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Nonlinear Dynamics

, Volume 89, Issue 4, pp 2785–2793 | Cite as

Random perturbations of parametric autoresonance

  • Oskar SultanovEmail author
Original Paper

Abstract

We consider a system of two nonlinear differential equations describing the capture into autoresonance in nonlinear oscillators under small parametric driving. Solutions with an infinitely growing amplitude are associated with the autoresonance phenomenon. Stability of such solutions is of great importance because only stable solutions correspond to physically observable motions. We study stability of autoresonant solutions with power asymptotics and show that the random fluctuations of the driving cannot destroy the capture into the parametric autoresonance.

Keywords

Nonlinear system Autoresonance Random perturbation Stability analysis 

1 Introduction

Autoresonance is a persistent phase-locking phenomenon occurring in resonantly driven nonlinear oscillatory systems. The essence of this phenomenon is that the nonlinear oscillator automatically adjusts to slowly varying driving and remains in a resonance for a long time. This results in a considerable growth of the amplitude of oscillations. Autoresonance was first suggested in [1, 2] to accelerate relativistic particles. Later, a wide range of applications of autoresonance phenomena to various physical problems associated with nonlinear oscillators and waves were observed in [3]. The capture into the resonance in nonlinear systems with slowly varying parameters was described in [4, 5]. Mathematical theory of autoresonance was developed in [6].

Autoresonance phenomenon in a parametrically driven oscillator is called parametric. The parametric autoresonance was studied both by means of mathematical models and in physical experiments. In particular, the first experimental and theoretical study of the parametric autoresonance phenomenon was described in [7]. The existence of autoresonant solutions was established in [8] for the system of nonlinear oscillator. The theory of parametric autoresonance was developed for nonlinear Faraday waves in the papers [9, 10]. Asymptotic analysis of the capture into the parametric autoresonance was discussed in [11]. Most recent results in the field and the analysis of the transition from the classical parametric autoresonance to quantum ladder climbing were presented in [12].

In spite of the extensive studies of the autoresonance, the question of stability has not been considered closely. The main obstacles to analyzing the stability of autoresonance are that there are no explicit formulas for exact solutions and the linear stability analysis based on the asymptotic solutions usually fails: The linearized system has purely imaginary eigenvalues [13, 14]. In this paper, we provide a careful stability analysis of the parametric autoresonance with respect to random perturbations. In particular, we show that the random fluctuations of the driving cannot destroy the capture into the parametric autoresonance.

The paper is organized as follows. In Sect. 1, we give the mathematical formulation of the problem. In Sect. 2, the stability of autoresonant solutions with respect to initial data disturbances is discussed. In Sect. 3, we describe a class of random perturbations preserving stability of autoresonant solutions. Examples of admissible perturbations are contained in Sect. 4. In Sect. 5, we illustrate our results with numerical simulations. The paper concludes with a brief discussion of the results obtained.

2 Problem statement

We consider the primary parametric resonance equation:
$$\begin{aligned} \begin{array}{l} {\displaystyle \frac{\text {d}r}{\text {d}\tau }= r\sin \psi -\nu r, } \quad {\displaystyle \frac{\text {d}\psi }{\text {d}\tau }=r-\lambda \tau + f \cos \psi , } \end{array} \end{aligned}$$
(1)
where \(\nu >0\) is a dissipation parameter, \(f\ne 0\) and \(\lambda > 0\) are parameters related to amplitude and frequency of driving. The real-valued functions \(r(\tau )\) and \(\psi (\tau )\) represent the amplitude and phase shift of harmonic oscillations. Solutions with an infinitely growing amplitude \(r(\tau )=\lambda \tau +{{\mathcal {O}}}(1)\) as \(\tau \rightarrow \infty \) are associated with the capture of an oscillatory nonlinear system into the parametric autoresonance.
System (1) is of universal character in the mathematical description of parametric autoresonance. It describes long-term evolution of nonlinear oscillations under small parametric driving. As one example, let us consider Duffing’s equation with dissipation:
$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\text {d}^2x}{\text {d}t^2}+ \beta \frac{\text {d}x}{\text {d}t}+[1+\varepsilon \cos (2\phi (t))]\,x-\gamma x^3=0, \end{array} \end{aligned}$$
(2)
where \(\phi (t)=t-\alpha t^2\), \(\alpha , \beta , \varepsilon \) are small positive parameters, and \(\gamma ={\hbox {const}}> 0\). Solutions whose amplitude increase with time from small values to quantities of order one are associated with the autoresonance phenomenon. For studying such solutions, it is convenient to use the method of two scales [15]. We introduce a slow time: \(\tau =\varepsilon t/2\). Then the asymptotics of the resonant solutions can be constructed with using the asymptotic substitution
$$\begin{aligned} x(t)=\varepsilon ^{1/2}\sqrt{\frac{2}{3\gamma } r (\tau ) } \sin \left( \phi (t)-\frac{\psi (\tau )}{2}\right) +{{\mathcal {O}}}(\varepsilon ) \end{aligned}$$
in Eq. (2). The averaging procedure over the fast variable \(\phi (t)\) in the leading-order term in \(\varepsilon \) leads to primary parametric resonance Eq. (1) for the slow varying functions \(r(\tau )\) and \(\psi (\tau )\) with \(\lambda =16\alpha \varepsilon ^{-2}\), \(\nu =2\beta \varepsilon ^{-1}\), and \(f=1\).

2.1 Autoresonant solutions

The asymptotic solutions of (1) can be constructed in the form:
$$\begin{aligned} \begin{array}{l} \displaystyle R(\tau )=\lambda \tau + \sum _{j=0}^{\infty } r_j\, \tau ^{-j}, \ \ \varPsi (\tau )=\sum _{j=0}^{\infty }\psi _j\, \tau ^{-j} \end{array} \end{aligned}$$
(3)
as \(\tau \rightarrow \infty \). Substituting these series in (1) and equating the expressions of the same powers give the recurrence relations for determining the constant coefficients \(r_j\) and \(\psi _j\). Following this approach, in case of \(0<\nu <1\), we find two asymptotic solutions distinguished by the choice of a root to the equation: \(\sin \psi _0=\nu \). Indeed, \(\psi _1=1/\cos \psi _0\), \(r_0=-f\cos \psi _0\), \(r_1=f \tan \psi _0\), etc. The asymptotics in the form of series with constant coefficients can be justified using the results of [16]. Note that Eq. (1) has many autoresonant solutions with more complicated asymptotic behavior at infinity. In this paper, we focus only on the solutions with power asymptotics (3).

