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Chaos is considered to be essential for life, since it can bring diversity to the world. However, it is difficult to detect the pattern of chaos due to its apparent randomness. In this paper, a nonlinear system based on a Hénon-like map is studied to show the semi-ordered structure of its chaotic sequences. The periodic sequences of such a map are solved analytically, and the stability is analyzed. Multiple coexisting unstable sequences with different periods and the chaotic attractor are obtained. An evidence is found that chaos is not purely apparently random, and it is weakly attracted by different unstable periodic sequences. Due to the weak attraction, it transits from one unstable periodic sequence to another at random instances, and it appears unpredictable.

We examine the invariant set of the Newton–Leipnik attractor using the inverse limit of a related function. This is accomplished using a simpler model than that proposed in [LoFaro, 1997b], and we argue that the earlier model is suspect.

There are many methods for computing stable and unstable manifolds in autonomous flows. When the flow is nonautonomous, however, difficulties arise since the hyperbolic trajectory to which these manifolds are anchored, and the local manifold emanation directions, are changing with time. This article utilizes recent results which approximate the time-variation of both these quantities to design a numerical algorithm which can obtain high resolution in global nonautonomous stable and unstable manifolds. In particular, good numerical approximation is possible locally near the anchor trajectory. Nonautonomous manifolds are computed for two examples: a Rossby wave situation which is highly chaotic, and a nonautonomus (time-aperiodic) Duffing oscillator model in which the manifold emanation directions are rapidly changing. The numerical method is validated and analyzed in these cases using finite-time Lyapunov exponent fields and exactly known nonautonomous manifolds.

In this work, we study, from a numerical point of view, the dynamics of a specific hybrid map which is that of the kicked van der Pol system. The dynamics of this system is generated by a two-stage procedure: the first stage is the time-$\tau$ map of the vector field associated with the van der Pol equation, and the second stage is a translation. We propose a numerical method for computing local (un)stable manifolds of a given fixed point, which leads to high order polynomial parameterization of the embedding. Such a representation of the dynamics on the manifold is obtained in terms of a simple conjugacy relation and by constructing two contracting operators. We illustrate our techniques by plotting (for specific values of the parameters) both stable and unstable manifolds displaying homoclinic intersection. Furthermore, our numerical study reveals the presence of a strange attractor included in the closure of the unstable manifold of the fixed point.