On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrödinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing.We give sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform.This analysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.     

Insight into Phenomena of Symmetry Breaking Bifurcation

We show that symmetry-breaking (SB) bifurcation is just a transition of different forms of symmetry, while still preserving system's symmetry. SB bifurcation always associates with a periodic saddle-node bifurcation, identifiable by a zero maximum of the top Lyapunov exponent of the system. In addition, we show a significant phase portrait of a newly born periodic saddle and its stable and unstable invariant manifolds, together with their neighbouring flow pattern of Poincaré mapping points just after the periodic saddle-node bifurcation, thus gaining an insight into the mechanism of SB bifurcation.     

Nonlinear Saffman-Taylor instability

We show, both theoretically and experimentally, that the interface between two viscous fluids in a Hele-Shaw cell can be nonlinearly unstable before the Saffman-Taylor linear instability point is reached. We identify the family of exact elastica solutions [Nye et al., Eur. J. Phys. 5, 73 (1984)]] as the unstable branch of the corresponding subcritical bifurcation which ends up at a topological singularity defined by interface pinchoff. We devise an experimental procedure to prepare arbitrary initial conditions in a Hele-Shaw cell. This is used to test the proposed bifurcation scenario and quantitatively asses its practical relevance.     

Coexisting tori and torus bubbling in non-smooth systems

Pulse modulated power electronic converters represent an important class of piecewise-smooth dynamical systems with a broad range of applications in modern power supply systems. The paper presents a detailed investigation of a number of unusual bifurcation phenomena that can occur in power converters with multilevel control. In the first example a closed invariant curve arises in a border-collision bifurcation as a period-6 saddle cycle collides with a stable fixed point of focus type and transforms it into an unstable focus point. The second example involves the formation of a structure of coexisting tori through the interplay between border-collision and global bifurcations. We examine the behavior of the system in the presence of two coexisting stable resonance tori and finally show how an existing torus can develop heteroclinic bubbles that connect the points of a stable resonance cycle with an external pair of saddle and focus cycles. The appearance of these structures is explained in terms of a sequence torus-birth bifurcations with pairs of stable and unstable tori folding one over the other.     

Control of degenerate Hopf bifurcations in three-dimensional maps

A feedback control method is proposed to create a degenerate Hopf bifurcation in three-dimensional maps at a desired parameter point. The particularity of this bifurcation is that the system admits a stable fixed point inside a stable Hopf circle, between which an unstable Hopf circle resides. The interest of this solution structure is that the asymptotic behavior of the system can be switched between stationary and quasi-periodic motions by only tuning the initial state conditions. A set of critical and stability conditions for the degenerate Hopf bifurcation are discussed. The washout-filter-based controller with a polynomial control law is utilized. The control gains are derived from the theory of Chenciner's degenerate Hopf bifurcation with the aid of the center manifold reduction and the normal form evolution.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

基于延时干扰的感应电能传输系统分岔频率输送控制

本文针对感应电能传输系统分岔频率的输送控制问题, 提出一种基于延时干扰的变轨控制方法. 该方法在反馈控制环节中加入一段延时干扰, 通过调节延时参数, 可使系统相轨迹流在各稳定极限环吸引子间自由切换. 文中以原副边均为串联谐振的感应电能传输系统为例, 对该方法的机理及实现方案进行了研究, 并通过仿真和实验验证了其有效性. 论文的研究结果对类似多吸引子分岔行为的输送控制可提供一定的理论参考. 关键词： 感应电能传输 频率分岔 输送控制 延时干扰     

Surface solitons in trilete lattices

Fundamental solitons pinned to the interface between three semi-infinite one-dimensional nonlinear dynamical chains, coupled at a single site, are investigated. The light propagation in the respective system with the self-attractive on-site cubic nonlinearity, which can be implemented as an array of nonlinear optical waveguides, is modeled by the system of three discrete nonlinear Schrödinger equations. The formation, stability and dynamics of symmetric and asymmetric fundamental solitons centered at the interface are investigated analytically by means of the variational approximation (VA) and in a numerical form. The VA predicts that two asymmetric and two antisymmetric branches exist in the entire parameter space, while four asymmetric modes and the symmetric one can be found below some critical value of the inter-lattice coupling parameter—actually, past the symmetry-breaking bifurcation. At this bifurcation point, the symmetric branch is destabilized and two new asymmetric soliton branches appear, one stable and the other unstable. In this area, the antisymmetric branch changes its character, getting stabilized against oscillatory perturbations. In direct simulations, unstable symmetric modes radiate a part of their power, staying trapped around the interface. Highly unstable asymmetric modes transform into localized breathers traveling from the interface region across the lattice without significant power loss.     

