Symmetry-Breaking Homoclinic Bifurcations in Diffusively Coupled Equations

In this study we examine a symmetry-breaking bifurcation of homoclinic orbits in diffusively coupled ordinary differential equations. We prove that asymmetric homoclinic orbits can bifurcate from a symmetric homoclinic orbit when the equilibria to which the latter is homoclinic undergoes a pitchfork bifurcation. A condition which defines the direction of the bifurcation in a parameter space is given. All hypotheses of the main theorem are verified for a diffusively coupled logistic system and the twistedness of the bifurcating homoclinic orbits is computed for a range of coupling strengths.     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.     

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov–Takens bifurcation with $Z_2$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.Partially supported by DARPA and NSF.Partially supported by the Deutsche Forschungsgemeinschaft and by Konrad-Zuse-Zentrum für Informationstechnik Berlin.     

Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems

The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.     

非惯性参考系中弹性薄板动力系统在纵横振动相互耦合时的全局分岔及混沌性质

讨论非惯性参考系中弹性薄板动力系统１∶１内共振时的全局分岔及其混沌性质．首先对系统的奇点进行了分析，进而得到了奇点附近同宿轨的参数方程，再用Ｍｅｌｎｉｋｏｖ方法研究了系统的同宿轨分岔及其混沌运动．研究表明，对各种不同共振情形，系统将由同宿轨分岔过渡到混沌运动．最后用数值仿真证实了理论分析的结果．     

A group of chaotic motion of soft spring quadratic duffing equations

In this paper,we use the Melnikov function method to study a kind of soft Duffing equations(?) Af((?),x) x-x~(2k 1)=r[M(x,(?))cosωt N(x,(?))sinωt](k=1,2,3…)and give the condition that the equations have chaotic motion and bifurcation.The method used in this paper is effective for dealing with the Melnikov function integral of the system whose explict expression of the homoclinic or heteroclinic orbit cannot be given.     

Homoclinic flip bifurcation with a nonhyperbolic equilibrium

In this paper, the problem of homoclinic bifurcation accompanied by a transcritical bifurcation is investigated for high-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. Under certain nongeneric conditions (orbit flip and inclination flip homoclinic orbits), the existence, nonexistence, coexistence and uniqueness of homoclinic and periodic orbits are studied. Some known results are extended.     

Chaotic motions of the L-mode to H-mode transition model in tokamak

The chaotic dynamics of the transport equation for the L-mode to H-mode near the plasma in a tokamak is studied in detail with the Melnikov method. The transport equations represent a system with external and parametric excitation. The critical curves separating the chaotic regions and nonchaotic regions are presented for the system with periodically external excitation and linear parametric excitation, or cubic parametric excitation, respectively. The results obtained here show that there exist uncontrollable regions in which chaos always take place via heteroclinic bifurcation for the system with linear or cubic parametric excitation. Especially, there exists a controllable frequency, excited at which chaos does not occur via homoclinic bifurcation no matter how large the excitation amplitude is for the system with cubic parametric excitation. Some complicated dynamical behaviors are obtained for this class of systems.     

Codimension-two bursting analysis in the delayed neural system with external stimulations

In the neural system, action potentials play a crucial role in many mechanisms of information communication. The quiescent state, spiking and bursting activities are important biological behaviors with the different neurocomputational properties. In this paper, based on the bifurcation mechanisms involved in the generation of action potentials, an interesting mathematical study of bursting behavior is obtained. The transition between the bursting and quiescence state is investigated,which shows that the time delay must be large enough for bursting behavior to occur in a delayed system. Two types of the codimension-two bifurcation, i.e., Bogdanov–Takens (BT) bifurcation and saddle-node homoclinic (SNH) bifurcation are investigated also. The bifurcation curves of the parameters and the phase portraits for the different regions are shown. The local existence of the homoclinic curve is achieved by using the center manifold reduction and normal form method. For occurrence of a periodic stimulation in the neighborhood of the SNH bifurcation, the system can switch over from an equilibrium state to an oscillatory state either through saddle-node on an invariant circle bifurcation (called circle bifurcation for simplicity) or saddle-node (SN) bifurcation, and back from the oscillatory state to the equilibrium state through the circle or homoclinic bifurcation. Complex bursting phenomena are displayed for the different values of delay couplings and stimulation intensities. Some types of bursting behaviors, such as Circle/Circle (Type II or parabolic bursting), Circle/Homoclinic, SN/Circle (triangular bursting), SN/Homoclinic (Type I or square-wave bursting), and Fold/Hopf bursting are obtained in the firing area. The results show that the different burstings are related to the delay coupling and external inputs.     

Predicting Homoclinic Bifurcations in Planar Autonomous Systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.     

电磁场下神经元模型中混合簇的分岔机制

Pre-B?tzinger复合体是新生哺乳动物呼吸节律起源的关键部位, 是呼吸节律产生的中枢. 忆阻器的功能类似于神经元突触的可塑性, 可用其模拟磁通量.本文在Butera动力学模型的基础上引入刺激电流和磁通控制忆阻器, 分别研究这两个因素对单个pre-B?tzinger复合体神经元中混合簇放电模式的影响.通过无量纲化的方法对变量进行时间尺度分析, 结果表明, 模型包含3个不同的时间尺度.通过快慢分解和分岔分析研究了神经元混合簇放电产生和转迁的动力学机制.电流和磁通量都可以影响混合簇中胞体簇的个数, 减小电流和磁通量的值, 混合簇中胞体簇的个数也会相应减少, 并使簇的类型由"fold/homoclinic"型簇放电转迁为经由"fold/homoclinic"滞后环的"Hopf/Hopf"型簇放电.双参数分岔分析表明, 随着钙离子浓度的逐渐增加, 全系统轨线在鞍结分岔曲线和同宿轨分岔曲线之间来回跃迁, 是混合簇的产生分岔机制.全系统轨线在鞍结分岔曲线和同宿轨分岔曲线之间跃迁的次数, 与混合簇中胞体簇的个数相对应.      

Lorenz type attractors from codimension one bifurcation

Lorenz type attractors are found from a codimension one bifurcation of a system on the boundary of Morse-Smale systems. Conditions of their emerging are formulated in terms of conventional Floquet exponents of homoclinic orbits—a new characteristic of homoclinic orbits at the bifurcation moment.     

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.     

Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.     

Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Concerns double homoclinic loops with orbit flips and two resonant eigen- values in a four-dimensional system.We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium,and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops,then compose them to get the important Poincarémap.A simple cal- culation gives explicitly an expression of the associated successor function.By a delicate analysis of the bifurcation equation,we obtain the condition that the original double homoclinic loops are kept,and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface,2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions.We also locate the large periodic orbits and large homoclinic orbits and their number.