Narrow Results

With amplitude modulated excitation, the effect on chaotic behavior of Φ⁶-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitudes, when both frequencies of modulated and unmodulated forces are different, on bifurcation diagram and Poincaré map is also investigated, in addition to the surface of Maximal Lyapunov exponent versus modulated and unmodulated parameters for suppressing chaos being shown.

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

In order to extend the working range of the energy harvester, a nonlinear spring with multistable property is usually introduced to realize the structural design optimization. The nonlinear multistability property might increase the chance of inter-well chaotic responses, which leads to low-power output and unpredictability in motions. Quantitative determination of the parameter space in which chaos occurs is important. We investigated the chaos threshold of a multistable piezoelectric energy harvester subjected to wake-galloping in this paper. The homoclinic-like and heteroclinic-like orbits are obtained analytically for this system. This allows us to employ the Melnikov method to obtain chaos criterion from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external wake-galloping. The threshold of the excitation amplitude for snap-through motion is verified by numerical calculation. We observed that the nonlinear parameter can be effectively used to control the dynamical behavior of the system. The normalized voltage increases significantly when an inter-well chaotic response occurs. The results provide a guideline for the design and optimization of the multistable energy harvester subjected to wake-galloping.

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

Solitary wave solutions of two-component Drinfel’d–Sokolov–Wilson system with Kuramoto–Sivashinsky perturbation are considered. We first employ geometric singular perturbation theory to reduce the higher-dimensional system of equations to the perturbed planar system. We then further exploit the Melnikov method to explore the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit, indicating the existence of single- and double-peak solitary waves. Of particular interest is the appearance of the double-peak solitary wave solution. Finally, we include the numerical simulations to verify the theoretical results.

In this paper, a criteria of suppressing chaos for a kind of nonlinear oscillators is established by the theory of the strange attractor. The oscillators considered include Duffing, van der Pol, Duffing–van der Pol and pendulum. According to this criteria, we analyze the phase effect using two methods, one by adding the second external force term and the other by adding parametric excitation, both of which may be used to suppress chaos in the systems. Some examples are used to illustrate the validity of the criteria and the importance of phase effect in suppressing chaos.

In this paper, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three domains separated by two switching manifolds $x=a$ and $x=b$. The dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we wish to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter $\epsilon$ near $\epsilon =0$. We denote the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.

The discovery of Pluto’s small moons in the last decade has brought attention to the dynamics of the dwarf planet’s satellites. With such systems in mind, we study a planar $N$-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar $N$-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid ‘Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits nonspherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in $N$-body systems such as the Pluto system where some of the bodies are highly nonspherical.

In this paper, theoretical framework and numerical verification for suppressing homoclinic chaos of a class of vibro-impact oscillators are discussed by adding parametric excitations in the form of $xfcos(\omega \tau +\phi )$ as the control item. The analytical Melnikov method for planar vibro-impact systems is employed to obtain the corresponding thresholds of parameters as sufficient conditions for suppressing chaos. Two typical oscillators are presented to show the effectiveness of theoretical analysis for suppressing homoclinic chaos by tuning the amplitudes, frequencies and phases of the parametric excitations.

In the present work, the chaotic wave motions and the chaotic dynamic responses are investigated for a four-edge simply supported piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Based on the reductive perturbation method, the complicated partial differential nonlinear governing equation of motion for the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations is transformed into an equivalent and soluble nonlinear wave equation. The heteroclinic orbit and resonant torus are obtained for the unperturbed nonlinear wave equation. The topological structures of the unperturbed and the perturbed nonlinear wave equations are investigated on the fast and the slow manifolds. The persistence of the heteroclinic orbit is studied for the perturbed nonlinear wave equation through the Melnikov method. The geometric analysis is utilized to prove that the heteroclinic orbit goes back to the stable manifold of the saddle point on the slow manifold under the perturbations. The existence of the homoclinic orbit is conformed for the perturbed nonlinear wave equation by the first and the second measures. When the homoclinic orbit is broken, the chaotic motions occur in the Smale horseshoe sense for the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Numerical simulations are finished to study the influence of the damping coefficient on the propagation properties of the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Both theoretical study and numerical simulation results indicate the existence of the chaotic wave motions and the chaotic dynamic responses of the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations.

The chaotic behavior of a carbon nanotube with waviness along its axis is investigated. The equation of motion involves a quadratic and cubic terms due to the curved geometry and the mid-plane stretching. Melnikov method is applied for the system, and Melnikov criterion for global homoclinic bifurcations is obtained analytically. The numerical solution of the system using a fourth-order-Runge–Kutta method confirms the analytical predictions and shows that the transition from regular to chaotic motion is often associated with increasing the energy of an oscillator. Moreover, a detailed numerical study of the periodic attractor in the period window is also carried out.

This article extends the range of applications of a previous paper [Leung & Liu, 2003]. Some new methods to suppress chaos are proposed so that the Melnikov function is modified only slightly for the easy elimination of simple zeros. The reasonability of the methods is analyzed by the previously established criteria. Some examples are also given.

The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases.

We study the chaotic dynamics of one-degree-of-freedom nonlinear oscillator representing a density perturbation in plasma device model excited by parametric and external driven forces. Critical parameters for the onset of chaotic motions are specified using Melnikov method. The analytical results are confirmed by numerical simulations. The global dynamical changes of the system have been examined by evaluating parametric changes of the bifurcation diagrams, maximum Lyapunov exponent, Poincaré map and the basin boundaries of attraction. The transitions to chaos caused by the cascade bifurcation and intermittency are clearly shown by graphical methods.

In this paper, the chaotic behavior of a driven planar system with two discontinuous terms and a pseudo-equilibrium point in the intersection of the discontinuity surfaces is analyzed. This scenario is not covered by smooth techniques of chaos analysis or other techniques like the extension of Melnikov's method for nonsmooth systems. In consequence, we propose to use an approximate model of the discontinuous system for which this technique can be applied, and compare the responses of both systems, the discontinuous and the approximate, when this last model is close, in a certain way, to the discontinuous system. One of the discontinuous terms, given by a sign function, is approximated by a saturation with high slope at the equilibrium point. Some conditions that determine the chaotic behavior of the approximate system are formally established, and the convergence of its chaotic orbits to some orbits of the discontinuous system, when the slope of the approximation is large enough, is shown. In particular, we show the similarity of the dynamical behavior of both systems where a chaotic behavior can be displayed, for a parameter region determined by the application of the Melnikov technique to nonsmooth systems. A comparison of the Feigenbaum diagrams, for a parameter range obtained from the application of this technique, shows the similarity of their dynamics and the chaotic nature of the discontinuous system.

In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map $f$ and check that $f$ is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

We study Melnikov conditions predicting appearance of chaos in Duffing oscillator with hardening type of non-linearity under two-frequency excitation acting in the vicinity of the principal resonance. Since Hamiltonian part of the system contains no saddle points, Melnikov method cannot be applied directly. After separating the external force into two parts, we use a perturbation analysis that allows recasting the original system to the form suitable for Melnikov analysis. At the initial step, we perform averaging at one of the frequencies of the external force. The averaged equations are then analyzed by traditional Melnikov approach, considering the second frequency component of the external force and the dissipation term as perturbations. The numerical study of the conditions for homoclinic bifurcation found by Melnikov theory is performed by varying the control parameters of amplitudes and frequencies of the harmonic components of the external force. The predictions from Melnikov theory have been further verified numerically by integrating the governing differential equations and finding areas of chaotic behavior. Mismatch between the results of theoretical analysis and numerical experiment is discussed.