2.2 Perturbed equations

Along with (1), we consider the perturbed system of the form:
$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{\text {d}r}{\text {d} \tau }&{}=(1+\mu \xi )r\sin \psi -\nu r, \\ \displaystyle \frac{\text {d}\psi }{\text {d} \tau }&{}=r-\lambda \tau + \mu \zeta +(f+ \mu \eta )\cos \psi , \end{array} \end{aligned}$$
(4)
where \(\xi (r,\psi ,\tau ,\omega )\), \(\eta (r,\psi ,\tau ,\omega )\), and \(\zeta (r,\psi ,\tau ,\omega )\) (\((r,\psi )\in {\mathbb {R}}^2\), \(\tau >0\)) are measurable stochastic processes defined on a probability space \((\varOmega , {\mathcal {F}}, {\mathbb {P}})\) with values in \({\mathbb {R}}\). Here \(\varOmega =\{\omega \}\) is the sample space, \({\mathcal {F}}\) is a \(\sigma \)-algebra, and \({\mathbb {P}}\) is a probability measure. A small parameter \(0<\mu \ll 1\) controls the intensity of the perturbation. We assume that the random processes \(\xi ,\eta ,\zeta \) are well defined with well-behaved sample properties so that the perturbed system can be analyzed by the ordinary rules of calculus. Here we do not consider white noise perturbations. Our goal is to identify a class of functions \((\xi ,\eta ,\zeta )\) such that any solution \(r_\mu (\tau ,\omega )\), \(\psi _\mu (\tau ,\omega )\) of system (4) starting in some neighborhood of the autoresonant solution behaves like \(r_\mu (\tau ,\omega )\sim \lambda \tau , \psi _\mu (\tau ,\omega )\sim {{\mathcal {O}}}(1)\) with high probability.
The structure of the perturbed system is motivated by the fact that the functions \(\xi ,\eta ,\zeta \) correspond to perturbation of the parametric driving of a nonlinear oscillator. Indeed, consider the perturbed equation (cf. (2)):
$$\begin{aligned} \begin{array}{ll} &{} \displaystyle \frac{\text {d}^2x}{\text {d}t^2}+\beta \displaystyle \frac{\text {d}x}{\text {d}t} \\ &{} \quad +\,[1+\varepsilon (1+ \mu \varGamma )\cos (2\phi + \mu \varphi )]x-\gamma x^3=0,\\ \end{array} \end{aligned}$$
(5)
where \(\varGamma =\varGamma (t,\tau ,\omega )\), \(\varphi =\varphi (t,\tau ,\omega )\). For simplicity suppose that \(\langle \varGamma \cos 2\phi \rangle \equiv \langle \varGamma \cos 4\phi \rangle \equiv \langle \varGamma \sin 4\phi \rangle \equiv 0\), where the corner brackets denote the mean value integral with respect to the rapid variable \(\phi \), for instance,
$$\begin{aligned} \langle \varGamma \cos 2\phi \rangle = \frac{1}{2\pi }\int \limits _0^{2\pi } \varGamma (\phi +4\alpha \tau ^2\varepsilon ^{-2},\tau ,\omega ) \cos 2\phi \, \text {d}\phi . \end{aligned}$$
Then the averaging procedure over the fast time leads to (4) with \(\xi \equiv \langle \varGamma \rangle \), \(\eta \equiv \langle \varGamma \rangle \), \(\zeta \equiv -4\varepsilon ^{-1} \langle \partial _t\varphi \rangle \).

3 Lyapunov stability

The autoresonant solution with asymptotics (3), \(\psi _0=\arcsin \nu \) is unstable. It follows from linear stability analysis: One of the eigenvalues has positive real part. However, the linearization fails to determine stability of the solution (3) with \(\psi _0=\pi -\arcsin \nu \):
$$\begin{aligned} \begin{array}{ll} \displaystyle &{}R_0(\tau )=\lambda \tau +f \sigma +{{\mathcal {O}}}(\tau ^{-1}), \\ &{}\varPsi _0(\tau )=\pi -\arcsin \nu -(\sigma \tau )^{-1}+{{\mathcal {O}}}(\tau ^{-2}), \\ &{}\sigma =\sqrt{1-\nu ^2}, \end{array} \end{aligned}$$
(6)
since the main asymptotic terms for both eigenvalues are purely imaginary. In this case, the stability property depends on high-order terms of equation (see [14]). We have

Theorem 1

If  \(0<\nu <1\) and \(f>0\), then the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) with asymptotics (6) is asymptotically stable.

