The genesis of period-adding bursting without bursting-chaos in the Chay model

According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding “fold/homoclinic” bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to period-7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence.     

一类含五次非线性恢复力的Duffing系统共振与分岔特性分析

考虑一类含有外激力和五次非线性恢复力的Duffing系统,运用多尺度法求解得到该系统的幅频响应方程,给出不同参数变化下的幅频特性曲线及变化规律,同时利用奇异性理论得到该系统在3种情形下的转迁集及对应的拓扑结构.其次确定系统的不动点,运用Hamilton函数给出该系统的异宿轨,在此基础上,利用Melnikov方法得到该系统在Smale马蹄意义下发生混沌的阈值.而后通过数值仿真给出了系统随外激力、五次非线性项系数变化下的动态分岔与混沌行为,发现存在周期运动、倍周期运动、拟周期运动及混沌等非线性现象.最后运用Lyapunov指数、相轨图和Poincaré截面等非线性方法对理论的正确性进行验证.上述研究结论为进一步提升对Duffing系统非线性特性及其演化规律的认识提供了一定的理论参考.     

Omnidirectional walking of legged robots with a failed leg

This paper studies omnidirectional walking of a hexapod robot with a locked joint failure by proposing crab gaits and turning gaits. Due to the reduced workspace of a failed leg, fault-tolerant gaits have limitations in their mobility. As for crab gaits, an accessible range of the crab angle is derived for a given configuration of the failed leg. As for turning gaits, the conditions on turning trajectories guaranteeing fault tolerance are derived for spinning gaits and circling gaits. Based on the principles of fault-tolerant gait planning, periodic crab gaits and turning gaits are proposed in which a hexapod robot realizes tripod walking after a locked joint failure, having a reasonable stride length and stability margin. The proposed fault-tolerant gaits are then applied to an obstacle avoidance problem of a hexapod robot with a locked joint failure. The kinematic constraints of fault-tolerant gaits should be considered in planning the robot trajectory.     

Bifurcation of Periodic Orbits and Chaos in Duffing Equation

Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.     

Transitions to chaos in squeeze-film dampers

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

Nonlinear responses of a rub-impact overhung rotor

For a rotor system with bearings and step-diameter shaft in the oxygen pump of an engine, the contact between the rotor and the case is considered, and the chaotic response and bifurcation are investigated. The system is divided into elements of elastic support, shaft and disk, and based on the transfer matrix method, the motion equation of the system is derived, and solved by Newmark integration method. It is found that hardening the support can delay the occurrence of chaos. When rubbing begins, the grazing bifurcation will cause periodic motion to become quasi-period. With variation of system parameters, such as rotating speed, imbalance and external damping, chaotic response can be observed, along with other complex dynamics such as period- doubling bifurcation and torus bifurcation in the response.     

The genesis of period-adding bursting without bursting-chaos in the Chay model

According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding “fold/homoclinic” bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to 7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence.     

Complex dynamics in a two-dimensional noninvertible map

A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors.     

Diffusive Lorenz dynamics: Coherent structures and spatiotemporal chaos

In this paper, we are interested in collective behaviors of many interacting Lorenz strange attractors. With an intermediate diffusion coupling between the attractors, a new remarkable synchronization of well organized structures merges as a result of two competing mechanisms: temporal chaos and spatial diffusive stabilization. A window of the coupling parameter for coherent structures is found numerically. Different from all existing scenarios of routes to chaos (period doubling, intermittency and strange attractors), an algorithmetic increase of wavenumbers before an abrupt change to chaos (compared to the periodic doubling geometrical) is unexpectedly discovered. Meta-stable states axe also observed in simulations.     

The Effects of Continuously Varying the Fractional Differential Order of Chaotic Nonlinear Systems

We consider nonlinear third order differential equations which are known to exhibit chaotic behaviour, and amend their order using fractional calculus techniques. By doing this we demonstrate that by continuously increasing the order of differentiation for those systems from 2 to 3, a period doubling route to chaos ensues. This period doubling begins at a system specific order value between 2 and 3.     

Horseshoe chaos in a class of simple Hopfield neural networks

A class of new simple Hopfield neural networks is revisited. To confirm the chaotic behavior in these Hopfield neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps.     

Bifurcation of the period-4 orbits in the standard map

The period doubling bifurcation process in the two-dimensional area preserving mapping is investigated on the basis of symmetry structure analysis. In particular a case of the peirod-4 orbits in the standard map has been studied thoroughly to analyze boundary islands formation around the principal period-4 island, and the onset of the hyperbolic bifurcation without reflection. It is illustrated explicity that the hyperbolic bifurcation without reflection gives rise to the birth of twin orbits with the periodicity of the mother orbit.     

Chaos synchronization of resistively coupled Duffing systems: Numerical and experimental investigations

This paper studies chaos synchronization dynamics of two resistively coupled Duffing systems, through numerical and experimental investigations. Various bifurcation structures are derived and it is found that chaos appear suddenly, through period doubling cascades. The experimental study of these systems is carried out with appropriate software electronic circuit, proposed using the BSIMV3.3 parameters for the investigation of the dynamical behavior. The appropriate coupled coefficient for chaos synchronization is found using numerical and experimental simulations. The reliability of the analytical formulas is approved by the good agreement with the results obtained by both numerical and experiment simulations.     

Investigation of optimal bipedal walking gaits subject to different energy-based objective functions

Optimal bipedal walking gaits subject to different energy-based objective functions are investigated using a simple planar rigid body model of a bipedal robot with upper body, thighs and shanks. The robot's segments are connected by revolute joints actuated by electric motors. The actuators' torques are generated by a trajectory tracking controller to produce periodic walking gaits. A numerical optimization routine is used to find optimal reference trajectories for average speeds in the range of 0.3 – 2.3 m/s to investigate the influence of different objective functions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation and chaos in discrete-time predator–prey system

The discrete-time predator–prey system obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3, 5, 6, 7, 8, 9, 10, 12, 18, 20, 22, 30, 39-orbits in different chaotic regions, attracting invariant circle, period-doubling bifurcation from period-10 leading to chaos, inverse period-doubling bifurcation from period-5 leading to chaos, interior crisis and boundary crisis, intermittency mechanic, onset of chaos suddenly and sudden disappearance of the chaotic dynamics, attracting chaotic set, and non-attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.     

Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot

This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

关于连续区间映射的敏感依赖性

首先证明:若区间映射f是敏感依赖的,则f的拓扑熵ent(f)>0.然后通过引入一种扩张映射进一步证明了敏感依赖的区间映射的拓扑熵的下确界为0,即,上式中拓扑熵的下界0是最优的.最后通过实例展示稠混沌、Spatio-temporal混沌、Li-Yorke敏感及敏感性之间是几乎互不蕴含的.