Shock and Vibration

Volume 2016 (2016), Article ID 9471423, 10 pages

http://dx.doi.org/10.1155/2016/9471423

## Novel Hyperbolic Homoclinic Solutions of the Helmholtz-Duffing Oscillators

^{1}Key Laboratory of Vibration Control and Structural Safety, Guangzhou University, Guangzhou 510405, China^{2}Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, China

Received 28 May 2015; Revised 23 August 2015; Accepted 26 August 2015

Academic Editor: Chia-Ming Chang

Copyright © 2016 Yang-Yang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The exact and explicit homoclinic solution of the undamped Helmholtz-Duffing oscillator is derived by a presented hyperbolic function balance procedure. The homoclinic solution of the self-excited Helmholtz-Duffing oscillator can also be obtained by an extended hyperbolic perturbation method. The application of the present homoclinic solutions to the chaos prediction of the nonautonomous Helmholtz-Duffing oscillator is performed. Effectiveness and advantage of the present solutions are shown by comparisons.

#### 1. Introduction

It has been widely accepted that homoclinic solutions play a fundamental role in global bifurcations and chaos predictions of dynamical systems [1, 2]. For instance, the experimental study of certain magnetic pendulum verified the homoclinic solutions as the precursors to chaotic vibration [3]. Some occurrences of homoclinic solutions can be regarded as the criterion from single well chaos to cross well chaos motion of oscillators [4], or as the onsets of chaotic vibrations of asymmetric nonconservative oscillators [5]. Homoclinic solutions were also adopted in bifurcation and chaotic vibration controls for beam structures [6, 7]. Another typical application of homoclinic solutions aims at solitary wave studies. For instance, a proper homoclinic solution can govern the solitary roll waves down an open inclined channel [8], or optical solitary waves propagating in fibers [9, 10]. The association between the singular solitary waves and homoclinic solutions can be interpreted based on phase plane analysis [11].

Because of their importance in nonlinear systems, many homoclinic solutions have been derived in the past few decades. Such works include but are not limited to the following: Xu et al. [12] proposed the perturbation-incremental method for homoclinic solutions; Chan et al. [13] applied the perturbation-incremental method to study the stability and the homoclinic bifurcations of limit cycles; Belhaq et al. [14] analytically developed criterions for predicting homoclinic connection of limit cycle. Mikhlin and Manucharyan [15] and Manucharyan and Mikhlin [16] applied the Padé and quasi-Padé approximants for homo- and heteroclinic solutions. Y. Y. Chen and S. H. Chen [17] and Chen et al. [18] developed perturbation techniques by hyperbolic functions for homoclinic solutions of strongly nonlinear oscillators. Cao et al. [19] improved the perturbation-incremental homoclinic solutions for strongly nonlinear oscillators. Recently, Li et al. [20] improved the perturbation method based on harmonic functions to derive homoclinic solutions of Helmholtz-Duffing oscillators.

Nevertheless, to the best of our knowledge, the completely analytical, exact, and explicit homoclinic solution of the strongly nonlinear Helmholtz-Duffing oscillators has not yet been derived, in spite of the wide application of its equation for many engineering problems such as ship dynamics, oscillation of the human ear drum, oscillations of one-dimensional structural system with an initial curvature, some electrical circuits, microperforated panel absorber, and heavy symmetric gyroscope [21–29]. It should be pointed out that the previous typical solutions [14–18] become invalid for such mix-parity systems. Even for the conservative Helmholtz-Duffing oscillator, solutions by the perturbation methods [12, 13, 19, 20] based on generalized harmonic functions can only be obtained implicitly, in which the infinite time domain of a homoclinic motion has to be transformed into a finite period of the harmonic. Moreover, for strongly nonlinear oscillators, as the perturbation-incremental method [12, 13, 19] consists of perturbation procedure with the incremental harmonic balance method, their solutions are always expressed by harmonic functions with numerical coefficients. That means such implicit solutions are semianalytical and seminumerical and cumbersome for practical application.

This paper aims to present new homoclinic solutions of the Helmholtz-Duffing oscillators. The completely analytical, exact, and explicit homoclinic solution of the conservative Helmholtz-Duffing oscillator will be derived by a hyperbolic function balance procedure. Then the homoclinic solution of the self-excited Helmholtz-Duffing oscillator will also be obtained by an extended hyperbolic perturbation method. The application of the present solutions to the chaos prediction of the nonautonomous Helmholtz-Duffing oscillator is performed. The preference of the present solution will be illustrated by comparison.

#### 2. The Explicit and Exact Homoclinic Solution of the Undamped Helmholtz-Duffing Oscillator

Consider the homoclinic solution of the undamped Helmholtz-Duffing equation

If , (1) becomes the classical Duffing equation, which possesses a homoclinic solution with and . Such homoclinic solution of classical Duffing equation has been discussed in detail in [17], in which the solution can be written as

If , (1) becomes the classical Helmholtz equation, which possesses a homoclinic solution. Such homoclinic solution of classical Helmholtz equation has been discussed in detail in [18], in which the solution can be written asNoting the relationship as below,we can rewrite (2) and (3), respectively; that is,Here, to find a proper trial solution form for (1), we can observe the two special cases above. It can be seen that the two expressions above are similar, because they have the common form expressed asin which, when , the constants of (6) are , , , and . While when , , and , the constants of (6) are , , , and .

Thus, the time derivative of (6) isNote that is the homoclinic point. For and , we adopt (6) as a trial solution for the homoclinic solution of (1) and try to determine all its constants by substituting (6) into (1); that is,In order to balance (8) for all time , we equate coefficients of like powers of the hyperbolic function term and get the following nonlinear algebraic equations:From (12),orThe left hand side of (12) can be regarded as the restoring force of the oscillator with . In other words, (12) means that the displacement derivative of the potential energy curve at is zero. Furthermore, we have to make sure that the potential energy curve at is not concave. Thus, the displacement derivative of the restoring force at will not be positive; that is,Therefore, can be determined by (13)–(15) and then, (9)–(11) can be discussed, respectively, in the two cases as follows.

*Case 1 (). *Equations (9)–(11) can be rewritten as belowby which one can obtain

*Case 2 (). *As and are nonzero, multiplying (12) by and then adding it to (9) yieldMultiplying (12) by and then adding it to (10), the latter becomesMultiplying (12) by and then adding it to (11), we also getThus from (18)–(20),

Furthermore, noting that, for the homoclinic solution, the potential energy values at and at should be equal, we have by which the final values of , , and expressed by (17) or (21) can be selected.

*Example 1. *Here we apply the method for equationwhich is a case of (1) with , , and . From (14), (21), and (22), we can determine all the constants and get the homoclinic solution asThe time histories and the phase portraits of the solutions by different methods are shown in Figures 1 and 2, respectively. It can be seen from the figures that the present method yields accurate and explicit solutions in both the figures, while the generalized harmonic function perturbation method can only provide valid solution in Figure 2. The reason is that based on harmonic functions [12, 13, 19, 20] the homoclinic solutions can only be expressed implicitly by the nonlinear time scale they adopted and be investigated only in phase planes. Such implicit solutions are too abstract or cumbersome to use in some practical problems. Therefore, the present explicit solutions in respect to time are more applicable.