Proof

By the change of variables
$$\begin{aligned} {\displaystyle r=R_0(\tau )+\sqrt{\lambda \tau }\, R, \quad \psi =\varPsi _0(\tau )+\varPsi } \end{aligned}$$
(7)
system (1) can be rewritten in the form
$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{1}{\sqrt{\lambda \tau }}\frac{\text {d} R}{\text {d}\tau }=-\partial _\varPsi H+F, \quad \frac{1}{\sqrt{\lambda \tau }} \frac{\text {d}\varPsi }{\text {d}\tau }=\partial _R H, \end{array} \end{aligned}$$
(8)
where
$$\begin{aligned} \begin{array}{lll} \displaystyle H(R,\varPsi ,\tau ) &{} = &{} \displaystyle \frac{R^2}{2}+ \left[ \frac{R_0(\tau )}{\lambda \tau } +\frac{f R}{\sqrt{\lambda \tau }}\right] \\ &{} &{}\displaystyle \times \, [\cos (\varPsi +\varPsi _0(\tau ))-\cos \varPsi _0(\tau )] \\ &{} &{} \displaystyle +\, R_0(\tau )\sin \varPsi _0(\tau ) \frac{\varPsi }{\lambda \tau }, \\ \displaystyle F(R,\varPsi ,\tau )&{}= &{} \displaystyle -\frac{R}{\sqrt{\lambda \tau }} [\nu +(f-1)\sin (\varPsi +\varPsi _0(\tau ))]\\ &{} &{} \displaystyle -\,\tau ^{-1} \frac{R}{2}. \end{array} \end{aligned}$$
For the new variables R, \(\varPsi \) we study stability of the equilibrium (0; 0) by Lyapunov’s second method. To construct a Lyapunov function for system (8), the asymptotics of the right-hand sides in a neighborhood of the equilibrium (as \(\rho \mathop {=}\limits ^{def}\sqrt{R^2+\varPsi ^2}\rightarrow 0\)) and at infinity (as \(\tau \rightarrow \infty \)) are used. Notice that all asymptotic estimates written out below in the form \({\mathcal {O}}(\rho ^s)\) and \({\mathcal {O}}(\tau ^{-s})\) (\(s={\hbox {const}}>0\)) are uniform with respect to \(R,\varPsi ,\tau \) in the domain
$$\begin{aligned} {\mathcal {D}}(\rho _*,\tau _*)\mathop {=}\limits ^{def}\{(R,\varPsi ,\tau )\in {\mathbb {R}}^3: \rho <\rho _*, \, \tau >\tau _*\}, \end{aligned}$$
\(\rho _*,\tau _*={\hbox {const}}>0\). It can easily be checked that the Hamiltonian has a positive quadratic form as the leading term of the asymptotic expansion:
$$\begin{aligned} H=\frac{R^2}{2}+\sigma \frac{\varPsi ^2}{2}+{\mathcal {O}}(\rho ^3)+{\mathcal {O}}(\rho ^2){\mathcal {O}}(\tau ^{-1/2}) \end{aligned}$$
as \(\rho \rightarrow 0\), \(\tau \rightarrow \infty \). By taking into account (6), one can readily write out the asymptotics of the derivatives:
$$\begin{aligned} \displaystyle \partial _R H= & {} \displaystyle R+ \frac{f}{\sqrt{\lambda \tau }} [\sigma (1-\cos \varPsi )-\nu \sin \varPsi ] \\&\displaystyle +\, {{\mathcal {O}}}(\rho ){{\mathcal {O}}}(\tau ^{-3/2}), \\ \displaystyle \partial _\varPsi H= & {} \displaystyle \sigma \,\sin \varPsi +\nu (1-\cos \varPsi ) \\&\displaystyle +\, \frac{R}{\sqrt{\lambda \tau }} [ \sigma \sin \varPsi +\nu (1-\cos \varPsi )]\\&+\,{\mathcal {O}}(\rho ){\mathcal {O}}(\tau ^{-1}), \\ \displaystyle \partial _\tau H= & {} \displaystyle {\mathcal {O}}(\rho ^2){\mathcal {O}}(\tau ^{-3/2}). \end{aligned}$$
The non-Hamiltonian part \(F(R,\varPsi ,\tau )\) tends to zero as \(\tau \rightarrow \infty \):
$$\begin{aligned} F=-m [R+{{\mathcal {O}}}(\rho ^2)] \tau ^{-1/2}+{{\mathcal {O}}}(\rho ){{\mathcal {O}}}(\tau ^{-1}), \end{aligned}$$
where \(m={\nu f}/{\sqrt{\lambda }}>0\). The Lyapunov function is constructed of the form
$$\begin{aligned} \displaystyle V(R,\varPsi ,\tau )=H(R,\varPsi ,\tau )+ \frac{m}{2} R \varPsi \, \tau ^{-1/2}. \end{aligned}$$
(9)
The derivative of the function \(V(R,\varPsi ,\tau )\) with respect to time along the trajectories of system (8) has a quadratic form in the leading term of its asymptotic expansion:
$$\begin{aligned} {\displaystyle \frac{1}{\sqrt{\lambda \tau }}\frac{\text {d}V}{\text {d}\tau }\Big |_{{}(8) }}= & {} {\displaystyle \frac{ \partial _\tau V}{\sqrt{\lambda \tau }}+\partial _ R V [-\partial _\varPsi H+ F] + \partial _\varPsi V \partial _R H } \\ {\displaystyle }= & {} \displaystyle -\frac{m}{2\sqrt{\tau }} \big [R^2+\sigma \,\varPsi ^2\big ]\\&\times [1+{{\mathcal {O}}}(\rho )+{{\mathcal {O}}}(\tau ^{-1})]. \end{aligned}$$
Note that the sum \({{\mathcal {O}}}(\rho )+{{\mathcal {O}}}(\tau ^{-1})\) can be made arbitrarily small by choosing a suitable domain \({\mathcal {D}}(\rho _*,\tau _*)\). It follows that there exist \(\rho _1>0\) and \(\tau _1>0\) such that the inequality
$$\begin{aligned} \frac{\text {d}V}{\text {d}\tau }\Big |_{{(8)}}\le -\frac{m\sqrt{\lambda }}{4}(R^2+\sigma \varPsi ^2) \end{aligned}$$
holds for all \((R,\varPsi ,\tau )\in {\mathcal {D}}(\rho _1,\tau _1)\). Similarly, there exists \(\rho _2>0\), \(\tau _2>0\) such that
$$\begin{aligned} {\displaystyle \frac{1}{4} (R^2+\sigma \varPsi ^2)\le V(R,\varPsi ,\tau )\le \frac{3}{4} (R^2+\sigma \varPsi ^2) } \end{aligned}$$
(10)
for all \((R,\varPsi ,\tau )\in {\mathcal {D}}(\rho _2,\tau _2)\). Thus,
$$\begin{aligned} {\displaystyle \frac{\text {d}V}{\text {d}\tau }\Big |_{{(8)}}\le -\frac{m\sqrt{\lambda }}{3} V, \quad \forall \,(R,\varPsi ,\tau )\in {\mathcal {D}}(\rho _0,\tau _0), } \end{aligned}$$
(11)
where \(\rho _0=\min \{\rho _1,\rho _2\}\) and \(\tau _0=\max \{\tau _1,\tau _2\}\). Let \(\epsilon \) be an arbitrary positive constant such that \(0<\epsilon <\rho _0\); then
$$\begin{aligned}&\displaystyle \sup _{\rho \le \delta , \tau>\tau _0} V(R,\varPsi ,\tau ) \le \frac{3\delta ^2}{4} < \frac{\sigma \epsilon ^2}{4}\\&\quad \le \inf _{\rho =\epsilon , \tau >\tau _0} V(R,\varPsi ,\tau ), \end{aligned}$$
\(\delta \mathop {=}\limits ^{def}\epsilon \sqrt{\sigma /6}<\epsilon \). Hence, any solution \(R(\tau )\), \(\varPsi (\tau )\) of system (8) with initial data \([R^2(\tau _0)+\varPsi ^2(\tau _0)]^{1/2}\le \delta \) cannot leave \(\epsilon \)-neighborhood of the equilibrium (0; 0) as \(\tau >\tau _0\): \([R^2(\tau )+\varPsi ^2(\tau )]^{1/2}<\epsilon \).
Integrating (11) with respect to \(\tau \), we obtain
$$\begin{aligned} 0\le V(R,\varPsi ,\tau )\le C \text {e}^{-\nu f \tau /3}, \end{aligned}$$
for all \((R,\varPsi ,\tau )\in {\mathcal {D}}(\rho _0,\tau _0)\). The constant \(C>0\) depends on \(R(\tau _0),\varPsi (\tau _0)\). Therefore, the Lyapunov function \(V(R,\varPsi ,\tau )\) tends exponentially to zero along the trajectories of system (8). Combining this with (7) and (10), we obtain asymptotic estimates for the solutions of system (1) with initial data from a neighborhood of \(R_0(\tau _0)\), \(\varPsi _0(\tau _0)\):
$$\begin{aligned} \begin{array}{ll} \displaystyle r(\tau )&{}=R_0(\tau )+{{\mathcal {O}}}(\tau ^{1/2}\text {e}^{-\nu f \tau /6}), \\ \displaystyle \psi (\tau )&{}=\varPsi _0(\tau )+{{\mathcal {O}}}(\text {e}^{-\nu f \tau /6}), \end{array} \end{aligned}$$
(12)
as \(\tau >\tau _0\). This completes the proof. \(\square \)

