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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.

In this letter, the characteristics of a novel switched circuit is presented. Its pseudo chaotic signal is digitally programmable. Robust synchronization between two mismatched circuits is demonstrated by simulation results. Lastly, applications of this circuit on chaotic communication are suggested.

In this paper, a nonlinear system aiming at reducing the signal transmission rate in a networked control system is constructed by adding nonlinear constraints to a linear feedback control system. Its stability is investigated in detail. It turns out that this nonlinear system exhibits very interesting dynamical behaviors: in addition to local stability, its trajectories may converge to a nonorigin equilibrium or be periodic or just be random. Furthermore it exhibits sensitive dependence on initial conditions — a sign of chaos. Complicated bifurcation phenomena are exhibited by this system. After that, control of the chaotic system is discussed. All these are studied under scalar cases in detail. Some difficulties involved in the study of this type of systems are analyzed. Finally an example is employed to reveal the effectiveness of the scheme in the framework of networked control systems.

This work investigates maps T of the plane with points p ∈ ℝ² where T is undefined. Through the study of certain families with a singularity at the origin two very different dynamics will be illustrated. The first example will show that a singularity has no apparent effect on forward mapping. In this example there exists also an attractive fixed point. The second example will illustrate a singularity that exhibits attracting fixed point-like behavior.

In a broad sense, model reduction means producing a low-dimensional dynamical system that replicates either approximately, or more strictly, exactly and topologically, the output of a dynamical system. Model reduction has an important role in the study of dynamical systems and also with engineering problems. In many cases, there exists a good low-dimensional model for even very high-dimensional systems, even infinite dimensional systems in the case of a PDE with a low-dimensional attractor. The theory of global attractors approaches these issues analytically, and focuses on finding (depending on the question at hand), a slow-manifold, inertial manifold, or center manifold, on which a restricted dynamical system represents the interesting behavior of the dynamical system; the main issue depends on defining a stable invariant manifold in which the dynamical system is invariant. These approaches are analytical in nature, however, and are therefore not always appropriate for dynamical systems known only empirically through a dataset. Empirically, the collection of tools available are much more restricted, and are essentially linear in nature. Usually variants of Galerkin's method, project the dynamical system onto a function linear subspace spanned by modes of some chosen spanning set. Even the popular Karhunen–Loeve decomposition, or POD, method is exactly such a method. As such, it is forced to either make severe errors in the case that the invariant space is intrinsically a highly nonlinear manifold, or bypass low-dimensionality by retaining many modes in order to capture the manifold. In this work, we present a method of modeling a low-dimensional nonlinear manifold known only through the dataset. The manifold is modeled as a discrete graph structure. Intrinsic manifold coordinates will be found specifically through the ISOMAP algorithm recently developed in the Machine Learning community originally for purposes of image recognition.

In this paper, we study the dynamical behavior of nonautonomous, almost-periodic discrete FitzHugh–Nagumo system defined on infinite lattices. We prove that the nonautonomous infinite-dimensional system has a uniform attractor which attracts all solutions uniformly with respect to the translations of external terms. We also establish the upper semicontinuity of uniform attractors when the infinite-dimensional system is approached by a family of finite-dimensional systems. This paper is based on a uniform tail method, which shows that, for large time, the tails of solutions are uniformly small with respect to bounded initial data as well as the translations of external terms. The uniform tail estimates play a crucial role for proving the uniform asymptotic compactness of the system and the upper semicontinuity of attractors.

A particularly simple and mathematically elegant example of chaos in a three-dimensional flow is examined in detail. It has the property of cyclic symmetry with respect to interchange of the three orthogonal axes, a single bifurcation parameter that governs the damping and the attractor dimension over most of the range 2 to 3 (as well as 0 and 1) and whose limiting value b = 0 gives Hamiltonian chaos, three-dimensional deterministic fractional Brownian motion, and an interesting symbolic dynamic.

To detect determinism and instability geometrically, the Lorenz map z_n+1 = f(z_n), which projects a local maximum z_n to the next local maximum z_n+1 of the time series, is useful for such a thin attractor as the Lorenz attractor, but is not applicable for such a "fat" attractor as the Chua circuit. Hence, this study proposes a novel and simple method that is applicable for various attractors. It shows that a T-delayed Lorenz map z_n+T = f_T(z_n), which projects a local maximum z_n to the T-delayed local maximum z_n+T, is more useful. It was found that the appearance of a parabolic image in a T-delayed Lorenz map is a sign of chaos.

In this paper, we investigate the intertwined basins of attraction. Using a general definition, we discuss the local intertwining properties and present an easy condition to guarantee the existence of intertwined basins for planar dynamical systems.

A modified generalized Lorenz system in a canonical form extended from the generalized Lorenz system is proposed in this paper. This novel system has a folded factor and can display complex 2-scroll folded attractors and 1-scroll folded attractors at different parameter values. Three typical normal forms, called Lorenz-like, Chen-like and Lü-like chaotic system respectively, of three-dimensional quadratic autonomous chaotic systems are derived, and their dynamical behaviors are further investigated by employing Lyapunov exponent spectrum, bifurcation diagram, Poincaré mapping and phase portrait, etc. Of particular interest is the fact that the folded factor makes Chen-like and Lü-like chaotic systems exhibit complicated nonlinear dynamical phenomena.

Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is expressed as a multilinear mapping. Under this framework, the dynamics of a multivalued logical network is converted into a standard discrete-time linear system. Analyzing the network transition matrix, easily computable formulas are obtained to show (a) the number of equilibriums; (b) the numbers of cycles of different lengths; (c) transient period, the minimum time for all points to enter the set of attractors, respectively. A method to reconstruct the logical network from its network transition matrix is also presented. This approach can also be used to convert the dynamics of a multivalued control network into a discrete-time bilinear system. Then, the structure and the controllability of multivalued logical control networks are revealed.

This letter introduces a hyperchaotic system from the Lü system [Lü et al., 2004] with a linear state feedback controller. This hyperchaotic system has more complex dynamical behaviors, and can generate 4-scroll hyperchaotic attractor and 2-scroll chaotic attractor under different control parameters. In particular, the system can also exhibit novel coexisting intermittent chaotic orbits. Theoretical analyses and simulation experiments are conducted to investigate the dynamical behaviors of the proposed hyperchaotic system.

Flux dynamics in an annular long Josephson junction is studied. Three main topics are covered. The first is chaotic flux dynamics and its prediction via Melnikov integrals. It turns out that DC current bias cannot induce chaotic flux dynamics, while AC current bias can. The existence of a common root to the Melnikov integrals is a necessary condition for the existence of chaotic flux dynamics. The second topic is on the components of the global attractor and the bifurcation in the perturbation parameter measuring the strength of loss, bias and irregularity of the junction. The global attractor can contain coexisting local attractors, e.g. a local chaotic attractor and a local regular attractor. In the infinite dimensional phase space setting, the bifurcation is very complicated. Chaotic attractors can appear and disappear in a random fashion. Three types of attractors (chaos, breather, spatially uniform and temporally periodic attractor) are identified. The third topic is ratchet effect. Ratchet effect can be achieved by a current bias field which corresponds to an asymmetric potential, in which case the flux dynamics is ever lasting chaotic. When the current bias field corresponds to a symmetric potentially, the flux dynamics is often transiently chaotic, in which case the ratchet effect disappears after sufficiently long time.

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

With the abundance of chaotic systems that have now been identified and studied, it is prudent to establish a standard for the publication of new examples of such systems and to develop acceptable criteria for their characterization.

A three-degree-of-freedom vibro-impact system with symmetric two-sided constraints is considered. The system is strongly nonlinear and symmetric. The symmetric fixed point of the Poincaré map is deduced analytically, and the existence conditions of the symmetric fixed point are obtained. The six-dimensional Poincaré map can be expressed as the second iteration of another unsymmetric implicit map, which implies the symmetry of the Poincaré map. When the control parameter changes successively, symmetry-breaking bifurcation and symmetry-restoring bifurcation will occur at some point, and the attractor may change between symmetry and antisymmetry repeatedly. When a symmetry breaking bifurcation occurs, the symmetry is still the intrinsic property of the vibro-impact system. Here the Poincaré map cannot reflect the symmetry itself. However, the unsymmetric implicit map can capture a pair of antisymmetric ω-limit sets, which reflects the symmetry of the vibro-impact system. Different Poincaré sections are locally conjugate about a diffeomorphism. Therefore, as long as the perturbation is sufficiently small, changing the Poincaré section does not have any effect on the dynamical behavior. The transition to symmetric chaos is represented by numerical simulations.

In this paper, we investigate the problem of dynamical behaviors and numerical simulation of Lorenz systems for the incompressible flow between two concentric rotating cylinders. A spectral Galerkin method is used to derive a model system of axisymmetric Couette–Taylor flow, a three-mode system, which is structurally similar to the Lorenz system, is obtained by a suitable three-mode truncation of the Navier–Stokes equations for the incompressible flow between two concentric rotating cylinders. The stability of the three-mode system is discussed, the existence of its attractor is given. Moreover, numerical simulation results indicate that this low-dimensional model exhibits a route to chaos via a period doubling cascade. Using these numerical results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment.

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝⁿ.

In this paper, we present a simple and accessible way to enhance the stable behaviors of a chaotic dynamical system which models a cancer growth, as presented in [Itik & Banks, 2010]. The algorithm presented in [Danca et al., 2012], approximates numerically any attractor of a system belonging to a defined class of dynamical systems, by alternating the control parameter in relatively short periods of time. When switching the control parameter within a set of values corresponding to some chaotic behaviors, the result may be a stable evolution or, reversely a chaotic behavior may be obtained by switching the parameter within a set of values corresponding to stable evolutions. This apparently surprising phenomenon is, in fact, a generalization of the known Parrondo's paradox.

In the survey, we consider bifurcations of attracting (or repelling) invariant sets of some classical dynamical systems with a discrete time.