New Homoclinic and Heteroclinic Solutions for Zakharov System

A new type of homoclinic and heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions, for Zakharov system are obtained using extended homoclinic test and two-soliton methods, respectively. Moreover, the homoclinic and heteroclinic structure with local oscillation and mechanical feature different from homoclinic and heterocliunic solutions are investigated. Result shows complexity of dynamics for complex nonlinear evolution system. Moreover, the similarities and differences between homoclinic (heteroclinic) breather and homoclinic (heteroclinic) tube are exhibited. These results show that the diversity of the structures of homoclinic and heteroclinic solutions.     

New Homoclinic and Heteroclinic Solutions for Zakharov System

A new type of homoclinic and heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions, for Zakharov system are obtained using extended homoclinic test and two-soliton methods, respectively. Moreover, the homoclinic and heteroclinic structure with local oscillation and mechanical feature different from homoclinic and heterocliunic solutions are investigated. Result shows complexity of dynamics for complex nonlinear evolution system. Moreover, the similarities and differences between homoclinic (heteroclinic) breather and homoclinic (heteroclinic) tube are exhibited. These results show that the diversity of the structures of homoclinic and heteroclinic solutions.     

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.     

Chaos in the second—order autonomous Birkhoff system with a heteroclinic circle

Chaotic behaviour in a second-order autonomous Birkhoff system with a heteroclinic circle under weakly periodic perturbation is studied using the Melnikov method.The equations of heteroclinic orbits and the criteria for chaos are given.One example is also presented to illustrate the application of the results.     

一类非线性相对转动系统的组合共振分岔与混沌

研究一类具有异宿轨道的非线性相对转动系统的分岔与混沌运动. 应用耗散系统的拉格朗日方程建立一类组合谐波激励作用下非线性相对转动系统的动力学方程. 利用多尺度法求解相对转动系统发生组合共振时满足的分岔响应方程并进行奇异性分析, 得到了系统稳态响应的转迁集. 根据相对转动系统异宿轨道参数方程, 求解了异宿轨道的Melnikov函数, 并给出了系统发生Smale马蹄变换意义下混沌的临界条件. 最后采用数值方法, 通过分岔图, 最大Lyapunov指数图, 相轨迹图和庞加莱截面图研究系统参数对混沌运动的影响.     

A hierarchical heteroclinic network

We consider a heteroclinic network in the framework of winnerless competition of species. It consists of two levels of heteroclinic cycles. On the lower level, the heteroclinic cycle connects three saddles, each representing the survival of a single species; on the higher level, the cycle connects three such heteroclinic cycles, in which nine species are involved. We show how to tune the predation rates in order to generate the long time scales on the higher level from the shorter time scales on the lower level. Moreover, when we tune a single bifurcation parameter, first the motion along the lower and next along the higher-level heteroclinic cycles are replaced by a heteroclinic cycle between 3-species coexistence-fixed points and by a 9-species coexistence-fixed point, respectively. We also observe a similar impact of additive noise. Beyond its usual role of preventing the slowing-down of heteroclinic trajectories at small noise level, its increasing strength can replace the lower-level heteroclinic cycle by 3-species coexistence fixed-points, connected by an effective limit cycle, and for even stronger noise the trajectories converge to the 9-species coexistence-fixed point. The model has applications to systems in which slow oscillations modulate fast oscillations with sudden transitions between the temporary winners.     

一类约瑟夫森结混沌系统的谐和共振控制

引入外激和参激两种不同形式的谐和共振激励,探讨了一类约瑟夫森结(Josephson junction)系统的混沌控制问题.利用Melnikov方法研究了异宿混沌的生成和抑制,得到了在一定的控制激励振幅范围内,能确保异宿混沌被控制住,而且推导出控制激励与系统的激励两者之间的相位差和两者频率之间的共振阶数应满足的关系式.从定性的角度说明相位差在异宿混沌的控制中确实有着至关重要的影响,而且,数值方法的研究表明可通过调节相位来控制非自治系统中的稳态混沌.通过分析、比较外激和参激两种不同的共振激励对约瑟夫森结系统的异宿混沌的控制效果,得到对于较小的共振频率,宜采用参激激励,而对于较大的共振频率,宜采用外激激励. 关键词： 混沌控制 谐和共振激励 相位控制 Melnikov方法     

A geometric criterion for positive topological entropy

We prove that a diffeomorphism possessing a homoclinic point with a topological crossing (possibly with infinite order contact) has positive topological entropy, along with an analogous statement for heteroclinic points. We apply these results to study area-preserving perturbations of area-preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections. In the heteroclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection. In the homoclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection. These results significantly simplify the application of the Poincaré-Arnold-Melnikov-Sotomayor method. The results apply even when the contraction and expansion at the fixed point is subexponential.The first author was partially supported by a Sloan Foundation Fellowship and an N.S.F. grant. The second author was partially supported by a National Science Foundation Postdoctoral Research Fellowship. Both authors would like to thank MSRI for their support during the period that much of this paper was written.     

