Homoclinic Bifurcations in Simple Parametrically Driven Systems

In this paper we consider the onset of chaotic particle motion in a perturbed Morse potential and the homoclinic bifurcations in a parametrically driven Lorenz system. The Melnikov-Keener method is used to derive bifurcation conditions for the parameters of the dynamical systems. For some selected parameter values the theoretical predictions are checked by numerical experiments.     

Activation barrier scaling and crossover for noise-induced switching in micromechanical parametric oscillators

We explore fluctuation-induced switching in parametrically driven micromechanical torsional oscillators. The oscillators possess one, two, or three stable attractors depending on the modulation frequency. Noise induces transitions between the coexisting attractors. Near the bifurcation points, the activation barriers are found to have a power law dependence on frequency detuning with critical exponents that are in agreement with predicted universal scaling relationships. At large detuning, we observe a crossover to a different power law dependence with an exponent that is device specific.     

Subharmonic and homoclinic bifurcations in a parametrically forced pendulum

Depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the Melnikov method and of averaging methods. It is shown that, as a parameter is varied, repeated resonances of successively higher periods occur culminating in homoclinic orbits. According to the theorem of Smale homoclinic bifurcation is the source of the unstable chaotic motions observed. For some selected parameter sets the theoretical predictions are tested by numerical calculations. Very good agreement is found between analytical and numerical results.     

Recognition of resonance type in periodically forced oscillators

This paper deals with families of periodically forced oscillators undergoing a Hopf-Ne?marck-Sacker bifurcation. The interest is in the corresponding resonance sets, regions in parameter space for which subharmonics occur. It is a classical result that the local geometry of these sets in the non-degenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our main contribution is providing corresponding recognition conditions, that determine to which of these cases a given family of periodically forced oscillators corresponds. The conditions are constructed from known results for families of diffeomorphisms, which in the current context are given by Poincaré maps. Our approach also provides a skeleton for the local resonant Hopf-Ne?marck-Sacker dynamics in the form of planar Poincaré-Takens vector fields. To illustrate our methods two case studies are included: A periodically forced generalized Duffing-Van der Pol oscillator and a parametrically forced generalized Volterra-Lotka system.     

Pattern formation in forced reaction diffusion systems with nearly degenerate bifurcations

The existence and stability of stable standing-wave patterns in an assembly of spatially distributed generic oscillators governed by a couple of complex Ginzburg-Landau equations, subjected to parametric forcing, are reported. The mechanism of a dispersion-induced pattern in dissipative oscillators parametrically forced near the degenerate Turing-Hopf bifurcation is also illustrated. We show that, when excitation occurs just after the Turing bifurcation and before the Hopf instability, the system exhibits a new type of stable standing-wave structures, namely the mixed-mode solutions. The Brussellator-model, parametrically forced below the threshold of oscillations, is analyzed as an example of calculation.     

Routes to bursting in a periodically driven oscillator

The dynamical behaviors of a periodic excited oscillator with multiple time scales in the form that order gap exists between the frequency of the excitation and the natural frequency, are investigated in this Letter. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamics. Different types of bursting phenomena, such as fold/Hopf bursting, fold/Hopf/homoclinic bursting and Hopf/homoclinic bursting, are presented, the mechanism of which is obtained based on the bifurcations of the generalized autonomous system as well as the introduction of the so-called transformed phase portraits. Furthermore, the evolution of the bursting is discussed in details, in which one may find that when the two limit cycles caused by the Hopf bifurcations of the two related equilibrium points interact with each other, homoclinic bifurcation may occur, leading to the merge of the two cycles to form a large amplitude cycle. The homoclinic bifurcation may cause the two asymmetric bursters to merge into a symmetric enlarged burster, in which the large amplitude of the spiking state agrees well with the amplitude of the cycle caused by the homoclinic bifurcation.     