The sufficient conditions obtained in Theorem 1 are almost necessary; this can be seen from the following theorem.

Theorem 2

If \(0<\nu <1\) and \(f<0\), then the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) with asymptotics (6) is unstable.

Proof

We use \(V(R,\varPsi ,\tau )\), defined by (9), as a Chetaev function candidate. Since \(f<0\), we have
$$\begin{aligned} \frac{\text {d}V}{\text {d}\tau }\Big |_{{(8)}}\ge -\frac{\nu f}{4}(R^2+\sigma \varPsi ^2)>0, \end{aligned}$$
for all \((R,\varPsi ,\tau )\in {\mathcal {D}}(\rho _0,\tau _0)\). It follows from the Chetaev instability theorem (see [17]) that the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) is unstable. \(\square \)

4 Stability under random perturbations

The previous section shows that the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) of Eq. (1) is locally asymptotically stable if \(f>0\). In this section, we study stability of this solution with respect to persistent random perturbations (4). Our goal is to find out what kind of perturbations \((\xi ,\eta ,\zeta )\) preserve stability of the autoresonant solution on a finite but asymptotically long time interval \((0,\mathcal T_\mu )\), \(\mathcal T_\mu \rightarrow \infty \) as \(\mu \rightarrow 0\). The restriction of the time interval is motivated by the fact that mathematical model (1) is valid only as \( \tau \in [0,\tau _\varepsilon ]\), where \(\tau _\varepsilon \rightarrow \infty \) as \(\varepsilon \rightarrow 0\) [\(0<\varepsilon \ll 1\) is the driving parameter in (2)]. In order to study the parametric autoresonance phenomenon for \(\tau >\tau _\varepsilon \), another mathematical model should be considered (see [6]).

Now we shall give the following concept of stability in probability (see [18, Ch. 9]).

Definition 1

The solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) to system (1) is said to be stable in probability with respect to persistent random perturbations on the time interval \((0, \mathcal T_\mu )\) uniformly for \((\xi ,\eta ,\zeta )\in \mathcal {R}\), if \(\forall \, \epsilon , \upsilon>0 \ \ \exists \, \delta ,\varDelta >0 \mathrm{:}\) \(\forall \,\varrho _0, \phi _0: \) \(|\varrho _0-R_0(0)|+|\phi _0-\varPsi _0(0)|\le \delta \), \(\mu <\varDelta \), and \((\xi ,\eta ,\zeta )\in \mathcal R\) the solution \(r_\mu (\tau ,\omega )\), \(\psi _\mu (\tau ,\omega )\) to system (4) with initial data \(r_\mu (0,\omega )=\varrho _0\), \(\psi _\mu (0,\omega )=\phi _0\) satisfies the inequality \( {\mathbb {P}}\big (\sup _{0<\tau <\mathcal T_\mu }\big \{|r_\mu (\tau ,\omega )-R_0(\tau )| \tau ^{-1/2} + |\psi _\mu (\tau ,\omega )-\varPsi _0(\tau )|\big \}\ge \epsilon \big )\le \upsilon . \)

It should be noted that Definition 1 weakens a classical concept of stability due to the presence of the factor \(\tau ^{-1/2}\) in the inequality for the amplitude. This factor allows us to take into account the character of stability in the unperturbed system (see (12)) and to set the admissible rate of growth for the perturbed solutions. So, we say that the autoresonant solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) is stable in probability under persistent perturbations, if the trajectory of the stochastic process \(r_\mu (\tau ,\omega )\), \(\psi _\mu (\tau ,\omega )\) with initial values sufficiently close to \(R_0(0), \varPsi _0(0)\) does not leave the expanding neighborhood of the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) with probability tending to one.

The influence of stochastic perturbations on dynamical systems has attracted the attention of many authors (see, for instance, [19, 20, 21]). The systematic exposition of stability theory for randomly perturbed differential equations is presented in [22]. It was shown in [22, §1.6] that a stochastic perturbation with uniformly bounded expectation preserve stability of asymptotically stable equilibrium in a dissipative system.1 However, these results do not apply here, since the unperturbed system has solutions of two types, bounded and unbounded at infinity (see Fig. 1 and [6]). The stability of non-dissipative systems under persistent random perturbations was discussed in the papers [23, 24]. Here we extend these results to study the stability on a finite time interval.
Fig. 1

Evolution of the amplitude \(r(\tau )\) for solutions of (1) with different initial data, \(f = 0.2\), \(\nu =0.2\), and \(\lambda = 1\)

For all \((a,b,c)\in {\mathbb {R}}^3\) define a class \(\mathcal {R}_{a,b,c}\) of random processes \((\xi ,\eta ,\zeta )\) such that \(\forall \, (\xi ,\eta ,\zeta )\in \mathcal {R}_{a,b,c}\) there exists at least one random process \(S(\tau ,\omega )\) satisfying the following conditions:
  • \(\displaystyle \sup _{(r,\psi )\in {\mathbb {R}}^2} \big (|\xi |\tau ^{-a}+|\eta |\tau ^{-b}+|\zeta |\tau ^{-c}\big )<S(\tau ,\omega )\), for all \(\tau >0, \, \omega \in \varOmega \);

