# Asymptotic analysis of the autoresonance capture in oscillating systems with combined excitation

Abstract. A mathematical model of autoresonance in nonlinear systems with combined parametric and external chirped frequency excitation is considered. Solutions with a growing amplitude and a bounded phase mismatch are associated with the autoresonant capture. By applying Lyapunov function method we investigate the conditions for the existence and stability of autoresonant modes and construct long-term asymptotics for stable solutions. In particular, we show that unstable regimes become stable when the system parameters pass through certain threshold values.

Keywords: nonlinear oscillations, autoresonance, stability, asymptotics, averaging method, Lyapunov function

Mathematics Subject Classification: 34C15, 34D05, 37B25, 37B55, 93D20

## Introduction

Autoresonance is a phenomenon of persistent phase synchronization between an oscillatory nonlinear system and a small resonant chirped frequency perturbation that leads to a significant increase in the amplitude of the oscillator [1]. Autoresonance was fist realized in relativistic particles accelerators in the middle of the twentieth century, and nowadays, it is considered as a universal phenomenon that occurs in a wide range of physical systems [2, 3, 4, 5, 6]. In recent times, the mathematical models of autoresonance are actively studied numerically and analytically (see [7], and references therein). However, up to now the autoresonance in nonlinear systems with parametric [8, 9] and external [10] driving has been treated separately. To the best of our knowledge, the effect of the combined excitation on the capture into autoresonance has not previously been discussed. In this paper, parametric and external perturbations are considered together, and all possible autoresonant modes are described with a help of the Lyapunov function technique.

The paper is organized as follows. In section 1, the mathematical formulation of the problem is given. In section 2 particular isolated autoresonant solutions are constructed. Section 3 provides linear and nonlinear stability analysis of these solutions. Asymptotic analysis of general autoresonant solutions is contained in section 4. The paper concludes with a brief discussion of the results obtained.

## 1. Problem statement

Consider the system of two differential equations:

(1) |

with the parameters , and a smooth bounded given function . This model system arises in the study of the autoresonance phenomena in a wide class of nonlinear oscillators with a time-periodic slowly-varying perturbation and describes the principal terms of the asymptotic behaviour of the oscillators at the initial step of the capture. We assume that the function , corresponding to the amplitude of a parametric driving, has a power-law asymptotics:

The unknown functions and play the role of the amplitude and the phase mismatch, respectively. The solutions with and as are associated with the phase-locking [11] and the autoresonance [1]. In addition to autoresonant modes, system (1) also has solutions with a limited amplitude and an increasing phase mismatch. Such solutions correspond to the phase-slipping phenomenon and are not investigated in the framework of the present paper.

Note that (1) with corresponds to the case of a pure external excitation, see [12, 10]. If , the action of the external driving becomes insignificant and system (1) reduces to a weak perturbation of the model of parametric autoresonance [8, 13]. The combined excitation takes effect only in the case when .

In this paper, we investigate the conditions for the existence and stability of autoresonant solutions to system (1) and construct long-term asymptotics for such solutions. In the first step, particular autoresonant solutions with special power-law asymptotics at infinity are constructed and their Lyapunov stability is investigated. Linear stability analysis allows us to identify only the conditions that guarantee the instability of solutions; the stability at this level cannot be determined due to certain features of the equations. To solve the stability problem, it is necessary to carry out a careful nonlinear analysis with the Lyapunov function method. The presence of stability will ensure the existence of a two-parameter family of autoresonant solutions. For such solutions, the asymptotics at infinity will be constructed at the last step.

As but one example we consider Duffing’s oscillator with small combined parametric and external excitation:

(2) |

where , , , , , . It is easy to see that equation (2) without perturbation () has stable fixed point and periodic solutions. Consider the solutions of the perturbed equation with small enough initial values . Numerical analysis shows that for some initial data there are solutions whose the energy significantly increases with time and the phase is synchronised with the pumping: . Such solutions are associated with the capture into the autoresonance. For non-captured solutions, the energy remains small and the phase mismatch increases: , see. Fig. 1. For the description of a long-term evolution of solutions to equation (2), the method of two scales can be used. We introduce a slow time , () and a fast variable , then the asymptotic substitution in equation (2) and the averaging over the fast variable in the leading-order term in lead to system (1) for the slowly varying functions and with and . Similarly, system (1) appears in many other problems of nonlinear physics related to autoresonance.

## 2. Particular autoresonant solutions

The simplest asymptotic expansion for autoresonant solutions is constructed in the form of power series with constant coefficients:

(3) |

Substituting this series in system (1) and grouping the terms of the same power of , we can determine all the coefficients and . In particular, , , and satisfies the following equation:

(4) |

that has a different number of roots depending on the values of the parameters and . Besides, if the inequality holds, the remaining coefficients , as are determined from the chain of linear equations:

(5) |

where

etc. Note that the pair of equations and defines the bifurcation curves

on the parameter plane , where . It can easily be checked that system (5) is solvable whenever . The bifurcation curves divide the parameter plane into the following parts (see Fig. 2):

If , the equation has four different roots on the interval . In the case , there are only two different roots.

Thus we have

###### Theorem 1.

The existence of solutions , with the asymptotics (3) as follows from [14, 15], and comparison theorems [16] applied to system (1) ensure that the solutions can be extended to the semi-axis. In the next sections, the stability of such solutions is discussed, as well as the asymptotics for two-parameter family of autoresonant solutions to system (1) is constructed.