Chaotic dynamics of a whirling pendulum

A physical system is considered consisting of a rigid frame which is free to rotate about a vertical axis and to which is attached a planar simple pendulum. This system has “one and a half” degrees of freedom due to the fact that the frame and pendulum may freely rotate about the vertical axis, i.e., conservation of angular momentum holds for the “ideal”, or unperturbed, system. Using a Hamiltonian formulation we reduce the unperturbed equations of motion to a conservative planar system in which the constant angular momentum plays the role of a parameter. This system is shown to possess one or two sets of homoclinic motions depending on the level of the angular momentum. When this system is perturbed by external excitations and dissipative forces these homoclinic motions can break into homoclinic tangles providing the conditions for chaotic motions of the horseshoe type to exist. The criteria for this to occur can be formulated using a variation of Melnikov's method developed for slowly varying oscillators [1, 2]. For the present problem, the angular momentum becomes a slowly varying parameter upon addition of the disturbances. These ideas are used to rigorously prove the existence of chaotic motions for this system and to compute, to first order, global bifurcation parameter conditions. Since two types of homoclinic motions can occur, two different chaotic modes of motion can result and physical interpretations of these motions are given. In addition, a limiting case is considered in which the system becomes a single degree of freedom oscillator with parametric excitation.     

耦合非线性振子系统的同步研究

研究了考虑振子振幅效应的耦合极限环系统的同步.研究表明，耦合极限环系统的序参量随耦合强度的增加呈现非单调变化，并且出现若干不可微的点；平均频率随耦合强度的变化过程表现为同步分岔树结构；在临界点处出现了相速度的滑移、锁定和相速度差的开关阵发现象，开关阵发的平均周期具有很好的标度关系；振子的平均振幅随相同步的进程实际上是由均匀化逐渐分岔而达到非均匀化的过程，振子振幅的变化范围在临界点处突然减小. 关键词： 耦合极限环系统 同步 振幅效应     

脉冲激励下环形耦合Duffing振子间的瞬态同步突变现象

研究非周期信号激励下Duffing振子动力学行为变化特征时,发现处于倍周期分岔的环形耦合Duffing振子系统,在一定的参数条件下,脉冲信号能引起其中一个振子与其他振子运动轨迹间出现短暂失同步的现象即瞬态同步突变现象.利用这种现象可以快速检测出强噪声背景中的微弱脉冲信号,从而扩展了现有的Duffing振子对非周期信号的检测范围及应用领域. 关键词： 瞬态同步突变 微弱信号检测 脉冲信号 Duffing振子     

Characterizing strange nonchaotic attractors

Strange nonchaotic attractors typically appear in quasiperiodically driven nonlinear systems. Two methods of their characterization are proposed. The first one is based on the bifurcation analysis of the systems, resulting from periodic approximations of the quasiperiodic forcing. Second, we propose to characterize their strangeness by calculating a phase sensitivity exponent, that measures the sensitivity with respect to changes of the phase of the external force. It is shown that phase sensitivity appears if there is a nonzero probability for positive local Lyapunov exponents to occur. (c) 1995 American Institute of Physics.     

Šilnikov manifolds in coupled nonlinear Schrödinger equations

We consider two coupled nonlinear Schrödinger equations with even, periodic boundary conditions, that are damped and quasiperiodically forced. We prove the existence of invariant manifolds with Šilnikov-type dynamics that are homoclinic to a spatially independent invariant torus. Such manifolds appear to induce complex behavior in numerical experiments.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

Oscillation death in a coupled van der Pol–Mathieu system

We report an investigation of the oscillation death (OD) of a parametrically excited coupled van der Pol–Mathieu (vdPM) system. The system can be considered as a pair of harmonically forced van der Pol oscillators under a double-well potential. The two oscillators are coupled with a cubic nonlinearity. We have shown that the system arrives at an OD regime when coupling strength crosses a threshold value at which the system undergoes saddle-node bifurcation and two limit cycles coalesce onto a fixed point of the system. We have further shown that this nonautonomous system possesses a centre manifold corresponding to the OD regime.     

Critical curves and coexisting attractors in a quasiperiodically forced delayed system

We study three critical curves in a quasiperiodically driven system with time delays, where occurrence of symmetry-breaking and symmetry-recovering phenomena can be observed. Typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished. A striking phenomenon that can be discovered is multistability and coexisting attractors in some tongues surrounding by critical curves. The blowout bifurcation accompanying with on-off intermittency can also be observed. We show that collision of attractors at a symmetric invariant subspace can lead to the appearance of symmetry-breaking.     

