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.     

非线性切换系统的动力学行为分析

建立了周期切换下的非线性电路模型,基于子系统平衡点及其稳定性分析,分别给出了其相应的fold分岔和Hopf分岔条件,讨论了子系统在不同平衡态下由周期切换导致的各种复杂行为,指出切换系统的周期解随参数的变化存在着倍周期分岔和鞍结分岔两种失稳情形,并相应地导致不同的混沌振荡,进而结合系统轨迹及其相应的分岔分析,揭示了各种振荡模式的动力学机理. 关键词： 周期切换 倍周期分岔 鞍结分岔 混沌     

混沌吸引子的对称破缺激变

许多非线性动力系统都有某种对称性,在不同情形下可有不同的表现形式,但始终保持其对称的特点.不同对称形式间的转变导致对称破缺分岔或激变.关于非线性动力系统中相空间运动轨道的对称破缺分岔,已有大量研究工作,但绝大多数是指周期或拟周期相轨的对称破缺,偶尔提到对称系统中的混沌相轨也存在“对偶性”.最近,在简谐外激Duffing系统周期轨道对称破缺引发鞍-结分岔的研究中,得到了分岔后由Poincaré映射点间断流构成的图像,其中包括两个稳定周期结点、一个周期鞍点,及其稳定流形与不稳定流形,均较规则.本工作研究了正弦 关键词： 对称破缺 混沌 激变 分形吸引域     

Quantitative calculation and bifurcation analysis of periodic solutions in a driven Josephson junction including interference current

The calculation scheme for periodic solutions in an rf-driven Josephson junction including interference current is derived by using the incremental harmonic balance method. The approximate analytical expressions of stable and unstable periodic orbits are obtained. The stability and bifurcation of the periodic solutions are analyzed based on Floquet theory. The results show that, with the increase of the driving amplitude, one of the periodic solutions undergoes symmetry-breaking and period-doubling bifurcation, which leads to chaos eventually. However, the other periodic solution of the system disappears via a saddle-node bifurcation.     

A normal form for excitable media

We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.     

Delay control of coherence resonance in type-I excitable dynamics

We consider a simple paradigmatic system of type-I excitability subject to noise and time-delayed feedback. This system is governed by a global bifurcation, namely a saddle-node bifurcation on a limit cycle. In the absence of noise, delay can induce complex dynamics including multiple stable and unstable periodic orbits. Random fluctuations result in coherence resonance in dependence on the noise strength. We show that this effect can be enhanced by delayed feedback control with suitably chosen feedback strength and time delay.     

From multi-layered resonance tori to period-doubled ergodic tori

The Letter presents a number of new bifurcation structures that can be observed when a multi-dimensional period-doubling system is subjected to a periodic forcing. We show how multi-layered tori arise through transverse period-doubling bifurcations of the resonant saddle and node cycles, and how these multi-layered tori transform into period-doubled ergodic tori through sets of saddle-node bifurcations.     

Bifurcation analysis of a periodically forced pair of tubular catalytic combustors

This paper presents a numerical study of the dynamics of a reactor network (RN) made of a pair of tubular catalytic combustors. The RN is forced with a periodic change of the feed position, which emulates a moving bed. The distributed model is discretized and the resulting, rather large, dynamical system is studied via bifurcation analysis of a proper discrete system, related to its Poincaré map through spatiotemporal symmetry. The analysis is made possible by parallel computation. An operating parameter, the switch time, is chosen as the bifurcation parameter. A wide operation region of high-conversion periodic regimes is found, delimited by two saddle-node bifurcations. The influence of the heat capacity of the catalyst phase is also reported and discussed. Particularly, higher heat capacity corresponds to wider stable regions of operation. In the low range of the switch time complex dynamical regimes are detected, including symmetric and non symmetric spatiotemporal patterns.     

Beyond the odd number limitation of time-delayed feedback control of periodic orbits

We discuss the stabilization of odd-number orbits by time-delayed feedback control. In particular, we review the stabilization of odd-number orbits born in a subcritical Hopf bifurcation or a saddle-node bifurcation of periodic orbits. These examples refute the often invoked odd-number theorem.     

Modulated amplitude waves and the transition from phase to defect chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures evolve towards defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.     

