Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force

Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov's method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. (Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soc. A 366 635).     

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.     

Bifurcations near homoclinic orbits with symmetry

The effect of symmetry on bifurcations associated with homoclinic orbits to saddle-foci is analysed. With symmetry each homoclinic bifurcation contributes three periodic orbits to the global bifurcation picture as opposed to a single orbit in the general case. Bifurcations on these orbits are studied: there are sequences of saddle-node and period-doubling bifurcations, chaos and more complicated homoclinic orbits.     

An integrable model in general relativity

Melnikov-method-based theoretical results are demonstrated concerning the relative effectiveness of any two weak excitations in suppressing homoclinic/heteroclinic chaos of a relevant class of dissipative, low-dimensional and non-autonomous systems for the main resonance between the chaos-inducing and chaos-suppressing excitations. General analytical expressions are derived from the analysis of generic Melnikov functions providing the boundaries of the regions as well as the enclosed area in the amplitude/initial phase plane of the chaos-suppressing excitation where homoclinic/heteroclinic chaos is inhibited. The relevance of the theoretical results on chaotic attractor elimination is confirmed by means of Lyapunov exponent calculations for a two-well Duffing oscillator. Received 21 May 2002 / Received in final form 13 September 2002 Published online 29 November 2002     

待定固有频率法在分析系统混沌临界值问题中的应用

利用规范形理论与待定固有频率寻求改善Melnikov函数分析非线性振动系统混沌阈值的简单方法,着重讨论参数与周期激励联合作用下具有主参数共振的三阱势能系统,建立了其Melnikov函数积分式.引入由待定固有频率形成的时间尺度变换,从同宿及异宿分岔两个角度获取系统的混沌临界值,使得非线性扰动量对于基频的影响有效地体现于Melnikov函数表达式中,进而结合相应的分析过程提高所得结果的计算精度.作为算例,对解析解与数值积分结果进行了对比,以验证提出方法的有效性与可行性. 关键词： 规范形 Melnikov方法 混沌 同宿分岔     

Sil'nikov chaos of the Liu system

A new chaotic attractor is discovered for the Liu system. The homoclinic and heteroclinic orbits in the Liu system have been found by using the undetermined coefficient method. It analytically demonstrates that there exists one heteroclinic orbit of the Sil'nikov type that connects two nontrivial equilibrium points, and therefore Smale horseshoes and the horseshoe chaos occur for this system via the Sil'nikov criterion. In addition, there also exists one homoclinic orbit joined to the origin. The convergence of the series expansions of these two types of orbits is proved.     

谐和激励与有界噪声作用下具有同宿和异宿轨道的Duffing振子的混沌运动

研究了具有同宿轨道、异宿轨道的双势阱Duffing振子在谐和激励与有界噪声摄动下的混沌运动.基于同宿分叉和异宿分叉，由Melnikov理论推导了系统出现混沌运动的必要条件及出现分形边界的充分条件.结果表明：当Wiener过程的强度参数大于某一临界值时，噪声增大了诱发混沌运动的有界噪声的临界幅值，相应地缩小了参数空间的混沌域，且产生混沌运动的临界幅值随着噪声强度的增大而增大.同时数值计算了最大Lyapunov指数，由最大Lyapunov指数为零从另一角度得到了系统出现混沌运动的有界噪声的临界幅值，发现在Wi 关键词： 混沌 同宿和异宿分叉 随机Melnikov方法 最大Lyapunov指数     

基于Coulomb摩擦效应的一类非线性相对转动系统的混沌研究

研究一类非线性相对转动系统在负载Coulomb摩擦效应下的混沌运动行为. 根据Lagrange方程建立一类含非线性负载Coulomb摩擦阻尼的两个质量相对转动系统的动力学方程. 利用Cardano公式讨论自治系统的特征值, 在此基础上, 应用待定系数法给出系统同宿轨道的存在性, 并借助Silnikov定理研究了系统的混沌行为. 最后数值模拟了给定参数下系统的混沌运动, 并给出在Coulomb摩擦阻尼变化下系统由周期、倍周期通向混沌的途径, 验证了理论分析的正确性.     

Analysis of the T-point-Hopf bifurcation

In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-2 heteroclinic cycle occurs. If the parameter space is three-dimensional, such a bifurcation is located generically on a curve. A more degenerate scenario appears when this curve reaches a surface of Hopf bifurcations of one of the equilibria involved in the heteroclinic cycle. We are interested in the analysis of this codimension-3 bifurcation, which we call T-point-Hopf. In this work we propose a model, based on the construction of a Poincaré map, that describes the global behavior close to a T-point-Hopf bifurcation. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol-Duffing electronic oscillator.     