  • \(\displaystyle \exists \, S_0(\omega )>0, \, T>0: \int \limits _{\tau }^{\tau +T} S(t,\omega )\, \text {d}t\le S_0(\omega )\), for all \(\tau >0,\, \omega \in \varOmega \);

  • \(\displaystyle \mathbb { E} [ S_0(\omega )]\mathop {=}\limits ^{def}\int \limits _{\varOmega } S_0(\omega ) {\mathbb {P}}\, (\text {d}\omega )<\infty .\)

Denote by \(\mathcal {R}_{a,b,c}^h\) the subset of \(\mathcal {R}_{a,b,c}\) such that
$$\begin{aligned} {\mathbb {E}} [S_0(\omega )]\le h. \end{aligned}$$
We also impose some additional restrictions on the functions \((\xi ,\eta ,\zeta )\in \mathcal {R}_{a,b,c}\) in order to ensure the global existence of solutions to perturbed Eq. (4), see, for instance [22, p. 5].

Let \(\varLambda \) denote the set of triples \((a,b,c)\in {\mathbb {R}}^3\) such that \(q\mathop {=}\limits ^{def}\max \{a+1/2,b,c\}>0\). Then we have

Theorem 3

If \(0<\nu <1\) and \(f>0\), then \(\forall \, h>0\), \((a,b,c)\in \varLambda \), and \(0<\varkappa <1/q\) the asymptotically stable solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) with asymptotics (6) is stable in probability with respect to persistent random perturbations on the interval \(0<\tau < {{\mathcal {O}}}(\mu ^{-\varkappa })\) uniformly by \((\xi ,\eta ,\zeta )\in \mathcal {R}^h_{a,b,c}\).

Proof

The change of variables (7) transforms (4) into the form
$$\begin{aligned} \begin{array}{ll} &{}\displaystyle \frac{1}{\sqrt{\lambda \tau }} \frac{\text {d}R}{\text {d}\tau } = -\partial _\varPsi H +F+\mu G, \\ &{}\displaystyle \frac{1}{\sqrt{\lambda \tau }} \frac{\text {d}\varPsi }{\text {d}\tau } = \partial _R H+ \mu Q, \end{array} \end{aligned}$$
(13)
where random functions \(G(R,\varPsi ,\tau ,\omega )\) and \(Q(R,\varPsi ,\tau ,\omega )\) are defined as follows:
$$\begin{aligned} \begin{array}{lll} G &{} = &{} \displaystyle (R_0(\tau )+R\sqrt{\lambda \tau })\sin (\varPsi +\varPsi _0(\tau ))\frac{ \xi }{\lambda \tau }, \\ Q &{} = &{} \displaystyle (f \eta \cos (\varPsi +\varPsi _0(\tau ))+ \zeta )\frac{1}{\sqrt{\lambda \tau }}. \end{array} \end{aligned}$$
In the new variables \((R,\varPsi )\), we study the stability of equilibrium point (0, 0) of system (8) with respect to random perturbations GQ.
Let \(\epsilon >0\), \(\upsilon >0\), \(h>0\), \((a,b,c)\in \varLambda \) be arbitrary constants. A Lyapunov function candidate for Eq. (13) is constructed of the form
$$\begin{aligned} U(R,\varPsi ,\tau ,\omega )=V(R,\varPsi ,\tau ) \exp \varPhi (\tau ,\omega ), \end{aligned}$$
where \(V(R,\varPsi ,\tau )\) is defined by (9), and a smooth (in \(\tau \)) function \(\varPhi (\tau ,\omega )\) is defined below. The derivative of \(U(R,\varPsi ,\tau ,\omega )\) with respect to \(\tau \) along the trajectories of (13) has the form:
$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\text {d}U}{\text {d}\tau }\Big |_{{(13)}} &{} = &{} \displaystyle \partial _\tau \varPhi \, U \left[ \frac{\text {d}V}{\text {d}\tau }\Big |_{{(8)}}\right. \\ &{} &{} \left. +\, \displaystyle \mu \sqrt{\lambda \tau }(G\, \partial _R V +Q\,\partial _\varPsi V)\right] \exp \varPhi . \end{array} \end{aligned}$$
From the definition of the class \(\mathcal R^h_{a,b,c}\) it follows that
$$\begin{aligned} |G|+|Q|\le M \tau ^{q-1/2} S(\tau ,\omega ), \quad \forall \, (R,\varPsi ,\tau )\in {\mathcal {D}}(\rho _0,\tau _0), \end{aligned}$$
where \(M={\hbox {const}}>0\). The partial derivatives \(\partial _R V\), \(\partial _\varPsi V\) satisfy the inequalities: \(|\partial _R V|,|\partial _\varPsi V|\le \ell V/\rho \) in the domain \({\mathcal {D}}(\rho _0,\tau _0)\), \(\ell ={\hbox {const}}>0\). Hence, in view of (10) and (11) the derivative of \(U(R,\varPsi ,\tau ,\omega )\) satisfies the estimate:
$$\begin{aligned} \frac{\text {d}U}{\text {d}\tau }\Big |_{{(13)}}\le \partial _\tau \varPhi \, U -\frac{m\sqrt{\lambda }}{3}\left( 1-\mu \tau ^{q} \frac{3 \ell M }{m \delta } S\right) U \end{aligned}$$
in the domain \(\delta<\rho <\epsilon \), \(\tau >\tau _0\). For any \(\varkappa \) such that \(0<\varkappa <1/q\), we define
$$\begin{aligned}&\varDelta _1(\omega )=\left[ \frac{\varDelta _0}{S_0(\omega )}\right] ^z,\quad \varDelta _0=\frac{m \delta }{3\ell M (2\tau _0)^q },\\&\quad z=\frac{1}{1-\varkappa q}. \end{aligned}$$
Then, for any \(\mu \le \varDelta _1(\omega )\) the inequality
$$\begin{aligned} \frac{\text {d}U}{\text {d}\tau }\Big |_{{(13)}}\le \partial _\tau \varPhi \, U - \frac{m\sqrt{\lambda }}{3 S_0}(S_0-S) U \end{aligned}$$
holds in the domain \(\delta<\rho <\epsilon \), \(0<\tau -\tau _0<\tau _0 \mu ^{-\varkappa }\). For a fixed \(\omega \in \varOmega \), we consider the integral
$$\begin{aligned} I(k,\omega ){=}\int \limits _{kT}^{(k+1)T}[S_0(\omega ){-}S(t,\omega )]\, \text {d}t, \quad k=0,1,2,\dots . \end{aligned}$$
From the definition of the class \(\mathcal R_{a,b,c}^h\), it follows that \(I(k,\omega )\ge 0\) for any \(k\ge 0\) and \(\omega \in \varOmega \). Define an auxiliary random process \(\theta (\tau ,\omega )\) such that
$$\begin{aligned} {\displaystyle \int \limits _{kT}^{(k+1)T}\theta (t,\omega )\text {d}t=I(k,\omega ), \quad k=0,1,2,\dots . } \end{aligned}$$
(14)
Since right-hand side of (14) is not negative, it follows that there exists a nonnegative function \(\theta (\tau ,\omega )\) satisfying (14) for all integers \(k\ge 0\). Without loss of generality, we can assume that for all \(\omega \in \varOmega \) the function \(\theta (\tau ,\omega )\) is continuous and \(\theta (kT,\omega )=0\) for \(k=0,1,2,\dots \). Let us define \(\varPhi (\tau ,\omega )\) as follows:
$$\begin{aligned} \varPhi (\tau ,\omega )\mathop {=}\limits ^{def}\frac{m \sqrt{\lambda }}{3 S_0(\omega )} \int \limits _{0}^{\tau } [S_0(\omega )-S(t,\omega )-\theta (t,\omega ) ]\, \text {d}t. \end{aligned}$$
Then,
$$\begin{aligned} \frac{\text {d}U}{\text {d}\tau }\Big |_{{(13)}}\le - \frac{m\sqrt{\lambda }}{3 S_0 } \theta U\le 0 \end{aligned}$$
for all \((R,\varPsi ,\tau )\) such that \(\delta<\rho <\epsilon \), \(0<\tau -\tau _0<\tau _0 \mu ^{-\varkappa }\). Taking into account the properties of the functions \(\theta (\tau , \omega )\) and \(S(\tau , \omega )\), we obtain \(|\varPhi (\tau , \omega )|\le \varPhi _0\), \(\varPhi _0=4m\sqrt{\lambda }/3\). Thus, for any \(\epsilon >0\), \(\omega \in \varOmega \), and \(\mu >0\) we have
$$\begin{aligned} \begin{array}{lll} \displaystyle \sup _{\rho \le \delta , \tau>\tau _0} U(R,\varPsi ,\tau ,\omega ) &{} \le &{} \displaystyle \frac{3\delta ^2}{4} \exp \varPhi _0\\ &{}<&{} \displaystyle \frac{\sigma \epsilon ^2}{4} \exp (-\varPhi _0) \\ &{} \le &{} \displaystyle \inf _{\rho =\epsilon , \tau >\tau _0} U(R,\varPsi ,\tau ,\omega ), \end{array} \end{aligned}$$
where \(\delta \mathop {=}\limits ^{def}\epsilon \, \exp (-\varPhi _0) \sqrt{\sigma /6}<\epsilon \).

