) has infinitely many zeros on $[t_{0}$ $\infty)$ for every $t_{0}\geq 0$ almost surely.

Proof. Theorem 4 states that, for properties holding almost surely, the analysis of the nonlinear oscillator (8) with growth condition (9) reduces to that of the harmonic oscillator (1). In this way, since by Theorem 2 each component of the harmonic oscillator (1) has infinitely many zeros on $[t_{0}$ $\infty)$, each component of the nonlinear oscillator (1) will also has infinitely many zeros on $[t_{0}$ $\infty)$ for every $t_{0}\geq 0$.

As example of equation (8), with condition (9) being satisfied, we can mention the equation of various type of coupled pendulums driven by random forces, as those of [16]: e.g., 1) a double pendulum (a pendulum with another pendulum attached to its end); and 2) a pair of identical pendulums connected by a weak spring. For the last one, we have the equation

[TABLE]

with $\alpha,$ $\beta,$ $\sigma_{1},$ $\sigma_{2}$ $\in\mathbb{R}^{+}$.

Clearly for this equation there is a function $\Phi_{t}$ satisfying (12). Thus, by Theorem 5, each component of this nonlinear equation has infinitely many zeros on $[t_{0}$ $\infty)$ for every $t_{0}\geq 0$ almost surely.

4 Simplicity of the zeros

This section deals with the simplicity of the zeros of coupled harmonic and coupled nonlinear stochastic oscillators considered in previous sections. That is, we will prove that the component $y^{i}$ of the oscillators does not vanish at the same time that the component $x^{i}$ does.

Theorem 6

The infinite many zeros of the coupled harmonic oscillator (1) are simple.

Proof. First, note that Theorem 4 implies that there is a probabilistic measure $\widetilde{\mathbb{P}}$ on $\left(\Omega,\mathfrak{F}\right)$ absolutely continuous with respect to $\mathbb{P}$ and an $m$-dimensional standard Wiener process $\widetilde{\mathbf{w}}_{t}$ on $\left(\Omega,\mathfrak{F,}\left(\mathfrak{F}_{t}\right)_{t\geq t_{0}},\widetilde{\mathbb{P}}\right)$ such that the solution $\left(x(t),y(t)\right)$ of the coupled harmonic oscillators (1) is also the unique solution of the discoupled oscillator

[TABLE]

on $[t_{0},T]$, with initial condition $\mathbf{x}(t_{0})=(x_{0},y_{0})^{\top}$, being $\mathbf{D}\in\mathbb{R}^{d\times d}$ a diagonal matrix and $\Pi$ defined as in (1).

When $\mathbf{w}_{t}$ in (1) is a one-dimensional standard Wiener process, the simplicity of the zeros of each component $x^{i}$ of (1) is a straightforward consequence of the mentioned in the previous paragraph and Theorem 4.1, pp. 280, in [13] for the simplicity of the zeros of the simple harmonic oscillator with one-dimensional Wiener process.

When $m>1$, by following similar ideas of the proof of Theorem 3.4, pp. 277, in [13], the simplicity of the zeros of the simple harmonic oscillator

[TABLE]

with $\alpha\in\mathbb{R},$ can be proved as follows.

We will first ensure the existence of a function $V(x,y)>0$ such that

[TABLE]

From the Itô-formula

[TABLE]

with the operator

[TABLE]

it is easy to check that if $LV(x,y)=0$ then $\mathbb{E}\left(V(x(t),y(t))\right)=\mathbb{E}\left(V(x_{0},y_{0})\right)$. In addition, note that the operator $L$ defines the Forward-Kolmogorov (Fokker-Planck) equation

[TABLE]

for the transition probability function $p$ corresponding to the solution of the linear SDE

[TABLE]

for some scalar Wiener process $\bar{W}_{t}$ with $\rho={\textstyle\sum\limits_{j=1}^{m}}\sigma_{j}^{2}$. Thus,

[TABLE]

with

[TABLE]

After some algebraic manipulation we obtain that

[TABLE]

with

[TABLE]

From this and (17), it is easy to check that

[TABLE]

satisfies the conditions (15).

Now, let us prove that for the stopping time

[TABLE]

$\widetilde{\mathbb{P}}\mathbb{(}\omega\in\Omega:\tau\leq T)=0$, for arbitrary $T>t_{0}$. For this, let us define additional stopping times

[TABLE]

for $n=1,2,...$. Using condition (15) and that $(x(\tau_{n}\wedge T),y(\tau_{n}\wedge T))=(x(\tau_{n}),y(\tau_{n}))$ for all $\omega\in\Gamma=\left\{\omega\in\Omega:\text{ }\tau\leq T\right\}\subset\left\{\omega\in\Omega:\tau_{n}\leq T\right\}$, we have

[TABLE]

In this way,

[TABLE]

From (15) and since $\left|x(\tau_{n})\right|+\left|y(\tau_{n})\right|\leq\sqrt{2\left(\left|x(\tau_{n})\right|^{2}+\left|y(\tau_{n})\right|^{2}\right)}\leq\sqrt{\frac{2}{n}}$, we have $\lim_{n\longrightarrow\infty}V(x(\tau_{n}),y(\tau_{n}))=\infty$. Thus, since $\mathbb{E}\left(V(x_{0},y_{0})\right)<\infty$, necessarily $\widetilde{\mathbb{P}}(\Gamma)=0$ holds. That is, all the zeros of (14) are simple. From this and taking into account that the coupled (1) and discoupled (13) harmonic oscillators with $m\geq 1$ have the same solution, almost surely, the proof is completed.

