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In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations.

We consider the Hénon attractor ℋ as a curve limit of continuous planar curves H(n), as n → ∞. We describe a set of tools for studying the Hausdorff dimension dim_H of a certain family of such curves, and we adapt these tools to the particular case of the Hénon attractor, estimating its Hausdorff dimension dim_H (ℋ) to be about 1.258, a number smaller than the usual estimates for the box-counting dimension of the attractor. We interpret this discrepancy.

We study the properties of a n²-dimensional Lotka–Volterra system describing competition among n different species with adaptive skills, i.e. whose interaction coefficients are time averages of the species level of interaction over their past.

Starting by the case of adaptive competition among species all having the same carrying capacities, we focus our attention on the model obtained on perturbing the carrying capacity of a fixed species, which is made more or less disadvantaged.

We prove the existence of a certain class of invariant subspaces and introduce a seven-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the system. The relevance of this reduced model to the complete one has also been found when the time dependent regimes have been investigated.

Ecologically relevant questions, i.e. species survival and the time dependent behavior of the system have also been analyzed focusing on the role of behavioral adaptation. In particular, we have found that competitive exclusion cannot occur but coexistence is possible in various forms (i.e. competitive stable equilibria and different periodic oscillations).

Database Tomography (DT) is a textual database analysis system consisting of two major components: (1) algorithms for extracting multiword phrase frequencies and phrase proximities (physical closeness of the multiword technical phrases) from any type of large textual database, to augment (2) interpretative capabilities of the expert human analyst. DT was used to derive technical intelligence from a Nonlinear Dynamics database derived from the Science Citation Index/Social Science Citation Index (SCI). Phrase frequency analysis by the technical domain experts provided the pervasive technical themes of the Nonlinear Dynamics database, and the phrase proximity analysis provided the relationships among the pervasive technical themes. Bibliometric analysis of the Nonlinear Dynamics literature supplemented the DT results with author/journal/institution publication and citation data.

We investigate the stability and bifurcations of parametrically excited thin liquid films. A recently derived nonlinear evolution equation for the two-dimensional spatio-temporal dynamics of falling liquid films on an oscillating vertical wall is expanded to low order Fourier modes. A fourth-order modal dynamical system is validated to yield the primary bifurcation structure of the fundamental falling film dynamics described by the Benney equation, and accurately predicts the quasi-periodic structure of the temporally modulated Benney equation (TMBE). The stability of fundamental steady and periodic solutions is analytically and numerically investigated so as to reveal the threshold for nonstationary and chaotic solutions corresponding to aperiodic modulated traveling waves. The reduced modal dynamical system enables construction of a comprehensive bifurcation structure, which is verified by numerical simulation of the evolution equation.

In this paper, we discuss a new mechanism for chaos in light of some new developments in the theory of dynamical systems. It was shown in [Wang & Young, 2002b] that strange attractors occur when an autonomous system undergoing a generic Hopf bifurcation is subjected to a weak external forcing that is periodically turned on and off. For illustration purposes, we apply these results to the Chua's system. Derivation of conditions for chaos along with the results of numerical simulations are presented.

The aim of this paper is to present results concerning a three-dimensional model including a prey, a predator and top-predator, which we have named the Volterra–Gause model because it combines the original model of V. Volterra incorporating a logisitic limitation of the P. F. Verhulst type on growth of the prey and a limitation of the G. F. Gause type on the intensity of predation of the predator on the prey and of the top-predator on the predator. This study highlights that this model has several Hopf bifurcations and a period-doubling cascade generating a snail shell-shaped chaotic attractor.

With the aim of facilitating the choice of the simplest and most consistent model a comparison is established between this model and the so-called Rosenzweig–MacArthur and Hastings–Powell models. Many resemblances and differences are highlighted and could be used by the modellers.

The exact values of the parameters of the Hopf bifurcation are provided for each model as well as the values of the parameters making it possible to carry out the transition from a typical phase portrait characterizing one model to another (Rosenzweig–MacArthur to Hastings–Powell and vice versa).

The equations of the Volterra–Gause model cannot be derived from those of the other models, but this study shows similarities between the three models. In cases in which the top-predator has no effect on the predator and consequently on the prey, the models can be reduced to two dimensions. Under certain conditions, these models present slow–fast dynamics and their attractors are lying on a slow manifold surface, the equation of which is given.

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk).

The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra–Gause are proposed.