Bifurcation and Stability Analysis of the Equilibrium States in Thermodynamic Systems in a Small Vicinity of the Equilibrium Values of Parameters

The dynamic behavior of thermodynamic system, described by one order parameter and one control parameter, in a small neighborhood of ordinary and bifurcation equilibrium values of the system parameters is studied. Using the general methods of investigating the branching (bifurcations) of solutions for nonlinear equations, we performed an exhaustive analysis of the order parameter dependences on the control parameter in a small vicinity of the equilibrium values of parameters, including the stability analysis of the equilibrium states, and the asymptotic behavior of the order parameter dependences on the control parameter (bifurcation diagrams). The peculiarities of the transition to an unstable state of the system are discussed, and the estimates of the transition time to the unstable state in the neighborhood of ordinary and bifurcation equilibrium values of parameters are given. The influence of an external field on the dynamic behavior of thermodynamic system is analyzed, and the peculiarities of the system dynamic behavior are discussed near the ordinary and bifurcation equilibrium values of parameters in the presence of external field. The dynamic process of magnetization of a ferromagnet is discussed by using the general methods of bifurcation and stability analysis presented in the paper.     

Stability of a detuned single mode homogeneously broadened ring laser

We study analytically the influence of detuning on the properties of a homogeneously broadened single mode ring laser. In the good cavity domain the stationary output is stable. In the bad cavity domain the stationary output becomes unstable via a Hopf bifurcation. Near line center this Hopf bifurcation is subcritical, leading to unstable small amplitude oscillations. Far from line center this Hopf bifurcation is supercritical and leads to stable small amplitude output.     

The effect of Lagrangian chaos on locking bifurcations in shear flows

The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.     

可兴奋性细胞混沌放电区间的识别机理

在神经起步点记录到加周期分岔过程的生理实验数据，在对此分岔过程中位于周期n爆发 和周期(n+1)爆发之间的混沌的峰峰间期数据检测不稳定的周期轨道时，发现从靠近周期 n爆发的混沌的峰峰间期数据中，可以检测出不稳定的周期n轨道；而从靠近周期(n+1)爆 发的混沌的峰峰间期数据中，不仅可以检测出不稳定的周期(n+1)轨道，还可以检测出不稳 定的周期n轨道.针对该现象，借助于Sherman建议的胰腺β细胞模型，从非线性动力 学角度给出了理论解释.指明了由鞍结分岔和倍周期分岔分别产生第一类阵发和第三类阵发 为出现该 关键词： 峰峰间期 不稳定的周期轨道 鞍结分岔 倍周期分岔     

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

Unstable Surface Waves in Running Water

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that the free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves is demonstrated at all wave numbers. We show the linear instability of small nontrivial waves that appear after bifurcation at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of free-surface water waves.     

Generators of quasiperiodic oscillations with three-dimensional phase space

Considering a family of three-dimensional oscillators originating in the field of radio-engineering, the paper describes three different mechanisms of torus formation. Particular emphasis is paid to a process in which a saddle-node bifurcation eliminates a stable cycle and leaves the system to find a stationary state between a saddle cycle and a pair of equilibrium points of unstable focus/stable node and unstable node/stable focus type.     

Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation

This Letter is concerned with bifurcation and chaos control in scalar delayed differential equations with delay parameter τ. By linear stability analysis, the conditions under which a sequence of Hopf bifurcation occurs at the equilibrium points are obtained. The delayed feedback controller is used to stabilize unstable periodic orbits. To find the controller delay, it is chosen such that the Hopf bifurcation remains unchanged. Also, the controller feedback gain is determined such that the corresponding unstable periodic orbit becomes stable. Numerical simulations are used to verify the analytical results.     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

A new hyperchaotic system and its linear feedback control

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter k. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.     

Instability of spatial patterns and its ambiguous impact on species diversity

Self-arrangement of individuals into spatial patterns often accompanies and promotes species diversity in ecological systems. Here, we investigate pattern formation arising from cyclic dominance of three species, operating near a bifurcation point. In its vicinity, an Eckhaus instability occurs, leading to convectively unstable "blurred" patterns. At the bifurcation point, stochastic effects dominate and induce counterintuitive effects on diversity: Large patterns, emerging for medium values of individuals' mobility, lead to rapid species extinction, while small patterns (low mobility) promote diversity, and high mobilities render spatial structures irrelevant. We provide a quantitative analysis of these phenomena, employing a complex Ginzburg-Landau equation.     

Complex patterns in reaction-diffusion systems: A tale of two front instabilities

Two front instabilities in a reaction-diffusion system are shown to lead to the formation of complex patterns. The first is an instability to transverse modulations that drives the formation of labyrinthine patterns. The second is a nonequilibrium Ising-Bloch (NIB) bifurcation that renders a stationary planar front unstable and gives rise to a pair of counterpropagating fronts. Near the NIB bifurcation the relation of the front velocity to curvature is highly nonlinear and transitions between counterpropagating fronts become feasible. Nonuniformly curved fronts may undergo local front transitions that nucleate spiral-vortex pairs. These nucleation events provide the ingredient needed to initiate spot splitting and spiral turbulence. Similar spatiotemporal processes have been observed recently in the ferrocyanide-iodate-sulfite reaction.