For the zeros of coupled nonlinear oscillators we have the following result.

Theorem 7

The infinite many zeros of the coupled nonlinear oscillator (8) are simple.

Proof. It is a straightforward consequence of Theorem 4 and Theorem 6 above.

5 The infinitely many zeros of the Local Linearized integrators for

coupled harmonic oscillators

Let $(t)_{h}=\{t_{n}=t_{0}+nh:n=0,1,\ldots\},$ $h>0,$ be a partition of the time interval $[t_{0},\infty)$. The Locally Linearized integrator for the equation (1) is defined by the recursive expression [5]

[TABLE]

for $n=0,1,\ldots$, with initial condition $\mathbf{x}_{0}=\mathbf{x}(t_{0})$, where $\mathbf{x}_{n}=(x_{n},y_{n})^{\top},$ $x_{n},y_{n}\in\mathbb{R}^{d}$, $\mathbf{u}_{n}=\mathbf{L}e^{\mathbf{C}_{n}h}\mathbf{r}$, and $\mathbf{z}_{n+1}=\mathbf{Q}\Delta\mathbf{w}_{n}$. Here

[TABLE]

$\mathbf{L}=\left[\mathbf{I}_{2d}\text{ \ }\mathbf{0}_{2d\times 1}\right]$, $\mathbf{r}=\left[\mathbf{0}_{1\times 2d}\text{ }1\right]^{\intercal}$, $\Delta\mathbf{w}_{n}=\mathbf{w}_{t_{n+1}}-\mathbf{w}_{t_{n}}$, and \mathbf{Q}=[\begin{array}[c]{c}Q^{1}\\ Q^{2}\end{array}] is a $2d\times m$ matrix with $Q^{1}$, $Q^{2}\in\mathbf{\mathbb{R}}^{d\times m}$. The Locally Linearized integrator $\mathbf{x}_{n+1}$ converges, strongly with order $1$, to the solution $\mathbf{x}(t_{n+1})$ of (1) at $t_{n+1}$ as $h$ goes to zero ([10], [5]).

Next theorem deals with the reproduction of the oscillatory behavior of coupled harmonic oscillators by the discrete dynamical system defined by the Locally Linearized integrator.

Theorem 8

Let $\lambda_{1},\ldots,\lambda_{d}$ be the eigenvalues of $\Lambda$, and $\left|\lambda\right|_{\max}=\max\limits_{k}\left(\left|\lambda_{k}\right|\right)$. For the coupled harmonic oscillator (1), each component of the Locally Linearized integrator switches signs infinitely many times as $n\rightarrow\infty$, almost surely, for any integration stepsize $h<\pi/\left|\lambda\right|_{\max}$.

Proof. Lemma 3.2 in [5] states that the Locally Linearized integrator (18) can be written as

[TABLE]

where

[TABLE]

Likewise in the proof of Theorem 2, by using the Spectral Theorem, the first component $x_{n+1}^{1}$ of $\mathbf{x}_{n+1}$ can be written

[TABLE]

where

[TABLE]

and

[TABLE]

being

[TABLE]

with $e_{j}^{l}=P_{1j}\left\langle P_{j},Q_{l}^{1}\right\rangle$, $f_{j}^{l}=P_{1j}\lambda_{j}^{-1}\left\langle P_{j},Q_{l}^{2}\right\rangle$ and $Q_{l}^{1},$ $Q_{l}^{2}$ the column vectors of $Q^{1}$ and $Q^{2}$, respectively.

Without loss of generality, let us assume that $\lambda_{k}>0$ and $\lambda_{k}\neq$ $\lambda_{r}$ for all $k\neq r$ with $k,r=1,\ldots,d$. Indeed, when there are only $d^{\ast}<d$ different values $\lambda_{j}^{\ast}$ of $\left|\lambda_{k}\right|$, $k=1,\ldots,d$ and $j=1,\ldots,d^{\ast}$, the expression (21) can be rewritten as

[TABLE]

where $E_{j}^{l}={\displaystyle\sum\limits_{k=1}^{d}}e_{k}^{l}\delta_{\lambda_{j}^{\ast}}^{\left|\lambda_{k}\right|}$, $F_{j}^{l}={\displaystyle\sum\limits_{k=1}^{d}}f_{k}^{l}\delta_{\lambda_{j}^{\ast}}^{\left|\lambda_{k}\right|}(1_{\lambda_{k}>0}-1_{\lambda_{k}<0})$, and $\delta$ is the Kronecker delta. For this expression of $V_{nr}$ the analysis below would be the same as that for (21) with the above mentioned assumptions on $\lambda_{k}$.