Sequential activity and multistability in an ensemble of coupled Van der Pol oscillators

In this paper collective dynamics of an ensemble of inhibitory coupled Van der Pol oscillators are studied. It was found that a stable heteroclinic contour and a stable heteroclinic channel between saddle cycles exist. These heteroclinic structures are responsible for the sequential activity of different oscillations. The corresponding bifurcations leading to the appearance of heteroclinic trajectories are analyzed.     

Construction and Verification of a Simple Smooth Chaotic System

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display different attractors with two unstable equilibrium points and four unstable equilibrium points respectively. Dynamical properties of this system are then studied. Furthermore, by applying the undetermined coefficient method, heteroclinic orbit of (S)hil'nikov's type in this system is found and the convergence of the series expansions of this heteroclinic orbit are proved in this article. The (S)hil'nikov's theorem guarantees that this system has Smale horseshoes and the horseshoe chaos.     

Construction and Verification of a Simple Smooth Chaotic System

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display different attractors with two unstable equilibrium points and four unstable equilibrium points respectively. Dynamical properties of this system are then studied. Furthermore, by applying the undetermined coefficient method, heteroclinic orbit of Shil＇nikov＇s type in this system is found and the convergence of the series expansions of this heteroclinic orbit are proved in this article. The Shil＇nikov＇s theorem guarantees that this system has Smale horseshoes and the horseshoe chaos.     

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.     

Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg-Landau equation

Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg-Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.     

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.     

Heteroclinic dynamics in a model of Faraday waves in a square container

We study periodic orbits associated with heteroclinic bifurcations in a model of the Faraday system for containers with square cross-section and single-frequency forcing. These periodic orbits correspond to quasiperiodic surface waves in the physical system. The heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable Hamiltonian limit, some of which persist in the presence of small damping. The dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate Poincaré maps, with predictions that agree with numerical computations. The results suggest a great richness of possible dynamics of Faraday waves even in simple geometries and with single-frequency forcing.     

Primary resonant optimal control for nested homoclinic and heteroclinic bifurcations in single-dof nonlinear oscillators

A unified control theorem is presented in this paper, whose aim is to suppress the transversal intersections of stable and unstable manifolds of homoclinic and heteroclinic orbits in the Poincarè map embedding in system dynamics. Based on the control theorem, a primary resonant optimal control technique (PROCT for short) is applied to a general single-dof nonlinear oscillator. The novelty of this technique is able to obtain the unified analytical expressions of the control gain and the control parameters for suppressing the homoclinic and heteroclinic bifurcations, where the control gain can guarantee that the control region where the homoclinic and heteroclinic bifurcations do not occur can be enlarged as much as possible at least cost. The technique is applied to a nonlinear oscillator with a pair of nested homoclinic and heteroclinic orbits. By the PROCT, the transversal intersections of homoclinic and heteroclinic orbits can be suppressed, respectively. The hopping phenomenon that there coexist two kinds of chaotic attractors of Duffing-type and pendulum-type can be suppressed. On the contrary, if the first amplitude coefficient is greater than the critical heteroclinic bifurcation value, then another degenerate hopping behavior of chaos will take place again. Therefore, the phenomenon of hopping is the dominant type of chaos in this oscillator, whose suppressing or inducing is admissible from the points of practical and theoretical view.     

New type of heteroclinic tangency in two-dimensional maps

A new mechanism of heteroclinic tangency is investigated by using two-dimensional maps. First, it is numerically shown that the unstable manifold from a hyperbolic fixed point accumulates to the stable manifold of a nearby period-2 hyperbolic point in a piecewise linear map and that the unstable manifold from a hyperbolic fixed point accumulates to the accumulation of the stable manifold of a nearby period-2 hyperbolic point in a cubic map. Second, a theorem on the impossibility of heteroclinic tangency (in the usual sense) is given for a particular type of map. The notions ofdirect andasymptotic heteroclinic tangencies are introduced and heteroclinic tangency is classified into four types.     

Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems

It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems(DPSs). This paper first provides a new class of four-dimensional(4 D) two-zone discontinuous piecewise affine systems(DPASs), and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.     

Comment on ‘Šilnikov-type orbits of Lorenz-family systems’ [Physica A 375 (2007) 438–446]

In the commented paper, the authors consider a four-parameter family of three-dimensional autonomous systems that includes the widely studied systems of Lorenz, Chen and Lü. Their goal is to rigorously prove the existence of Šilnikov chaos in this Lorenz-family via the Šilnikov criterion. They use an undetermined coefficient method to demonstrate the presence of heteroclinic orbits. In this comment we point out that their analysis is erroneous. The odd function they consider for the first component of the Šilnikov heteroclinic orbit cannot represent such global connection. Thus, the main result of their paper, Theorem 2, is incorrect.     

Transitions between chaos and order in rf-driven Josephson junction

We investigate transitions between the chaotic and regular states through a perturbed chaotic solution of a rf-driven Josephson system. It is shown that the transition from order to chaos may occur when the system initially near the heteroclinic points. The chaotic solution tends to an unstable periodic one for the initial values sufficiently nearing the heteroclinic orbit but going beyond the heteroclinic points. Thus the Josephson chaos can be analytically and numerically controlled, by adjusting the initial conditions.