Consider perturbations \((\xi ,\eta ,\zeta )\) from the class \(\mathcal R_{a,b,c}^h\) such that \(S_0(\omega )< h/\upsilon \) uniformly for all \(\omega \in \varOmega \). Then the parameter \(\mu \) can be bounded away from zero \(0<\mu<\varDelta \mathop {=}\limits ^{def}(\upsilon \varDelta _0/h)^z < \varDelta _1(\omega )\). In this case, any solution \(R_\mu (\tau ,\omega )\), \(\varPsi _\mu (\tau ,\omega )\) starting from the neighborhood \(\rho \le \delta \) of the equilibrium (0; 0) remains inside the ball: \([R_\mu ^2(\tau ,\omega )+\varPsi _\mu ^2(\tau ,\omega )]^{1/2}<\epsilon \) as \(0<\tau -\tau _0< {{\mathcal {O}}}(\mu ^{-\varkappa })\).

Let us consider perturbations \((\xi ,\eta ,\zeta )\in \mathcal R_{a,b,c}^h\) such that \(S_0(\omega )\ge h/\upsilon \). In this case, the solution of (13) starting from the ball \(\rho \le \delta \) can leave \(\epsilon \)-neighborhood of the equilibrium (0; 0). However, the probability of such outcomes is smaller than any fixed positive constant. Indeed,
$$\begin{aligned} \begin{array}{ll} &{}\displaystyle {\mathbb {P}}\left( \sup _{0<\tau -\tau _0< \mathcal T_\mu } [R_\mu ^2(\tau ,\omega )+\varPsi _\mu ^2(\tau ,\omega )]^{1/2}\ge \epsilon \right) \\ &{}\quad \displaystyle \le {\mathbb {P}}(S_0(\omega )\ge h/\upsilon )\le \frac{{\mathbb {E}}[ S_0(\omega )]}{h/\upsilon }\le \upsilon , \end{array} \end{aligned}$$
\(\mathcal T_\mu =\tau _0\mu ^{-\varkappa }\). The last estimate follows from the Chebyshev inequality.

The change-of-variables formula (7) implies the stability of the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) to system (1) with respect to random perturbations on the asymptotically long time interval \(0<\tau -\tau _0 < {{\mathcal {O}}}(\mu ^{-\varkappa })\). Stability on the interval \((0,\tau _0]\) follows from the continuity of solutions with respect to parameters (see, for instance, [19, Ch. 2]). This concludes the proof. \(\square \)

Theorem 4

If  \(0<\nu <1\) and \(f>0\), then \(\forall \, h>0\), \((a,b,c)\in {\mathbb {R}}^3\backslash \varLambda \) the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) with asymptotics (6) is stable in probability with respect to persistent random perturbations on the semi-infinite time interval \(\tau >0\) uniformly by \((\xi ,\eta ,\zeta )\in \mathcal {R}^h_{a,b,c}\).

Proof

The proof is similar to the proof of Theorem 3 with \(q=0\). \(\square \)

Note that the conditions, appeared in the definition of the class \(\mathcal R_{a,b,c}\), are sufficient (but not necessary) for stability of the autoresonance solution to Eq. (1). However, the weakening of these restrictions can lead to loss of stability. We illustrate this statement with a simple example.