As an example, let us fix , then , , . For , equation (4) has 4 roots: ; in the opposite case , there are only 2 roots: .

## 3. Stability analysis

1. Linearization. Let , be one of the particular autoresonant solutions with asymptotics (3). Then, the change of variables , leads to the following system

(6) |

with a fixed point at . Consider the linearized system:

It can easily be checked that the roots of the characteristic equation can be represented in the form , where

Therefore, if , the leading asymptotic terms of the eigenvalues are real of different signs. This implies that the fixed point of (6) is a saddle in the asymptotic limit, and the corresponding autoresonant solutions , to system (1) are unstable. In the opposite case, when , we have , as and the fixed point is a center in the asymptotic limit. Note that in the case as , linear stability analysis fails and it is necessary to consider nonlinear terms of equations. Indeed, let us consider as an example the non-autonomous system of differential equations:

(7) |

The solution has the form: , with arbitrary parameters and . These formulas indicate that the fixed point of system (7) is unstable. However, both roots of the characteristic equation for the corresponding linearized system are negative: .

Thus we have

###### Theorem 2.

If , the solution , with asymptotics (3) is unstable.

2. Lyapunov function. In the case when linear stability analysis fails, the stability problem can be solved using the Lyapunov function method. We have

###### Theorem 3.

###### Proof.

In system (1) we make the change of variables

(9) |

, and for new functions , we study the stability of the fixed point for the following system

(10) |

where

and

The construction of the Lyapunov function proposed here is based on the asymptotic behavior of the right-hand side of system (10) as . By taking into account (3) one can readily write out the asymptotics of the functions and :

where

Note that all asymptotic estimates written out here and bellow in the form and , , are uniform with respect to in the domain , where .

A Lyapunov function candidate is constructed of the form:

(11) |

where

Since and as , then for all there exist and such that

for . The derivatives of , , and with respect to along the trajectories of system (10) have the following asymptotics:

These formulas are used to calculate the expression for the total derivative of the function , which happens to have a sign-definite leading term of the asymptotics:

Since the remainders in the latter expression can be made arbitrarily small, it follows that for all there exist and such that

for all . In addition, the Lyapunov function has the following property: for all there exist such that

The last estimates and the negativity of the total derivative of the function ensure that any solution of system (10) with initial data cannot leave -neighborhood of the equilibrium as . Therefore, the fixed point is stable as . The stability on the finite time interval follows from the theorem on the continuity of the solution to the Cauchy problem with respect to the initial data.

Let us show that the fixed point is asymptotically stable. Indeed, consider the solution , to system (10) with initial data , then the function satisfies the inequality:

(12) |

Integrating the last expression with respect to , we obtain with the parameter , depending on and . This implies the asymptotic estimate: as . Returning to the original variables we derive asymptotic estimates (8) with for solutions to system (1) with initial data from a neighbourhood of the point . ∎

Let us consider again the case . If , system (1) has two particular autoresonant solutions with and , and if , there is an additional pair of solutions with . It is easy to calculate that . Hence the unstable solution , with becomes asymptotically stable when the parameter exceeds the threshold value . At the same time, , then the stability of the solution with persists as and is lost when . However, since , the additional solutions are unstable as . Thus the points and correspond to non-autonomous version of the center-saddle bifurcation: stable and unstable solutions coalesce and disappear. The situation is similar when , and the threshold value for unstable solutions can be found from the equation .

## 4. Asymptotic analysis

The stability of the particular solutions , ensures the existence of a two-parameter family of autoresonance solutions. At this section we construct the asymptotics for such solutions by averaging method [17] with some modifications, related to the involvement of the Lyapunov function, constructed in the previous section.

###### Theorem 4.

If , then system (1) has two-parameter family of autoresonant solutions , with the asymptotics

(13) |

where the functions , are -periodic in , and the function has the following asymptotics as :

, , , .

###### Proof.

We make change of variables (9) in system (1) and study the asymptotics for solutions to near-Hamiltonian system (10) in a neighbourhood of the stable fixed point . Let us consider the Hamiltonian system:

where , . It follows from the properties of the function that the level lines define a family of closed curves on the phase space parametrized by the parameter , (see Fig. 3). To each closed curve there corresponds a periodic solution , of period , where as .

Define auxiliary -periodic functions and , satisfying the equations:

and consider the transformation of variables

(14) |

in system (10). The Jacobian of this transformation is different from zero:

It can easily be checked that in new variables system (10) takes the form:

(15) |

where and are -periodic functions with respect to . Moreover, system (15) is asymptotically Hamiltonian with decaying non-Hamiltonian terms as . Solutions to such systems can be investigated by the averaging method. To simplify asymptotic constructions, we make one more transformation of variables. Introduce a new dependent variable associated with the Lyapunov function (11) such that

(16) |

where is -periodic function in . It follows from the properties of the Lyapunov function that and its total derivative with respect to have the following asymptotics

as , , uniformly with respect to . Hence the transformation is invertible: . System (15) in new variables takes the form:

(17) |

where

(18) |

It is not difficult to deduce from (18) and (12) the asymptotics of the functions and as :

where and are -parametric functions with respect to , , as . To construct asymptotic solutions to system (17), it is convenient to single out a Hamiltonian part:

(19) |

where

The asymptotic solution to the first equation in (19) is sought in the form [18]: , where is determined from the averaged equation

(20) |