A High-Order Melnikov Method for Heteroclinic Orbits in Planar Vector Fields and Heteroclinic Persisting Perturbations

This work extends the high-order Melnikov method established by FJ Chen and QD Wang to heteroclinic orbits, and it is used to prove, under a certain class of perturbations, the heteroclinic orbit in a planar vector field that remains unbroken. Perturbations which have this property together form the heteroclinic persisting space. The Van der Pol system is analysed as an application.

Complex Dynamics of a Four-Dimensional Circuit System

Combining qualitative analysis and numerical technique, the present work revisits a four-dimensional circuit system in [Ma et al., 2016] and mainly reveals some of its rich dynamics not yet investigated: pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, globally exponentially attractive set, invariant algebraic surface and heteroclinic orbit. The main contributions of the work are summarized as follows: Firstly, it is proved that there exists a globally exponentially attractive set with three different exponential rates by constructing a suitable Lyapunov function. Secondly, the existence of a pair of heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, numerical simulations not only are consistent with theoretical results, but also illustrate potential existence of hidden attractors in its Lorenz-type subsystem, singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small [Formula: see text], i.e. hyperchaotic attractors and nearby pseudo singularly degenerate heteroclinic cycles, i.e. a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity, or the true ones consisting of normally hyperbolic saddle-foci (or saddle-nodes) and stable node-foci, giving some kind of forming mechanism of hyperchaos.

NONLINEAR ROLLING STABILITY AND CHAOS RESEARCH OF TRIMARAN VESSEL WITH VARIABLE LAY-OUTS IN REGULAR AND IRREGULAR WAVES UNDER WIND LOAD

The trimaran vessel rolls strongly at low forward speed and may capsize in high sea conditions due to chaos and loss of stability, which is not usually considered in conventional limit-based criteria. In order to perfect the method of measuring roll performance of trimaran, a set of nonlinear roll motion stability analysis method based on Lyapunov and Melnikov theory was established. The nonlinear roll motion equation was constructed by CFD and high-order polynomial fitting method. The wave force threshold of rolling chaos in regular waves is calculated by Gauss-Legendre numerical integration method. The limited significant wave height of rolling chaos in random sea conditions is deduced by the phase space transfer rate, and the complex effect of wind load is superposed in the calculation. The influence of trimaran configuration on the roll system is analyzed through the state differentiation of homoclinic and heteroclinic orbit in phase portrait. The calculation of the maximum Lyapunov exponent further verified the applicability of Melnikov method, and the topological structure change of gradual failure of the rolling system is analyzed by the erosion of safe basin. The complex changes of the nonlinear damping coefficient and the nonlinear restoring moment coefficient caused by the change of the transverse lay-outs between the main hull and side hull have a significant influence on chaos and stability, and the existence of wind load has a certain weakening effect on the stability and symmetry of the system. The conclusion also further indicates the importance of the lay-outs to the dynamic stability of the trimaran vessel, which is significant for its seakeeping design.

MODEL PENULARAN PENYAKIT DEMAM BERDARAH DENGUE (DBD) DALAM SYSTEM DYNAMIK BERDIMENSI DUA

Here we examine the dynamic model of the spread of Dengue Hemorrhagic Fever (DHF) assuming a constant number of host and vector populations. In this paper, the model is reduced from a three-dimensional system to a two-dimensional system so that the dynamic behavior can be analyzed in the R2 plane. In the two-dimensional model, if the threshold parameter R > 1, the endemic state becomes globally asymptotically stable. During the analysis of its dynamic behavior, a trapping region is found which contains a heteroclinic orbit connecting the slowing point, namely the origin and the endemic point. By using heteroclinic orbits, it can be estimated the time period required from a state to reach a certain state.

Analysis of Tangential Nonlinear Vibration on Machine Hydrostatic Slide

In this paper, the nonlinear dynamic responses of the hydrostatic slide were investigated and the effects of damping and external force to control the vibration system were discussed. The dynamic model of the system was established, and the tangential vibration equation taking into account nonlinear factors was derived. The heteroclinic orbit parameter equations of the vibration system were solved, and the Melnikov function of vibration system is derived. And the chaos condition and judging criterion of the vibration system were obtained by Melnikov’s method. The vibration equation of the hydrostatic slide was solved using the numerical method. The bifurcation diagram, phase diagram, wave diagram of displacement, and Poincaré map were obtained, and the nonlinear dynamic responses were analyzed. Finally validation experiments were conducted, and the results agree well with the results obtained by the Melnikov method and numerical method.

Existence of Chaos in the Chen System with Linear Time-Delay Feedback

It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

Chaotic Wave Motions and Chaotic Dynamic Responses of Piezoelectric Laminated Composite Rectangular Thin Plate Under Combined Transverse and In-Plane 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.