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Novel Hyperbolic Homoclinic Solutions of the Helmholtz-Duffing Oscillators
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.
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 . Some occurrences of homoclinic solutions can be regarded as the criterion from single well chaos to cross well chaos motion of oscillators , or as the onsets of chaotic vibrations of asymmetric nonconservative oscillators . 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 , 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 .
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.  proposed the perturbation-incremental method for homoclinic solutions; Chan et al.  applied the perturbation-incremental method to study the stability and the homoclinic bifurcations of limit cycles; Belhaq et al.  analytically developed criterions for predicting homoclinic connection of limit cycle. Mikhlin and Manucharyan  and Manucharyan and Mikhlin  applied the Padé and quasi-Padé approximants for homo- and heteroclinic solutions. Y. Y. Chen and S. H. Chen  and Chen et al.  developed perturbation techniques by hyperbolic functions for homoclinic solutions of strongly nonlinear oscillators. Cao et al.  improved the perturbation-incremental homoclinic solutions for strongly nonlinear oscillators. Recently, Li et al.  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 , 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 , 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 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),
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.
3. Perturbation Homoclinic Solution of the Self-Excited Helmholtz-Duffing Oscillator
Consider the homoclinic solution of the self-excited Helmholtz equationwhere denotes constants. We assume the homoclinic solution of (26) can still be expressed in the similar form of (6); that is,however, the amplitude and the nonlinear time scale will depend upon the perturbation parameter . Thus and , respectively, can be expanded in the powers of ; that is,Then (27) can be rewritten aswhereThenAfter substituting (29) and (30) into (26), equating coefficients of like powers of yields the following equations:Then solutions can be determined by solving linear equations (33), (34), … one by one. It can be seen that (33) is obtained from (1) via the transformation in (29). Therefore, the homoclinic solution of (33) can be given by (6). Multiplying (34) by and integrating it from to , we obtainwhereNoting the properties of hyperbolic functions, we haveThus letting and in (35), we deriveEquation (38), which can also be derived by the classical Melnikov method, represents the critical condition under which the homoclinic bifurcation occurs. In other words, there exists a homoclinic solution once all the constants in (26) satisfy (38). Letting and in (35) givesFurthermore, substituting into (35) yieldsThe three equations above allow , , and to be determined one by one. As an illustration, here we considerin which , , and are constants. Noting (41), and substituting (6) and (7) into (36), the latter becomeswhere the expressions of the constants , , , , and are listed in Appendix. Noting that, in (41), is an even function and is an odd function with respect to , (38) can be rewritten as Thus substituting (42) into (43), one derivesEquation (44) is the condition under which homoclinic bifurcation occurs. Substituting (43) into (39) givesThen (40) can be rewritten asFinally, the expression for homoclinic solution can be expressed by
Example 2. Consider the equationwhich is a generalized Helmholtz-Duffing-Van der Pol equation with , , , , , and . Here is the classical damping coefficient which is considered as the bifurcation parameter. A limit cycle motion can break up by homoclinic bifurcation at a critical value of the damping coefficient. From (14), (15), (21), and (22), we have , , , and . From (44), the homoclinic bifurcation value of the damping coefficient isThen from (46) and (47), the homoclinic solution can be obtained as below,whereThe time history diagrams and the phase portraits of the solutions by different methods are shown in Figures 3 and 4, respectively. It can be seen from the figures that the present method still shows preference in the comparison. By the numerical method, the homoclinic bifurcation point can be found at , which are very close to those predicted by the present method.
4. Application to the Chaos Prediction of the Nonautonomous Helmholtz-Duffing Oscillator
Consider the nonautonomous Helmholtz-Duffing equationwhere is the external harmonic excitation with the amplitude and the frequency . According to the Melnikov method [1, 2], the Melnikov function of (52) can be written aswhere is the solution presented in (6) and denotes the vector cross product. Then the chaotic response of the system may occur if there exists a simple zero point of . Thus, we substitute (7) into (53), and let ; the latter yieldswhereTherefore, from (54) it can be seen that possesses simple zero points ifby which the onset of chaos can be evaluated with the threshold value .
Furthermore, if we rewrite (52) asand consider using the homoclinic solution expressed in (47) for Melnikov function, then we can deriveinstead. Note that is odd function of . For any real value of , there exists , where can be any integer number, to satisfy . That means the condition for chaos prediction becomes less restricted than that presented with the exact homoclinic solution of conservative system. Comparing (58) and (53) we can see that the difference is caused by omitting of the damping term in the Melnikov function calculation. In essence, the Melnikov method is based on approximation theory and, according to the method, it should be better to arrange all the perturbation terms at the right hand side of the equation to take them into account in the Melnikov function calculation. In other words, if we get the homoclinic solutions with perturbation method firstly and then implement the Melnikov function, we probably get a less restricted condition, because actually we conduct the perturbation approximations twice.
It is worth pointing out that as the present solutions by hyperbolic functions are explicit in respect to the real time , they are more applicable to the Melnikov method for chaos prediction than those derived implicitly by generalized harmonic functions [12, 13, 19, 20]. In particular, it should also be pointed out that the Melnikov method is only regarded as one of the conditions for chaotic prediction. At present, a chaotic motion should still be evaluated more thoroughly with qualitative theory and numerical method. Below are two examples which satisfy the Melnikov condition but with different characters of chaotic motions.
Example 3. Consider the equationwhich is the case of (52) with , , , , , and . From (17), (55), and (56) we can derive that when and when . Therefore it can be estimated that chaotic motion may happen if . The numerical results of homoclinic bifurcation by AUTO numerical method [30, 31] show that, with , the Lyapunov exponent value shows chaos behavior from to about and then converts to 0 gradually. That means the chaotic motion possesses dissipative chaos property. The Lyapunov exponent diagram is shown in Figure 5. The Lyapunov exponent value converts to less than 0.01 after . The phase portrait of the system after is shown in Figure 6, which shows that the motion converts to a limit cycle.
Example 4. Consider the equationwhich is the case of (52) with , , , , , and . From (17), (55), and (56) we can derive that when and when . Therefore it can be estimated that chaotic motion may happen if . The numerical results of homoclinic bifurcation by AUTO numerical method [30, 31] show that the Lyapunov exponent value stays more than 0 when . The Lyapunov exponent diagram is shown in Figure 7. The Poincaré projection of the system from to is shown in Figure 8. In the figure, the fractal character of a strange attractor can be observed, which supports the prediction of chaotic motion.
The present procedures are efficient for constructing homoclinic solutions of the Helmholtz-Duffing oscillator. The exact and explicit homoclinic solution of the undamped Helmholtz-Duffing oscillator is derived by a hyperbolic function balance procedure. The homoclinic solution of the self-excited system is then obtained by the extension of the hyperbolic perturbation procedure. The application to the chaos prediction of the nonautonomous Helmholtz-Duffing oscillator can also be conducted.
Conflict of Interests
The authors hereby declare that there is no conflict of interests regarding the publication of this paper.
The financial supports from the National Basic Research Program of China (2011CB013606), the National Natural Science Foundations of China (91315301-07, 11102045), and the National Plan for Science and Technology Support of China (2012BAJ07B02) are gratefully acknowledged.
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