Consider the following non-dissipative differential equation: \(\text {d}y/\text {d}\tau =-y+y^2\). It has two fixed points: The solution \(y(\tau )\equiv 0\) is exponentially stable and \(y(\tau )\equiv 1\) is unstable.2 It is easy to see that every trajectory with \(y(\tau _0)>1\) goes to infinity in finite time. The perturbed equation is considered in the form \(\text {d}y/\text {d}\tau =-y+y^2+\mu \xi _\mu \), where \(\xi _\mu (\tau ,\omega )=\mu ^{-2} h(\omega )\chi _{[0; \mu ]}(\tau -s(\omega )/\mu )\), \(0<\mu \ll 1\). Here \(\chi _{[0;\mu ]}(\cdot )\) is the indicator function, \(h(\omega )\) and \(s(\omega ): \varOmega \rightarrow {\mathbb {R}}\) are independent continuous random variables such that \(h(\omega )\ge 2\), \({\mathbb {E}}[h(\omega )]<\infty \), and the density function \(p_s(x)\) of random variable \(s(\omega )\) is assumed to be uniformly bounded. The perturbation \(\xi _\mu (\tau ,\omega )\) does not have the properties of the class \(\mathcal R_{a,b,c}\) uniformly in \(\mu \), since
$$\begin{aligned} {\mathbb {E}}\left[ \sup _{\tau >0}\int \limits _{\tau }^{\tau +1} \xi _\mu (t,\omega ) \text {d}t\right] = \mu ^{-1} {\mathbb {E}}[ h(\omega )] \rightarrow \infty \end{aligned}$$
as \(\mu \rightarrow 0\). However, this function satisfies a weaker condition:
$$\begin{aligned} \begin{array}{ll} \displaystyle {\mathbb {E}} [\xi _\mu (\tau ,\omega )]&{}= \frac{{\mathbb {E}}[h(\omega )]}{ \mu ^{2}}\int \limits _{\mu \tau -\mu ^2}^{\mu \tau } p_s(x)\,\text {d}x \\ \displaystyle &{}\le {\mathbb {E}}[h(\omega )] \sup _{-\mu ^2\le x-\mu \tau \le 0} p_s(x)<\infty \end{array} \end{aligned}$$
for all \(\tau >0\) and \(\mu >0\). Note that the condition \({\mathbb {E}}[\xi _\mu (\tau ,\omega )]<\infty \) is sufficient to preserve the stability of solution in a dissipative system (see [22, §1.6]). Nevertheless, it follows from Grönwall’s lemma that \(y_\mu (\tau ,\omega )\rightarrow \infty \) almost surely if \(y_\mu (0,\omega )\ge 0\). Indeed, \(y_\mu (\tau ,\omega )\ge h(\omega ) (1-\mu /2)>1\) as \(\tau \ge \mu +s(\omega )/\mu \). Hence, any solution \(y_\mu (\tau ,\omega )\) of the perturbed equation with initial data \(y_\mu (0,\omega )\ge 0\) leaves any bounded neighborhood of the stable equilibrium and goes to infinity.

5 Examples of admissible perturbations

Let \(\xi (\tau ,\omega )\equiv \tau ^{a} J(\tau ,\omega )\), \(\eta (\tau ,\omega )\equiv \tau ^{b} J(\tau ,\omega )\), \(\zeta (\tau ,\omega )\equiv \tau ^{c} J(\tau ,\omega )\).

Example 1

Define the function \(J(\tau ,\omega )\equiv j(\omega )\), where \(j: \varOmega \rightarrow {\mathbb {R}}\) is a real-valued random variable with a finite expectation \({\mathbb {E}} [| j(\omega ) |]<\infty \). It can easily be checked that \((\xi ,\eta ,\zeta )\in \mathcal R_{a,b,c}\).

Example 2

Define the random process
$$\begin{aligned} J(\tau ,\omega )=\sum \limits _{n=1}^{\infty }\frac{j_n(\omega )}{\mu } \chi _{[ \tau _n; \tau _n +\mu ]}(\tau ), \quad 0<\mu \ll 1, \end{aligned}$$
where \(\chi _{[ \tau _n; \tau _n +\mu ]}(\tau )\) is the indicator function, \(0<\tau _1<\tau _2<\dots<\tau _k<\dots \) are random variables such that \(\tau _{k+1}-\tau _k\ge 1\), and for all \(n>0\) the function \(j_n: \varOmega \rightarrow {\mathbb {R}}\) is a real-valued random variable. Assume that there exists a random variable \(j(\omega )>0\) such that \({\mathbb {E}}[j(\omega )]<\infty \) and \(|j_n(\omega )|<j(\omega )\) for all \(n>0, \omega \in \varOmega \). Since
$$\begin{aligned} \int \limits _\tau ^{\tau +1} | J(t,\omega ) | \text {d}t \le j(\omega ), \quad \forall \,\tau \ge 0,\mu >0 \end{aligned}$$
it follows that \((\xi ,\eta ,\zeta )\in \mathcal R_{a,b,c}\).

Example 3

Consider the jump process
$$\begin{aligned} J(\tau ,\omega )=\sum _{n=1}^\infty j_n(\omega ) \chi _{[\tau _n; \infty )} (\tau ), \end{aligned}$$
where \(0<\tau _1<\tau _2<\dots<\tau _k<\dots \) are random variables, \(j_n: \varOmega \rightarrow {\mathbb {R}}\) such that \({\mathbb {E}} [ | j_n(\omega ) |]\le 1/n^2\). It can readily be seen that
$$\begin{aligned} \int \limits _{\tau }^{\tau +T}| J(t,\omega ) | \text {d}t \le S_0(\omega )\mathop {=}\limits ^{def} T \sum _{n=1}^\infty |j_n(\omega )|,\quad \forall \,\tau \ge 0, \end{aligned}$$
and \({\mathbb {E}}[S_0(\omega )]\le T \pi ^2/6\). Hence, \((\xi ,\eta ,\zeta )\in \mathcal R_{a,b,c}\).
Fig. 2

Realization of \(\mu \xi \) (thick black curve), \(\mu \eta \) (thin black curve), and \(\mu \zeta \) (gray curve), for \(\mu =10^{-2}\), \(h_1=100\), and \(h_2=4\)

