International Journal of Bifurcation and Chaos

Over the last two decades, theoretical design and circuit implementation of various chaos generators have been a focal subject of increasing interest due to their promising applications in various real-world chaos-based technologies and information systems. In particular, generating complex multiscroll chaotic attractors via simple electronic circuits has seen rapid development. This article offers an overview of the subject on multiscroll chaotic attractors generation, including some fundamental theories, design methodologies, circuit implementations and practical applications. More precisely, the article first describes some effective design methods using piecewise-linear functions, cellular neural networks, nonlinear modulating functions, circuit component design, switching manifolds, multifolded tori formation, and so on. Based on different approaches, computer simulation and circuit implementation of various multiscroll chaotic attractors are then discussed in detail, with some theoretical proofs and laboratory experiments presented for verification and demonstration. It is then followed by some discussion on potential applications of multiscroll chaotic attractors, including secure and digital communications, synchronous prediction, random bit generation, and so on. The article is finally concluded with some future research outlooks, putting the important subject into engineering perspective.

Some basic dynamical behaviors and the compound structure of a new four-dimensional autonomous chaotic system with cubic nonlinearities are investigated. A four-wing chaotic attractor is observed numerically. This attractor, however, is shown to be an numerical artifact by further theoretical analysis and analog circuit experiment. The observed four-wing attractor actually has two coexisting (upper and lower) attractors, which appear simultaneously and are located arbitrarily closely in the phase space. By introducing a simple linear state-feedback control term, some symmetries of the system and similarities of the linearized characteristics can be destroyed, thereby leading to the appearance of some diagonal and anti-diagonal periodic orbits, through which the upper and lower attractors can indeed be merged together to form a truly single four-wing chaotic attractor. This four-wing attractor is real; it is further confirmed analytically, numerically, as well as electronically in the paper. Moreover, by introducing a sign-switching control function, the system orbit can be manipulated so as to switch between two equilibria or among four equilibria, generating two one-side double-wing attractors, which can also be merged to yield a real four-wing attractor.

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.

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z₂-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z₂-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

We employ Chirikov's overlap criterion to investigate interactions between subharmonic resonances of coherent structure solutions of the Gross–Pitaveskii (GP) equation governing the mean-field dynamics of cigar-shaped Bose–Einstein condensates in optical superlattices. We apply a standing wave ansatz to the GP equation to obtain a parametrically forced Duffing equation describing the BEC's spatial dynamics. We then investigate analytically the dependence of spatial resonances on the depth of the superlattice potential, deriving an order-of-magnitude estimate for the critical depth at which spatial resonances with respect to different lattice harmonics first overlap. We also derive a formula for the size of resonance zones and examine changes in our estimates as the relative superlattice amplitudes corresponding to the different harmonics are adjusted. We investigate the onset of global chaos and support our analytical work with numerical simulations.

The paper proposes a systematic solution to the problem of mixing different stochastic processes, each implied by a certain mode of operation of the system at hand and with a random duration whose distribution depends on the previous and present modes. We do so by widening the scope of an existing framework for the statistical characterization of finite valued processes with memory-one properties. The point of view is that of stochastic dynamics and the state space of the process is partitioned into regions (that we identify with modes) such that, if sojourn in a mode can be assumed, the statistical characterization is fully understood. The process is also allowed to stochastically move from one mode to another and the number of time steps for which it remains in each mode is a random variable whose distribution is a function only of the mode visited before. A general theoretical framework is developed here for the computation of any-order joint probabilities. The framework is then exemplified for the case of locally looping systems that are random sequences of modes comprising the cyclical execution of given atomic actions. They are the model of choice for complex appliances that operate following the steps of a communication protocol, and/or the various phases of a bus cycle, and/or the load-compute-store mechanism of a microprocessor, etc. Exploiting the theory put forward by the paper, we highlight how these processes could be generated by suitably designed 2-d chaotic maps and how their second- and third-order spectra may be obtained and interpreted when exponentially or polynomially decaying distributions are assumed for mode sojourn times.

A system of reaction–diffusion equations is considered in the unstirred chemostat with an inhibitor. Global structure of the coexistence solutions and their local stability are established. The asymptotic behavior of the system is given as a function of the parameters, and it is determined when neither, one, or both competing populations survive. Finally, the results of some numerical simulations indicate that the global stability of the steady-state solutions is possible. The main tools for our investigations are the maximum principle, monotone method and global bifurcation theory.