AN ANALYTICAL AND NUMERICAL STUDY OF A MODIFIED VAN DER POL OSCILLATOR

A three-dimensional system of differential equations that models an electronic oscillator is considered. The equations allow a variety of periodic orbits that originate from a degenerate Hopf bifurcation, which is analytically studied. Numerical results are presented that show the existence of saddle-node cusps of periodic orbits, as well as period-doubling bifurcations, that result in the coexistence of multiple “canard” orbits if one of the parameters is small. The presence of chaotic attractors is also detected.     

At the other side of a saddle-node

We describe phenomena occurring just before a saddle-node bifurcation for one-parameter families of interval maps. In particular, as a parameter approaches the bifurcation value, attracting periodic orbits of periodsk, k+1,k+2,k+3,... can appear. We make a detailed study of a family of cusp-shaped maps, where this phenomenon occurs in a pure form.     

Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe

We study a continuous and reversible transition between periodic tonic spiking and bursting activities in a neuron model. It is described as the blue-sky catastrophe, which is a homoclinic bifurcation of a saddle-node periodic orbit of codimension one. This transition constitutes a biophysically plausible mechanism for the regulation of burst duration that increases with no bound like 1/square root alpha-alpha0 as the transition value alpha0 is approached.     

Tracking unstable steady states and periodic orbits of oscillatory and chaotic electrochemical systems using delayed feedback control

Experimental results are presented on successful application of delayed-feedback control algorithms for tracking unstable steady states and periodic orbits of electrochemical dissolution systems. Time-delay autosynchronization and delay optimization with a descent gradient method were applied for stationary states and periodic orbits, respectively. These tracking algorithms are utilized in constructing experimental bifurcation diagrams of the studied electrochemical systems in which Hopf, saddle-node, saddle-loop, and period-doubling bifurcations take place.     

Travelling waves associated with saddle-node bifurcation in weakly coupled CML

Weakly coupled CML can be analytically solved by using perturbative methods. This technique has been recently used to deduce analytical expressions for travelling waves. Nonetheless, the results were limited for periodic solutions far away from saddle-node bifurcation. In this Letter, this problem is solved and periodic solutions, arising from the individual dynamics, are totally characterised.     

Subharmonic bifurcations and chaotic dynamics of an air damping completely inelastic bouncing ball

We investigate the dynamics of a plastic ball on a vibrated platform in air by introducing air damping effect into the completely inelastic bouncing ball model. The air damping gives rise to larger saddle-node bifurcation points and a chaos confirmed by the largest Lyapunov exponent of a one-dimensional discrete mapping. The calculated bifurcation point distribution shows that the periodic motion of the ball is suppressed and a chaos emerges earlier for an increasing air damping. When the reset mechanism and the linear stability which cause periodic motion of the ball both collapse, the investigated system is fully chaotic.     

脉冲作用下Chen系统的非光滑分岔分析

研究了脉冲作用下Chen系统的复杂动力学行为. 对脉冲作用下的Chen系统进行了非光滑分岔分析. 该系统可经级联倍周期分岔到达混沌, 也可由周期解经鞍结分岔直接到达混沌. 最后通过Floquet理论揭示了该系统周期解的非光滑分岔机理.     

Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of limit cycle oscillations (LCO) of a two-degree-of-freedom aeroelastic system containing a freeplay structural nonlinearity. Both stable and unstable LCOs can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Thus, the present method is capable of detecting complex aeroelastic responses such as periodic motion with harmonics, period-doubling (PD), saddle-node bifurcation, Neimark-Sacker bifurcation and the coexistence of limit cycles. Emanating branch from a PD bifurcation can be constructed. This method can also be applied to any piecewise linear systems.     

Symmetry breaking bifurcations in a circular chain of N coupled logistic maps

A circular chain of N cells with logistic dynamics, coupled together with symmetric nearest neighbor coupling and periodic boundary conditions is investigated. For certain coupling parameters we observe bifurcation of a stable state into two types of period two solutions. By using the symmetry of this Coupled Map Lattice model, we show that the bifurcated system only can have periodic solutions with symmetry group corresponding to certain subgroups of the full symmetry group of the system.