International Journal of Bifurcation and Chaos

This paper deals with the use of recent computational techniques in the numerical study of qualitative properties of two degrees of freedom of Hamiltonian systems. These numerical methods are based on the computation of the OFLI2 Chaos Indicator, the Crash Test and exit basins and the skeleton of symmetric periodic orbits. As paradigmatic examples, three classical problems are studied: the Copenhagen and the (n + 1)-body ring problems and the Hénón–Heiles Hamiltonian. All the numerical integrations have been done by using the state-of-the-art numerical library TIDES based on the extended Taylor series method.

The influence of the thalamus on the diversity of cortical activations is investigated in terms of the Ising model with respect to progressive levels of cortico-thalamic connectivity. The results show that better diversity is achieved at lower modulation levels, being higher than those obtained with counterpart network models.

This paper provides a practical implementation of a memristor based chaotic circuit. We realize a memristor using off-the-shelf components and then construct the memristor along with the associated chaotic circuit on a breadboard. The goal is to construct a physical chaotic circuit that employs the four fundamental circuit elements — the resistor, capacitor, inductor and the memristor. The central concept behind the memristor circuit is to use an analog integrator to obtain the electric flux across the memristor and then use the flux to obtain the memristor's characterstic function.

We try to stabilize steady solutions of a physical model described by wave equations with nonlinear boundary conditions. This system is a distributed parameter system in which ideal turbulence, introduced by Sharkovsky et al., occurs. Although the behavior of the system is quite intricate both in time and space, by using d'Alembert's solution, the analysis of the dynamic characteristics can be reduced to that of a finite-dimensional difference equation. In this report, based on this analytical method using d'Alembert's solution, we design control laws to stabilize steady solutions (equilibrium solutions and periodic solutions) and synchronize a pair of identical systems.

In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.

In this paper, the general perturbation of piecewise Hamiltonian systems on the plane is considered. When the unperturbed system has a family of periodic orbits, similar to the perturbations of smooth system, an expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits. As applications, the number of bifurcated limit cycles of several concrete piecewise systems are presented.

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that fold-Hopf singularities are O(h^p)-shifted and turned into fold-Neimark–Sacker points by one-step methods of order p. Then we analyze the effect of discretizations methods on the local bifurcation diagram near Bogdanov–Takens and fold-Hopf singularities. In particular, we prove that the discretized codimension one curves intersect at the singularities in a generic manner. The results are illustrated by a numerical example.

We cryptanalyze Fridrich's chaotic image encryption algorithm. We show that the algebraic weaknesses of the algorithm make it vulnerable against chosen-ciphertext attacks. We propose an attack that reveals the secret permutation that is used to shuffle the pixels of a round input. We demonstrate the effectiveness of our attack with examples and simulation results. We also show that our proposed attack can be generalized to other well-known chaotic image encryption algorithms.

In this paper, a symbolic dynamical model of the average queue size of the random early detection (RED) algorithm is proposed. The conditions for both the system parameters and the initial conditions that the average queue size of the RED algorithm would converge to a fixed point are derived. These results are useful for network engineers to design both the system parameters and the initial conditions so that internet networks can achieve a good performance.

We study the stochastically forced Chen system in its parameter zone under the transition to chaos via period-doubling bifurcations. We suggest a stochastic sensitivity function technique for the analysis of stochastic cycles. We show that this approach allows to construct the dispersion ellipses of random trajectories for any Poincaré sections, and these ellipses reflect the essential features of a spatial arrangement of random trajectories near deterministic cycles. For the Chen system, we demonstrate a growth of stochastic sensitivity of the forced cycles under transition to chaos.

The limit cycle bifurcations of a Z₂ equivariant planar Hamiltonian vector field of degree 7 under Z₂ equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.

In this paper, we consider compact, invariant sets in Hamiltonian systems in order to extend the concept of crisis to such systems. We focus on crisis-induced intermittency in several systems where two invariant sets merge, obtaining scaling laws for the residence times and for the probability distribution decay as a function of a critical parameter. The connection to hitherto known crisis-induced intermittency in dissipative systems is discussed.