Graph automata define symbol dynamics of graph structures, which are capable of generating structures in addition to describing state transition. Rules of the graph automata are uniform and are therefore suitable for evolutionary computation. This paper shows that various self-replications are possible in graph automata, and that evolutionary computation is applicable to automatic rule generation to obtain self-replicating patterns of graph automata.

In this letter we report a new class of chaotic three-neuron cellular neural networks that are described by special three-dimensional autonomous ordinary differential equations.

In this paper n-double scroll generating via diode-based PieceWise-Linear (PWL) circuit in State Controlled Cellular Neural Network (SC-CNN) is presented. It has been shown that by using simple diode-based configurations; alternative nonlinear circuit configurations for chaotic circuits and PWL-based systems can be used in the generation of n-double scrolls. With this study, while the analysis of the nonlinear block in the SC-CNN-based circuit is simplified, the implementation cost of the circuit is also reduced. Pspice simulations are proved with experimental studies.

Assuming the quantum mechanical "tight binding" of an electron to a nonlinear lattice with Morse potential interactions we show how electric conduction can be mediated by solitons. For relatively high values of an applied electric field the current follows Ohm's law. As the field strength is lowered the current takes a finite, constant, field-independent value.

As a special case of generalized synchronization, chaos anti-synchronization can be characterized by the vanishing of the sum of relevant variables. In this paper, based on Lyapunov stability theorem for ordinary differential equations, several sufficient conditions for guaranteeing the existence of anti-synchronization in a class of coupled identical chaotic systems via linear feedback or adaptive linear feedback methods are derived. Chua's circuit is presented as an example to demonstrate the effectiveness of the proposed approach by computer simulations.

A new method of projective synchronization of unidentical chaotic systems is proposed in this letter. This method is based on the stability criterion of linear systems. The response of two unidentical chaotic systems can synchronize up to any desired scaling factor by a suitable separation of the systems. The new method of projective synchronization is suitable not only for the three-dimensional coupled partially linear systems, but also for higher dimensional even hyperchaotic systems. The simplicity and effectiveness of the proposed method are illustrated by the Lorenz system, the four-dimensional partially linear system, the four-dimensional hyperchaotic Rösser system and Chua's circuit system as four numerical examples.

A qualitative model of cell cycle control is presented and its transition from bistability to limit cycle oscillations and vice versa is discussed. The origin of this model is the two-dimensional system of kinetic equations introduced by Novak–Tyson which is illustrated computationally and analytically. For this purpose a qualitative model is numerically reconstructed from the steady state behavior of the dynamical variables including the bifurcation parameter. Then, the reconstructed cubic polynomial model is generalized to an appropriate canonical form and is analyzed in terms of Lyapunov values. On this basis, the relationship between bistability and self-oscillatory behavior of mitotic cell cycle is approached qualitatively.

We suggest incorporating dynamical information such as locations of unstable fixed points into parameter estimation algorithms in order to improve the method of reconstructing dynamics from time series data. We show how the process of reconstruction using nonlinear filters such as the extended Kalman filter can be easily modified to take advantage of the additional information. We demonstrate the methods using data from two systems exhibiting chaotic dynamics — the Chua circuit and Chen's equations. In both cases we find the models reconstructed using constraints that better approximate the unstable fixed point structure of the underlying systems.

We answer a question posed by Buzzi about the quasi-compactness of the Frobenius–Perron operator of random β-transformations for β∈(1,∞).

This paper concerns a type of rotating machine (centrifugal vibrator), which is supported on a nonlinear spring. This is a nonideal kind of mechanical system. The goal of the present work is to show the striking differences between the cases where we take into account soft and hard spring types. For soft spring, we prove the existence of homoclinic chaos. By using the Melnikov's Method, we show the existence of an interval with the following property: if a certain parameter belongs to this interval, then we have chaotic behavior; otherwise, this does not happen. Furthermore, if we use an appropriate damping coefficient, the chaotic behavior can be avoided. For hard spring, we prove the existence of Hopf's Bifurcation, by using reduction to Center Manifolds and the Bezout Theorem (a classical result about algebraic plane curves).

In this paper, stabilization of fixed points of n-scroll Chua's circuit is investigated. Two adaptive control methods are proposed. One is based on an unstable low pass filter; the other is based on a stable and an unstable low pass filter. The simulation results verify the effectiveness of the two proposed control methods and performance comparisons show that the second control method is superior to the first one with regard to control speed and attraction basins.