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.     

一类相对转动非线性动力系统的混沌运动

研究一类具有同宿轨道、异宿轨道的相对转动非线性动力系统的混沌运动. 建立具有非线性刚度、非线性阻尼和外扰激励作用的一类两质量相对转动非线性动力系统的动力学方程. 利用Melnikov方法讨论了系统的全局分岔和系统进入混沌状态的可能途径,给出了系统发生混沌的必要条件,并利用最大Lyapunov指数图,分岔图,Poincare截面图和相轨迹图进一步分析了系统的混沌行为. 关键词： 相对转动 非线性动力系统 混沌 Melnikov方法     

Role of asymmetries in the chaotic dynamics of the double-well Duffing oscillator

Duffing oscillator driven by a periodic force with three different forms of asymmetrical double-well potentials is considered. Three forms of asymmetry are introduced by varying the depth of the left-well alone, location of the minimum of the left-well alone and above both the potentials. Applying the Melnikov method, the threshold condition for the occurrence of horseshoe chaos is obtained. The parameter space has regions where transverse intersections of stable and unstable parts of left-well homoclinic orbits alone and right-well orbits alone occur which are not found in the symmetrical system. The analytical predictions are verified by numerical simulation. For a certain range of values of the control parameters there is no attractor in the left-well or in the right-well.     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

Heteroclinic bifurcations and chaotic transport in the two-harmonic standard map

We study a two-parameter family of standard maps: the so-called two-harmonic family. In particular, we study the areas of lobes formed by the stable and unstable manifolds. Variational methods are used to find heteroclinic orbits and their action. A specific pair of heteroclinic orbits is used to define a difference in action function and to study bifurcations in the stable and unstable manifolds. Using this idea, two phenomena are studied: the change of orientation of lobes and tangential intersections of stable and unstable manifolds.     

Homoclinic (Heteroclinic) Orbit of Complex Dynamical System and Spiral Structure

Starting from iterated systems, it is shown that the homoclinic (heteroclinic) orbit is a kind of spiral structure. The emphasis is laid to show that there are homoclinic or heteroclinic orbits in complex discrete and continuous systems, and these homoclinic or heteroclinic orbits are some kind of spiral structure.     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

Chaotic attitude oscillations of a satellite filled with a rotating ellipsoidal mass of liquid subject to gravity-gradient torques

The Hamiltonian equations of a liquid-filled satellite subject to gravity-gradient torques in terms of the generalized Deprit variables are established for applying the Melnikov theory formally. The heteroclinic orbits of the torque-free symmetric liquid-filled satellite are found. A criterion for the heteroclinic transversal intersections at the onset of chaotic attitude is formulated using Melnikov's integral. The results from the Melnikov theory are crosschecked with simulation.     

Homoclinic bifurcation in Chua’s circuit

We report our experimental observations of the Shil’nikov-type homoclinic chaos in asymmetry-induced Chua’s oscillator. The asymmetry plays a crucial role in the related homoclinic bifurcations. The asymmetry is introduced in the circuit by forcing a DC voltage. For a selected asymmetry, when a system parameter is controlled, we observed transition from large amplitude limit cycle to homoclinic chaos via a sequence of periodic mixed-mode oscillations interspersed by chaotic states. Moreover, we observed two intermediate bursting regimes. Experimental evidences of homoclinic chaos are verified with PSPICE simulations.     

Scarring by homoclinic and heteroclinic orbits

In addition to the well-known scarring effect of periodic orbits, we show here that homoclinic and heteroclinic orbits, which are cornerstones in the theory of classical chaos, also scar eigenfunctions of classically chaotic systems when associated closed circuits in phase space are properly quantized, thus introducing strong quantum correlations. The corresponding quantization rules are also established. This opens the door for developing computationally tractable methods to calculate eigenstates of chaotic systems.     

连续时间系统同宿轨的搜索算法及其应用

同宿轨的求解是非线性系统领域的核心问题之一, 特别是对动力系统分岔与混沌的研究有重要意义. 根据同宿轨的几何特点, 采用轨线逼近的方式, 通过定义逼近轨线与鞍点的距离, 将同宿轨的求解转化为求距离最小值的无约束非线性优化问题. 为了提高优化结果的完整性, 还提出了基于区间细分的搜索算法和实现方法, 并找出了Lorenz系统, Shimizu-Morioka系统和超混沌Lorenz系统等的多个同宿轨道和对应参数, 验证了本文方法的有效性. 关键词： 混沌 同宿轨 非线性系统 数值计算