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利用Silnikov定理,讨论了具有自动频率跟踪功能电磁振动机械系统的混沌特性.借助卡尔达诺公式和微分方程组级数解分别讨论了该系统的特征值问题和同宿轨道的存在性,进而比较严密地证明了该系统Silnikov型Smale混沌的存在性,并给出发生Silnikov型Smale混沌所需条件.利用数值模拟得到该类机电耦合系统的相轨迹图、Lyaponov指数谱和Lyaponov维数,进一步验证了该非线性系统存在奇怪吸引子.
关键词:
混沌系统
Lyapunov指数
Silnikov定理
耦合 相似文献
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Stanisław Janeczko 《Letters in Mathematical Physics》1988,16(4):301-311
The topological type function for stationary probability density of stable stochastic dynamical systems is introduced. The corresponding bifurcation diagrams in the case of one dichotomic noise are derived. Examples encountered in physics and chemistry are given.The author was a visitor at the Department of Mathematics, Monash University, Australia, during part of the period when this paper was written. 相似文献
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Multistability or coexistence of different chaotic attractors for a given set of parameters depending on the initial condition only is one of the most exciting phenomenon in dynamical systems. The schemes to design multistability systems via coupling two identical or non-identical but the same-dimensional systems have been proposed earlier. Coupled different-dimensional systems are very useful to describe the real-world physical and biological systems. In this paper, a scheme for designing a multistable system by coupling two different-dimensional dynamical systems has been proposed. Coupled Lorenz and Lorenz–Stenflo systems have been considered to illustrate the scheme. The efficiency of the scheme is shown numerically, by presenting phase diagrams, bifurcation diagrams and variation of maximum Lyapunov exponents. 相似文献
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We construct new unidirectional coupling schemes for autonomous and nonautonomous drive systems, respectively. Each of these schemes makes the state of the response system asymptotically approach the first-order derivative of the state of the driver. From the point of view of geometry, the first-order derivative of the state of the driver can be viewed as a tangent vector of the trajectory of the driver, so the proposed schemes are named tangent response schemes. Numerical simulations of the Lorenz system and the forced Duffing oscillator verify the validity of the tangent response schemes. We further point out that the tangent response can be interpreted as a special kind of generalised synchronisation, thereby explaining why the response system can exhibit rich geometrical structures in its state space. 相似文献
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This Letter presents a method by which the mean field dynamics of a population of dynamical systems with parameter diversity and global coupling can be described in terms of a few macroscopic degrees of freedom. The method applies to populations of any size and functional form in the region of coherence. It requires linear variation or a narrow distribution for the dispersed parameter. Although an approximation, the method allows us to quantitatively study the transitions among the collective regimes as bifurcations of the effective macroscopic degrees of freedom. To illustrate, the phenomenon of oscillator death and the route to full locking are examined for chaotic oscillators with time scale mismatch. 相似文献
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We generalize previous systems of coupled oscillators possessing invariants. The method of generalizing these systems is used to construct invariants for new systems. 相似文献
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We investigate the dynamics generated by a type of equation which is common to a variety of physical systems where the undesirable effects of a number of self-consistent nonlinear forces are balanced by an externally imposed controlling harmonic force. We show that the equation presents a new sequence of bifurcations where periodic orbits are created and destroyed in such a nonsimultaneous way that may leave the appropriate phase-space occasionally empty of fundamental harmonic orbits and confined trajectories. A generic analytical model is developed and compared with a concrete physical example. 相似文献
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A density matrix first principles formalism is extended for use in coupled dynamical systems within the framework of the Zwanzig projection operator technique. Coupled linear integro-differential equations for the reduced density operators of two (or more) dynamical subsystems interacting with one (or more) dissipative subsystem(s) and weak driving fields are obtained. These coupled equations, which are highly problem independent and rendered in a form simple for applications, are applied to the well-known s-d exchange model in metals where the coupled Bloch-like equations obtained by others using many-body techniques are recovered. The problem of the instantaneous destination vector is discussed within the framework of our formalism using superoperator resolvent techniques. 相似文献
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In this Letter, we investigate the problem of impulsive synchronization of networked multi-agent systems, where each agent can be modeled as an identical nonlinear dynamical system. Firstly, an impulsive control protocol is designed for network with fixed topology based on the local information of agents. Then sufficient conditions are given to guarantee the synchronization of the networked nonlinear dynamical system by using algebraic graph theory and impulsive control theory. Furthermore, how to select the discrete instants and impulsive constants is discussed. The case that the topologies of the networks are switching is also considered. Numerical simulations show the effectiveness of our theoretical results. 相似文献
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M. Lakshmanan V. K. Chandrasekar 《The European physical journal. Special topics》2013,222(3-4):665-688
In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed. 相似文献
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We study localization and wave trapping in disordered, nonlinear dynamical systems. For some models of classical, disordered anharmonic crystal lattices, we prove that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space. Such vibrations remain localized, for all times, and there is no transport of energy through the lattice. Our general concepts and techniques extend to other systems, such as disordered, nonlinear Schrödinger equations, or randomly coupled rotors. 相似文献
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《Physica D: Nonlinear Phenomena》1997,104(1):61-74
We present a statistical approach for detecting the Markovian character of dynamical systems by analyzing their flow of information. Especially in the presence of noise which is mostly the case for real-world time series, the calculation of the information flow of the underlying system via the concept of symbolic dynamics is rather problematic since one has to use infinitesimal partitions. We circumvent this difficulty by measuring the information flow indirectly. More precisely, we calculate a measure based on higher order cumulants which quantifies the statistical dependencies between the past values of the time series and the point r steps ahead. As an extension of Theiler's method of surrogate data (Theiler et al., 1992) this cumulant based information flow (a function of the look-ahead r) is used as the discriminating statistic in testing the observed dynamics against a hierarchy of null hypotheses corresponding to nonlinear Markov processes of increasing order. This procedure is iterative in the sense that whenever a null hypothesis is rejected new data sets can be generated corresponding to better approximations of the original process in terms of information flow. Since we use higher order cumulants for calculating the discriminating statistic our method is also applicable to small data sets. Numerical results on artificial and real-world examples including non-chaotic, nonlinear processes, autoregressive models and noisy chaos show the effectiveness of our approach. 相似文献
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Maistrenko YL Maistrenko VL Popovych O Mosekilde E 《Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics》1999,60(3):2817-2830
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. 相似文献
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J. Heldstab H. Thomas T. Geisel G. Radons 《Zeitschrift für Physik B Condensed Matter》1983,50(2):141-150
We carry out a linear response theory for discrete dynanmical systems with periodic attractors. The symmetry properties of the susceptibility matrix are studied and its eigenvalues and eigenvectors are determined. Close to a period-doubling bifurcation where the susceptibility diverges, its half-width is related to the Lyapunov exponent. At the transition to chaos the susceptibility has some universal behaviour which is described by a critical exponent κ=1?(ln2/lnδ)=0.550193... At the bifurcation points where linear response theory becomes insufficient we also determine the nonlinear response. 相似文献
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Oleksandr BurylkoArkady Pikovsky 《Physica D: Nonlinear Phenomena》2011,240(17):1352-1361
We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs. 相似文献