Since $\Delta\mathbf{w}_{n-r}^{l}$ are independent Gaussian random variables with zero mean and variance $h$, $\{V_{nr}\}$ defines a sequence of $i.i.d.$ Gaussian random variable with zero mean and variance

[TABLE]

Thus, (20) can be rewritten as

[TABLE]

where $\eta_{0},...,\eta_{n}$ are $i.i.d.$ $\mathcal{N}(0,1)$ random variables. Thus, the variance $s_{n}^{2}$ of $S_{n}$ satisfies

[TABLE]

Note that, for all $j=1,...,d$,

[TABLE]

where $c_{j}^{2}=\left({\displaystyle\sum\limits_{l=1}^{m}}e_{j}^{l}\right)^{2}+\left({\displaystyle\sum\limits_{l=1}^{m}}f_{j}^{l}{}\right)^{2}$, $\alpha_{j}=\arctan\left({\displaystyle\sum\limits_{l=1}^{m}}f_{j}^{l}{}/{\displaystyle\sum\limits_{l=1}^{m}}e_{j}^{l}\right)$ for ${\displaystyle\sum\limits_{l=1}^{m}}e_{j}^{l}\neq 0$, and $\alpha_{j}=\frac{\pi}{2}$ for ${\displaystyle\sum\limits_{l=1}^{m}}e_{j}^{l}=0.$

From this and by using the identity $\cos(\theta)=\left(\exp(i\theta)+\exp(-i\theta)\right)/2$, we obtain that

[TABLE]

where $\operatorname{Re}$ denotes the real part of a complex number. Under the assumption $h<\pi/\left|\lambda\right|_{\max}$, it holds that $\theta\neq 0\operatorname{mod}2\pi$ for all $\theta=h\left(\lambda_{j}+\lambda_{k}\right)$ with $1\leq j,k\leq d$, and for all $\theta=h\left(\lambda_{k}-\lambda_{j}\right)$ with $1\leq j\neq k\leq d$. Therefore, from (23) and the known expression of the partial sum of the geometric series

[TABLE]

it is obtained that

[TABLE]

where $C_{n}$ is uniformly bound for all $n$. Thus, the assumption $h<\pi/\left|\lambda\right|_{\max}$ implies that

[TABLE]

Since $\sigma_{nr}^{2}$ is bounded for all $n$ and $r$, the Law of the Iterated Logarithms stated in Lemma 1 holds for $S_{n}$. Thus, for $0<\varepsilon<1$, (2) implies that

[TABLE]

In addition, since

[TABLE]

for all $n$, the fist component (19) of the Locally Linearized integrator (18) satisfies

[TABLE]

Similarly, (3) implies that

[TABLE]

for $0<\varepsilon<1$, and so

[TABLE]

We can proceed similarly to prove that the other components of $\mathbf{x}_{n+1}$ also change sign infinitely often. This completes the proof.

It was shown in [5] that, likewise the exact solution of the simple harmonic oscillator (equation (1) with $d=1)$, the path of the Local Linearized integrator (18) switches signs infinitely many times as $n\rightarrow\infty$ almost surely for any integration stepsize $h$. However, according to Theorem 8, in the case of the coupled oscillator (1), this dynamics of the Local Linearized integrator (18) is only guaranteed for stepsize $h<\pi/\max\left(\left|\lambda_{1}\right|,\ldots,\left|\lambda_{d}\right|\right)$, where $\lambda_{1},\ldots,\lambda_{d}$ are the eigenvalues of $\Lambda$.

Theorem 8 complements the results obtained in [5] that demonstrate the capability of the discrete dynamical system defined by the Local Linearized integrators for reproducing other essential continuous dynamics of the coupled harmonic oscillators: the linear growth of energy along the paths, and the symplectic structure of Hamiltonian oscillators.

Furthermore, since the exponential and trigonometric integrators considered in [18] and [4] reduce to the expression (18) when they are applied to equation (1), the Theorem 8 can be also applied for these integrators. In this way, these integrators with stepsize $h<\pi/\max\left(\left|\lambda_{1}\right|,\ldots,\left|\lambda_{d}\right|\right)$ also switch signs infinitely many times as $n\rightarrow\infty$ almost surely.

6 Conclusion

In this work, previous results concerning the infinitely many zeros of the single harmonic oscillators driven by random forces were extended to the general class of coupled harmonic oscillators. Furthermore, various classes of coupled nonlinear oscillators having this oscillatory behavior were identified. The ability of the discrete dynamical system defined by various numerical integrators for reproducing this oscillatory property of the continuous systems was also analyzed, which complements known results of these integrators for the simple harmonic oscillators driven by random forces.

Acknowledgements  The authors thank the financial support of a FGV/EMAp project, Brazil, and Centro de Investigación en Matemáticas (CIMAT), Mexico.