6 Numerical simulations

In order to illustrate our results, we solve the perturbed system (4) numerically with random functions \((\xi , \eta , \zeta )\in \mathcal R_{0,0,0}\). We take \(\xi \equiv J^1\), \(\eta \equiv J^2\), \(\zeta \equiv J^3\), where
$$\begin{aligned} J^k(\tau ,\omega )=\mu ^{-1}\sum \limits _{n=1}^{h_1} j^k_n(\omega )\chi _{[n,n+\mu ]}(\tau ). \end{aligned}$$
Here \(j^k_n(\omega )\equiv h_2 W^k_{n}(\omega )/\sqrt{n}\), \((W^1_t,W^2_t,W^3_t)\), \(t\ge 0\) are independent Wiener processes, \(h_1\in \mathbb N \), \(h_2>0\). It can easily be checked that \({\mathbb {E}}[j^k_n(\omega )]=0\), \({\mathbb {E}}[j^k_n(\omega )]^2=h_2^2\), and \({\mathbb {E}}[|j^k_n(\omega )|]=h_2(2/\pi )^{1/2}\), for all \(n>0\), \(k=1,2,3\). Note that \((\xi ,\eta ,\zeta )\in \mathcal R_{0,0,0}\). Indeed,
$$\begin{aligned} \int \limits _{\tau }^{\tau +1} |J^k(t,\omega )|\, \text {d}t\le S_0(\omega )\mathop {=}\limits ^{def} 2\sum _{k=1}^{3} \sum _{n=1}^{h_1} |j^k_n(\omega )|, \end{aligned}$$
\(\forall \, \tau >0\), and \({\mathbb {E}}[S_0(\omega )]= 3 h_1 h_2 (8/\pi )^{1/2}\).
We use particular realizations of the random processes \(\xi \), \(\eta \), \(\zeta \) (see Fig. 2) in the calculations. The numerical simulations (see Fig. 3) indicate that the stability of autoresonant solution is preserved at least over the interval \(0\le \tau \le {{\mathcal {O}}}(\mu ^{-1})\). This observation is consistent with Theorem 3.
Fig. 3

Sample path of the amplitude \(r_\mu (\tau ,\omega )\) for solution of (4) with \(f=1\), \(\nu =1/2\), and \(\lambda =1\)

7 Conclusion

The proposed theory for system (1) can be used in study of autoresonance phenomena in various nonlinear systems. In particular, the following statement is true.

Theorem 5

If \(0<2\beta <\varepsilon \), \(0<\alpha \le {{\mathcal {O}}}(\varepsilon ^2)\), then there exists a stable autoresonant solution \(x_{0}(t)\) to Eq. (2) with initial data \(|x_{0}(0)|+|\dot{x}_{0}(0)|\le {{\mathcal {O}}}(\varepsilon ^{1/2})\) as \(\varepsilon \rightarrow 0\). The solution has the asymptotic behavior \(x_{0}(t)= \sqrt{2 \varepsilon R_0(\varepsilon t/2)/3\gamma }\sin \left( \phi (t)-\varPsi _0(\varepsilon t/2)/2\right) +{{\mathcal {O}}}(\varepsilon )\) as \(\varepsilon \rightarrow 0\), \(0\le t\ll \varepsilon ^{-2}\), where \(R_0(\tau )=\lambda \tau +{{\mathcal {O}}}(1)\), \(\varPsi _0(\tau )=\pi -\arcsin (2\beta /\varepsilon )+{{\mathcal {O}}}(\tau ^{-1})\), \(\lambda =16\alpha \varepsilon ^{-2}\) as \(\tau \rightarrow \infty \). If, in addition, \(q=\max \{a+1/2,c\}>0\), \(\varGamma \equiv \xi (\tau ,\omega )\), \(\varphi \equiv \varepsilon t \zeta (\tau ,\omega )\), where \(\xi (\tau ,\omega )\), \(\zeta (\tau ,\omega )\) are continuous stochastic processes such that \((\xi ,\xi ,\zeta )\in \mathcal R_{a,a,c}\), then for all \(\mu >0\) there exists a stable solution \(x_{\mu }(t)\) to Eq. (5) such that \(x_{\mu }(t) = \sqrt{ {2 \varepsilon } r_\mu (\varepsilon t/2,\omega )/{3\gamma }} \sin (\phi (t) + \mu \varphi /2 - \psi _\mu (\varepsilon t/2,\omega )/2)+{{\mathcal {O}}}(\varepsilon )\) as \(\varepsilon \rightarrow 0\), \(0\le t\ll \varepsilon ^{-2}\), where \(|r_\mu (\tau ,\omega )-R_0(\tau )|\tau ^{-1/2}+|\psi _\mu (\tau ,\omega )-\varPsi _0(\tau )|\le o(1)\) as \(\mu \rightarrow 0\), \(0<\tau < {{\mathcal {O}}}(\mu ^{-\varkappa })\), \(0<\varkappa <1/q\), with probability tending to one.

In summary, we have investigated the stability problem of the capture into the parametric autoresonance in nonlinear oscillatory systems with a dissipation. We have considered the particular solutions with power asymptotics at infinity. It was shown that the solution \(R_0(\tau )\), \(\varPsi _0(\tau )\) is locally exponentially stable when \(f>0\). We have also described the class of random perturbations preserving the stability of the solution on asymptotically long time interval. The length of the stability interval is determined by the reciprocal quantity of the perturbation parameter \(0<\mu \ll 1\). The suggested method is based on the construction of the Lyapunov function for the perturbed system. White noise perturbation of parametric autoresonance has not been considered here. This will be discussed in further papers.

Footnotes

  1. 1.

    The system \(\dot{y}=Y(y,t)\) is said to be dissipative for \(t>0\) if there exists a positive number \(D>0\) such that for each \(d>0\), beginning from some time \(T(d,t_0)\ge t_0\), the solution \(y(t,y_0,t_0)\) to initial value problem \(y|_{t=t_0}=y_0\), \(|y_0|<d\), \(t_0>0\), lies in the domain \(\{y: |y|<D\}\). See [22, §1.2].

  2. 2.

    Compare it with system (1), where the solution \(R(\tau )\), \(\varPsi (\tau )\) with asymptotics (3), \(\psi _0 = \pi -\arcsin \nu \) is asymptotically stable and the solution with \(\psi _0=\arcsin \nu \) is unstable.

Notes

Acknowledgements

The publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement Number 02.A03.21.0008).

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, Ufa Scientific CenterRussian Academy of SciencesUfaRussia
  2. 2.Peoples Friendship University of Russia (RUDN University)MoscowRussia

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