环形耦合Duffing振子间的同步突变

以环形耦合Duffing振子系统为研究对象,分析了耦合振子间的同步演化过程.发现在弱耦合条件下,如果所有振子受到同一周期策动力的驱动,那么系统在经历倍周期分岔、混沌态、大尺度周期态的相变时,各振子的运动轨迹之间将出现由同步到不同步再到同步的两次突变现象.利用其中任何一次同步突变现象可以实现系统相变的快速判别,并由此补充了利用倍周期分岔与混沌态的这一相变对微弱周期信号进行检测的方法. 关键词： Duffing振子 同步突变 相变 微弱信号检测     

Slow passage through a transcritical bifurcation for Hamiltonian systems and the change in action due to a nonhyperbolic homoclinic orbit

One-degree of freedom conservative slowly varying Hamiltonian systems are analyzed in the case in which a saddle-center pair undergo a transcritical bifurcation. We analyze the case in which the method of averaging predicts the solution crosses the unperturbed homoclinic orbit at the precise time at which the transcritical bifurcation occurs. For the slow passage through the nonhyperbolic homoclinic orbit associated with a transcritical bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits surrounded by autonomous nonlinear saddle approaches. The change in action is computed by matching these solutions to those obtained by averaging, valid before and after crossing the nonhyperbolic homoclinic orbit. For initial conditions near the stable manifold of the nonhyperbolic saddle point, one saddle approach has particularly small energy and instead satisfies a nonautonomous nonlinear equation, which provides a transition between nonhyperbolic homoclinic orbits, centers, and saddles. (c) 2000 American Institute of Physics.     

Damping performance of two simple oscillators coupled by a visco-elastic connection

A simple dynamic system composed of two linear oscillators is employed to analyze the passive control performance that can be achieved through a visco-elastic damper connecting two adjacent free-standing structures. By extension, the model may also describe the energy dissipation which can be obtained by an internal coupling between two quasi-independent sub-systems composing a single complex structure. Two alternatives are evaluated for the linear coupling by considering either the serial or the parallel spring–dashpot arrangement known as the Kelvin–Voigt and the Maxwell damper model, which may synthetically reproduce the constitutive behavior of different industrial devices. The complex eigenvalues of the coupled system are parametrically analyzed to determine the potential benefits realized by different combinations of the coupling stiffness and damping coefficient. A design strategy to assess these parameters is outlined, driven by the relevant observation that a perfect tuning of the natural frequencies always corresponds, in the parameter space, to the maximum modal damping for one of the resonant modes, independent of the damper model. The effectiveness of the proposed strategy is discussed for different classes of the controlled system, depending on the mass and stiffness ratio of the component oscillators. As a major result, different design parameter charts for the two damper models are carried out and compared to each other. Performance indexes are introduced to quantitatively evaluate the passive control performance with respect to the mitigation of the system forced response under harmonic and seismic ground excitation. The analyses confirm the validity of the design strategy for a well-balanced mitigation of the displacement and acceleration response in both the oscillators.     

Non-smooth bursting analysis of a Filippov-type system with multiple-frequency excitations

The main purpose of this paper is to explore the patterns of the bursting oscillations and the non-smooth dynamical behaviours in a Filippov-type system which possesses parametric and external periodic excitations. We take a coupled system consisting of Duffing and Van der Pol oscillators as an example. Owing to the existence of an order gap between the exciting frequency and the natural one, we can regard a single periodic excitation as a slow-varying parameter, and the other periodic excitations can be transformed as functions of the slow-varying parameter when the exciting frequency is far less than the natural one. By analysing the subsystems, we derive equilibrium branches and related bifurcations with the variation of the slow-varying parameter. Even though the equilibrium branches with two different frequencies of the parametric excitation have a similar structure, the tortuousness of the equilibrium branches is diverse, and the number of extreme points is changed from 6 to 10. Overlying the equilibrium branches with the transformed phase portrait and employing the evolutionary process of the limit cycle induced by the Hopf bifurcation, the critical conditions of the homoclinic bifurcation and multisliding bifurcation are derived. Numerical simulation verifies the results well.     

Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays

Coupling delays may cause drastic changes in the dynamics of oscillatory networks. In the present paper we investigate how coupling delays alter synchronization processes in networks of all-to-all coupled pulse oscillators. We derive an analytic criterion for the stability of synchrony and study the synchronization areas in the space of the delay and coupling strength. Specific attention is paid to the scenario of destabilization on the borders of the synchronization area. We show that in bifurcation points the system possesses homoclinic loops, which give rise to complex long- or quasi-periodic solutions. These newly born solutions are characterized by a synchronous group, from which an oscillator periodically escapes, laps one period, and rejoins. We call such a dynamical regime “phase slip patterns”.     

Synchronization of diffusively coupled oscillators near the homoclinic bifurcation

It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak-coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram using the bifurcation analysis.