In this paper, we study the existence of a new class of chaotic attractors, namely the rank-one attractors, in the MLC (Murali–Lakshmanan–Chua) circuit [Murali et al., 1994] by numerical simulations based on a theory of rank-one maps developed in [Wang & Young, 2005]. With the guidance of the theory in [Wang & Young, 2005], weakly stable limit cycles, found through Hopf bifurcations and other numerical means, are subjected to periodic pulses with long relaxation periods to produce rank-one attractors. The periodic pulses are applied directly as an input. Periodic pulses have been used before in various schemes of chaos. However, for this scheme of creating rank-one attractors to work, the applied periodic pulses must have short pulse widths and long relaxation periods. This is one of the key components in creating this new class of chaotic attractors.

In this paper, we investigate the creation of strange attractors in a switch-controlled MLC (Murali–Lakshmanan–Chua) circuit. The design and use of this circuit is motivated by a recent mathematical theory of rank one attractors developed by Wang and Young. Strange attractors are created by periodically kicking a weakly stable limit cycle emerging from the center of a supercritical Hopf bifurcation, and are found in numerical simulations by following a recipe-like algorithm. Rigorous conditions for chaos are derived and various switch control schemes, such as synchronous, asynchronous, single-, and multi-pulse, are investigated in numerical simulations.

This work studies transitions to chaos and adaptive synchronization of a nonlinear emitter–receiver system in a drive–response framework. Safe bifurcations and explosive bifurcations are observed. A robust adaptive observer-based response system is designed to synchronize the emitter–receiver system with unknown parameters and external disturbances. Lyapunov stability ensures global synchronization between the drive and response systems even if Lipschitz constants on functions matrices and bound on uncertainties are unknown. Computer simulations are provided to illustrate the designed adaptive synchronization scheme.

The main purpose of this tutorial is to introduce to a more application-oriented audience a new chaos theory that is applicable to certain systems of differential equations. This new chaos theory, namely the theory of rank one maps, claims a comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain systems of differential equations, the existence of the indicated phenomenon of chaos can be verified through a well-defined computational process. Applications to the well-known Chua's and MLC circuits employing controlled switches are also presented to demonstrate the usefulness of the theory. We try to introduce this new chaos theory by using a balanced combination of examples, numerical simulations and theoretical discussions. We also try to create a standard reference for this theory that will hopefully be accessible to a more application-oriented audience.

In this communication the emergent behavior of clusters of robots with chaotic kinematics attractors is shown. Even if the aim of the research started from the study of the cooperative behavior of inspection robots, the reported results (including also the role of chaotic synchronization in the generation of the kinematics trajectory), show new aesthetic features of the motion in mechanical control systems.

Based on our recent work on Quantum Nambu Mechanics [Axenides & Floratos 2009], we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of noncommutative phase space coordinates as Hermitian N × N matrices in R³. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction, it violates the quantum commutation relations. We demonstrate that the Heisenberg–Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand, there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving nondissipative sector survive for long times.

A sinusoidally-driven conservative and dissipative system with signum nonlinearity is presented in this paper. The proposed system exhibits strange attractors, multistability, and megastability. The system exhibits both conservative and dissipative behaviors with the change of its parameters. The Lyapunov exponents, Lyapunov spectrum plots, bifurcation diagrams, and phase plots are used to show the special features of the proposed system.

In this paper, we present a 6D generalized Lorenz model for magnetoconvection of an electrically conducting viscous fluid layer subjected to horizontally imposed uniform magnetic field. It generalizes the 4D generalized Lorenz model of Macek and Strumik [Phys. Rev. E82, 027301 (2010)] taking into account high-wavenumber vertical Fourier modes of the temperature profile. These additional modes not only increase the feedback loop of the system but also subsequently affect the transitional processes. The boundedness, stability of solutions, bifurcation patterns enroute to chaos for the new 6D model are explored. Studies reveal that the stability of the quiescent state does not alter. But the stability of the steady convective state differs in comparison to the 4D model. The regions of aperiodic oscillation are suppressed which results in stabilization of the convective motion. Some new organized periodic structures embedded in chaotic domain appear in parameter space of the 6D model, and the transitional route to hyperchaos is altered owing to the inclusion of the high-order modes.

The method described here relies on interval arithmetic and graph theory to compute guaranteed coverings of strange attractors like Hénon attractor. It copes with infinite intervals, using either a geometric method or a new directed projective interval arithmetic.

The goal of the present investigation is to understand the typical bifurcation patterns organized around a Hopf-saddle-node (HSN) bifurcation of fixed points, defined as follows: a C^∞-family of diffeomorphisms F_α : ℝ³ → ℝ³, where α ∈ ℝ^P is a multi-parameter, is an HSN-family if

Note from Publisher: This article contains the abstract only.

In this paper, the linear feedback synchronization and adaptive feedback synchronization with only one controller for a new unified chaotic system are to applied to a unified chaotic system, which consists of three subsystems, constructed according to Vanecek and Celikovský criterion. Moreover, numerical simulations will be given to show the effectiveness of these methods.