Some classical chaotic systems such as the Lorenz system and Chua system have finite numbers of chaotic attractors. This letter develops a simple, effective method for constructing lower-dimensional autonomous systems with infinitely many chaotic attractors. As an application, a Lorenz-type system and a Rössler-type system with infinitely many chaotic attractors are constructed with bifurcation analysis, and with an extension to the fractional-order setting.
No abstract is provided for this article.
No abstract is provided for this article.
The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.
We investigate the cascading failure on weighted complex networks by adopting a local weighted flow redistribution rule, where the weight of an edge is (k(i)k(j))theta with k(i) and k(j) being the degrees of the nodes connected by the edge. Assume that a failed edge leads only to a redistribution of the flow passing through it to its neighboring edges. We found that the weighted complex network reaches the strongest robustness level when the weight parameter theta=1, where the robustness is quantified by a transition from normal state to collapse. We determined that this is a universal phenomenon for all typical network models, such as small-world and scale-free networks. We then confirm by theoretical predictions this universal robustness characteristic observed in simulations. We furthermore explore the statistical characteristics of the avalanche size of a network, thus obtaining a power-law avalanche size distribution together with a tunable exponent by varying theta. Our findings have great generality for characterizing cascading-failure-induced disasters in nature.
In this paper, the Timoshenko theory is applied to investigate a new mathematical model for the “shoulder-elbow-like” single flexible-link robot arm with dampings. Detailed analysis and derivation are given to support the mathematical modeling of this particular flexible mechanism. A new design of a fuzzy-logic-based (PI + D)2 control scheme is developed for both vibration suppression and set-point tracking. Computer simulation results for the modeling are performed to observe the significant vibration modes, and simulation results for the control scheme demonstrate that the controllers perform very well for the tracking based on this flexible-link model. A newly developed method for stability analysis using the “two-straight-lines” criterion is also presented.
This paper is concerned with invariance ( F 1 , F 2 ) -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of ( F 1 , F 2 ) -scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties P ( k ) and Q ( k ) of Furstenberg families are introduced. It is shown that, for any positive integer k, for any s ∈ [ 0 , 1 ] , Furstenberg family M ¯ ( s ) has properties P ( k ) and Q ( k ) , where M ¯ ( s ) denotes the family of all infinite subsets of Z + whose upper density is not less than s. Then, the following conclusion is obtained. D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ ) if and only if D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ [ m ] ) .
The well-known small-world network model was established by randomly rewiring edges, aiming to enhance the synchronizability of an undirected nearest-neighbor regular network. This paper demonstrates via extensive numerical simulations that randomly redirecting edges could enhance the robustness of the network controllability for directed snapback networks against both random and intentional node-removal and edge-removal attacks.
Over the last two decades, multi-scroll chaos generation has seen promising advances and becomes an active research field. This paper initiates a novel approach to design various grid multi-wing butterfly chaotic attractors from piecewise Lü system based on switching control and heteroclinic orbit. It should be especially pointed out that these generating multi-wing chaotic attractors are chaotic in the sense of Smale horseshoe from Shilnikov theorem. Moreover, there are some potential engineering applications in the future because of the simplicity of the proposed design approach.
It is known that certain physical systems, which do not generate deterministic chaos under conventional frameworks, may generate such complex behavior in a quantum mechanical setting. In this paper, it is proved that the annihilation operator of an unforced quantum harmonic oscillator admits an invariant distributionally ϵ -scrambled set for any 0 < ϵ < 2 , showing that this operator can exhibit maximal distributional chaos on an uncountable invariant subset.
We propose yet another feedback control algorithm for chaotification of an originally non chaotic system. The new controller uses the continuous sawtooth function instead of the modulo operation, which can generate discrete chaos in the sense of Li and Yorke, for arbitrarily given nonlinear autonomous systems of any dimensionalities.
Controlling chaos is a new concept. In this article,we survey and comment on the recent rapiddevelopment in control, synchronization and anti-control of chaotic dynamical systems. We believe that thestimulating and promising research direction of chaos control has an attractive future in both theory andapplication. This new scientific challenge calls for further effort and endeavor from communities of mathematics,physics,engineering,and many other interdisciplinary areas.
