Random perturbations of parametric autoresonance
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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 analysis1 Introduction
Autoresonance is a persistent phaselocking 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
2.1 Autoresonant solutions
2.2 Perturbed equations
3 Lyapunov stability
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
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
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 \(\forall \, \epsilon , \upsilon>0 \ \ \exists \, \delta ,\varDelta >0 \mathrm{:}\) \(\forall \,\varrho _0, \phi _0: \) \(\varrho _0R_0(0)+\phi _0\varPsi _0(0)\le \delta \), \(\mu <\varDelta \), 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.

\(\displaystyle \sup _{(r,\psi )\in {\mathbb {R}}^2} \big (\xi \tau ^{a}+\eta \tau ^{b}+\zeta \tau ^{c}\big )<S(\tau ,\omega )\), 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 \(\tau >0,\, \omega \in \varOmega \);

\(\displaystyle \mathbb { E} [ S_0(\omega )]\mathop {=}\limits ^{def}\int \limits _{\varOmega } S_0(\omega ) {\mathbb {P}}\, (\text {d}\omega )<\infty .\)
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
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 \(0<\tau \tau _0< {{\mathcal {O}}}(\mu ^{\varkappa })\).
The changeofvariables 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 semiinfinite 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.
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 realvalued 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
Example 3
6 Numerical simulations
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.
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.
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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