Effect of various periodic forces on duffing oscillator

Bifurcations and chaos in the ubiquitous Duffing oscillator equation with different external periodic forces are studied numerically. The external periodic forces considered are sine wave, square wave, rectified since wave, symmetric saw-tooth wave, asymmetric saw-tooth wave, rectangular wave with amplitude-dependent width and modulus of sine wave. Period doubling bifurcations, chaos, intermittency, periodic windows and reverse period doubling bifurcations are found to occur due to the applied forces. A comparative study of the effect of various forces is performed.     

Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs

The paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n-dimensional ODEs. We discuss the implementation and the performance of the algorithm in several examples, including an extended Lorenz-84 model and a laser system.     

On a special type of border-collision bifurcations occurring at infinity

In piecewise-smooth dynamical systems, the regions of existence of a periodic orbit are typically parameter sub-spaces confined by border-collision bifurcations of this orbit. We demonstrate that additionally to the usual border-collision bifurcations occurring at finite points in the state space there exist also border-collision bifurcations occurring at infinity.     

Analysis of grazing bifurcations of quasiperiodic system attractors

This paper presents the first application of the discontinuity-mapping approach to the study of near-grazing bifurcations of originally quasiperiodic, co-dimension-two system attractors. The paper establishes an exact formulation for the discontinuity-mapping methodology under the assumption that a Poincaré section can be found that is everywhere transversal to the grazing attractor. In particular, it is shown that, while a reduced formulation may be employed successfully in the case of co-dimension-one attractors, it fails to capture dynamics in directions transversal to the original quasiperiodic attractor. This shortcoming necessitates the full machinery presented here. The generality of the proposed approach is illustrated through numerical analysis of two nonlinear dynamical systems of dimension three and four.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

Localized bifurcations and defect instabilities in the convection of a nematic liquid crystal

The stationary and the time-dependent homogeneous ordered states in convection may both become unstable against localized perturbations. Defects are then created and they may contribute to the disorganization of the homogeneous state. We present an experimental study of defects in some homogeneous stationary structures as well as in the traveling-wave states of convection of a nematic liquid crystal. We show that the core of the defects is a germ of the unstable state and it can become unstable under the external stress. Then, either fully homogeneous states with the symmetry of the core, or complex disordered states can develop from the local instability of defects in processes quite similar to displacive transitions in solids. Some of the main features are qualitatively similar to numerical simulations of an appropriate Landau-Ginzburg equation.     

厄尔尼诺-南方涛动时滞海气振子耦合模型的周期解

运用重合度理论探讨了一类非线性问题的周期解的存在性.然后将其应用于一个厄尔尼诺-南方涛动时滞海气振子耦合模型周期解问题的研究,获得到了该模型存在周期解的新结果. 关键词： 非线性 厄尔尼诺-南方涛动 周期解     

Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems

A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.     

Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of ann-cycle (n > 1) approach the discontinuities of thenth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.     

Duffing振子系统周期解的唯一性与精确周期信号的获取方法

研究Duffing振子系统的周期解的唯一性与精确周期信号的获取方法.应用定性分析方法,获得了一类Duffing振子系统具有唯一周期解的必要条件,同时也得到了一类更广泛的非线性周期系统的周期解的唯一性.在一定条件下,给出了Duffing振子系统精确周期信号的获取方法。     

Non-smooth bifurcations in a double-scroll circuit with periodic excitation

The fast-slow effect can be observed in a typical non-smooth electric circuit with order gap between the natural frequency and the excitation frequency. Numerical simulations are employed to show complicated behaviours, especially different types of busting phenomena. The bifurcation mechanism for the bursting solutions is analysed by assuming the forms of the solutions and introducing the generalized Jacobian matrix at the non-smooth boundaries, which can also be used to account for the evolution of the complicated structures of the phase portraits with the variation of the parameter. Period-adding bifurcation has been explored through the computation of the eigenvalues related to the solutions. At the non-smooth boundaries the so-called `single crossing bifurcation' can occur, corresponding to the case where the eigenvalues jump only once across the imaginary axis, which leads the periodic burster to have a quasi-periodic oscillation.     

Stochastic analysis of symmetry-breaking bifurcations: Master equation approach

The multivariate master equation for a general reaction-diffusion system is solved perturbatively, in the vicinity of a bifurcation point leading to symmetry-breaking transitions. The possibility to express the result through a Brazovskii type of potential is examined, and a comparison with the Langevin analysis of Walgraefet al. [Adv. Chem. Phys. 49:311 (1982)] is performed.     

Dynamics and stability of a new class of periodic solutions of the optical parametric oscillator

The stability and dynamics of a new class of periodic solutions is investigated when a degenerate optical parametric oscillator system is forced by an external pumping field with a periodic spatial profile modeled by Jacobi elliptic functions. Both sinusoidal behavior as well as localized hyperbolic (front and pulse) behavior can be considered in this model. The stability and bifurcation behaviors of these transverse electromagnetic structures are studied numerically. The periodic solutions are shown to be stabilized by the nonlinear parametric interaction between the pump and signal fields interacting with the cavity diffraction, attenuation, and periodic external pumping. Specifically, sinusoidal solutions result in robust and stable configurations while well-separated and more localized field structures often undergo bifurcation to new steady-state solutions having the same period as the external forcing. Extensive numerical simulations and studies of the solutions are provided.     

Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems

This paper presents a unified framework for performing local analysis of grazing bifurcations in n-dimensional piecewise-smooth systems of ODEs. These occur when a periodic orbit has a point of tangency with a smooth (n−1)-dimensional boundary dividing distinct regions in phase space where the vector field is smooth. It is shown under quite general circumstances that this leads to a normal-form map that contains to lowest order either a square-root or a (3/2)-type singularity according to whether the vector field is discontinuous or not at the grazing point. In particular, contrary to what has been reported in the literature, piecewise-linear local maps do not occur generically. First, the concept of a grazing bifurcation is carefully defined using appropriate non-degeneracy conditions. Next, complete expressions are derived for calculating the leading-order term in the normal form Poincaré map at a grazing bifurcation point in arbitrary systems, using the concept of a discontinuity mapping. Finally, the theory is compared with numerical examples including bilinear oscillators, a relay feedback controller and general third-order systems.     

Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network

Recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators [R. Curtu, A. Shpiro, N. Rubin, J. Rinzel, Mechanisms for frequency control in neuronal competition models, SIAM J. Appl. Dyn. Syst. 7 (2) (2008) 609-649] showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics. Several types of behavior were identified such as: fusion, equivalent to a steady state of identical activity levels for both neural units; oscillations due to either an escape or a release mechanism; and a winner-take-all state of bistability. The model consists of two neural populations interacting through reciprocal inhibition, each endowed with a slow negative-feedback process in the form of spike frequency adaptation. In this paper we report the occurrence of another complex oscillatory pattern, the mixed-mode oscillations (MMOs). They exist in the model at the transition between the relaxation oscillator dynamical regime and the winner-take-all regime. The system distinguishes itself from other neuronal models where MMOs were found by the following interesting feature: there is no autocatalysis involved (as in the examples of voltage-gated persistent inward currents and/or intrapopulation recurrent excitation) and therefore the two cells in the network are not intrinsic oscillators; the oscillations are instead a combined result of the mutual inhibition and the adaptation. We prove that the MMOs are due to a singular Hopf bifurcation point situated in close distance to the transition point to the winner-take-all case. We also show that in the vicinity of the singular Hopf other types of bifurcations exist and we construct numerically the corresponding diagrams.     

器壁滑动摩擦力对受振颗粒体系中冲击力倍周期分岔过程的影响

颗粒物质由离散的固体颗粒组成, 受到周期性振动时可以表现出复杂的动力学行为. 这些行为往往受众多因素的影响, 如空气阻力和器壁摩擦力等. 针对受振颗粒体系中冲击力的倍周期分岔现象, 通过抽真空或将容器底镂空消除空气阻力, 单独研究器壁滑动摩擦力的影响. 结果表明在仅有器壁摩擦力作用的情况下, 倍周期分岔过程仅受约化振动加速度的控制, 与颗粒的尺寸、颗粒层数及振动频率无关. 将器壁摩擦力处理成一个大小恒定、方向与颗粒和器壁相对速度反向的阻力, 并包含到完全非弹性蹦球模型中, 能够对所观察到的现象给出很好的解释. 通过对倍周期分岔点测量平均值的拟合, 得到器壁滑动摩擦力的大小约为颗粒总重量的10%. 关键词： 颗粒物质 器壁摩擦力 倍周期分岔 冲击力     

Stability and bifurcations of a stationary state for an impact oscillator

The motion of a vibroimpacting one-degree-of-freedom model is analyzed. This model is motivated by the behavior of a shearing granular material, in which a transitional phenomenon is observed as the concentration of the grains decreases. This transition changes the motion of a granular assembly from an orderly shearing between two blocks sandwiching a single layer of grains to a chaotic shear flow of the whole granular mass. The model consists of a mass-spring-dashpot assembly that bounces between two rigid walls. The walls are prescribed to move harmonically in opposite phases. For low wall frequencies or small amplitudes, the motion of the mass is damped out, and it approaches a stationary state with zero velocity and displacement. In this paper, the stability of such a state and the transition into chaos are analyzed. It is shown that the state is always changed into a saddle point after a bifurcation. For some parameter combinations, horseshoe-like structures can be observed in the Poincare sections. Analyzing the stable and unstable manifolds of the saddle point, transversal homoclinic points are found to exist for some of these parameter combinations. (c) 1994 American Institute of Physics.     

Avalanches in a nonlinear oscillator chain in a periodic potential

We consider the dynamics of a chain of coupled units evolving in a periodic substrate potential. The chain is initially in a flat state and situated in a potential well. A bias force, acting as a weak driving mechanism, is applied at a single unit of the chain. We study the instigation of directed transport in two types of system: (i) a microcanonical situation associated with deterministic and conservative dynamics and (ii) the Langevin dynamics when the system is in contact with a heat bath. Interestingly, for the deterministic and conservative dynamics the directed transport is drastically enhanced compared with its Langevin counterpart. In particular, in the deterministic and conservative regime a self-organised redistribution of energy triggers huge-sized avalanches yielding ultimately accelerated transport of the chain. In contrast, in the thermally-assisted process between avalanches the chain settles always into a pinned metastable state impeding continual accelerated chain motion.