Hopf bifurcations in fast–slow systems of ordinary differential equations can be associated with a surprisingly rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space. A first-order asymptotic expansion of this location was found by Krupa and Szmolyan [2001a, 2001b, 2001c] in the framework of a "canard point"-normal-form for systems with one fast and one slow variable. We show how to compute the coefficients in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding the use of this normal form. Our results connect the theory of canard explosions with existing numerical software, enabling easier calculations of where canard explosions occur.

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

This paper presents a simple scheme for designing mixed-mode chaotic circuits using quadrature core oscillators. Two mixed-mode circuits are constructed by using a few operational amplifiers in suitable combination with some autonomous/nonautonomous second/third-order chaotic circuits. Simulation and experimental results have verified the simplicity and effectiveness of the new design.

A sinusoidally-driven system with a simple signum nonlinearity term is investigated through an analytical analysis as well as dynamic simulation. To obtain the correct Lyapunov exponents, the signum function is replaced by a sharply varying continuous hyperbolic tangent function. By phase portraits, Poincaré sections and bifurcation diagrams, the rich dynamic behaviors of this system are demonstrated, such as an onion-like strange attractor, pitchfork and attractor merging bifurcations, period-doubling routes to chaos, and chaotic transients in the case of small damping. Moreover, the chaos persists as the damping is reduced to zero.

The problem of synchronization of chaotic oscillators, using state observers, is handled through the use of a simplified perturbed linear integration model of the chaotic system output dynamics. The linear simplified model of the chaotic system does not entitle approximate linearizations, nor state coordinate transformations, but, simply, a pure linear integration model with additive unknown but bounded perturbation inputs lumping all the output dynamics nonlinearities. An extended linear state observer (here addressed as Generalized Proportional Integral (GPI) observer) is proposed for the accurate estimation of the phase variables and the perturbation input of the nonlinear output dynamics. The effectiveness of the approach is tested in the synchronization of two study cases: The Genesio–Tesi chaotic system and the Rossler oscillator. As an application of the estimation process, a coding–decoding process involving encrypted messages, in transmitted phase variables, is implemented using a Rossler chaotic system.

Recently, the concept of feedback passivity-based control has drawn attention to chaos control. In all existing papers, the implementations of passivity-based control laws require the system states for feedback. In this paper, a passivity-based control law which only requires the knowledge of the system output is proposed. Simulation results are provided to show the effectiveness of the proposed solution.

In this study, we numerically investigate the lowest instability boundary of nonaxisymmetric Taylor vortex flow (TVF) for different axial wavenumbers. The variation in the axial wavenumber of a supercritical TVF can affect the instability of the flow, because the wavelength of a Taylor vortex is constant only when the flow is axisymmetrical. When the nonaxisymmetric TVF is transformed to a wavy vortex flow (WVF), the instability boundary is changed with the variation in the axial wavenumber. We carry out an instability analysis of the nonaxisymmetric TVF between two concentric rotating cylinders, which have a radius ratio of 0.88.

In this paper, a modified degree-based strategy is proposed for pinning control of complex networks under constrained control capacity. Limited by the total control strength, the pinning strategy design problem is reconsidered as a constrained optimization problem. To provide a fast and effective solution, a decrease-and-conquer approach has been designed to assign the best control strengths to the pinned highest-degree nodes. Simulation results have demonstrated that the obtained strategy provides an outperformed control effect on various kinds of complex networks, as compared with strategies derived from degree, betweenness or closeness using evenly-distributed control strength.

Some sufficient conditions to ensure the existence, uniqueness and global exponential stability of the equilibrium point of cellular neural networks with variable delays are derived. These results extend and improve the existing ones in the literature. Two illustrative examples are given to demonstrate the effectiveness of our results.

In this paper, we consider a class of fuzzy BAM neural networks with distributed delays and variable coefficients. By using Brouwer fixed point theorem and differential inequality, we obtain some sufficient conditions for the existence and global exponential stability of periodic solutions for a class of fuzzy BAM neural networks with distributed delays and variable coefficients. In addition, we also present an illustrative example to show the effectiveness of obtained results.

A chaotic attractor has been observed with an autonomous circuit that uses only two energy-storage elements: a linear passive inductor and a linear passive capacitor. The other element is a nonlinear active memristor. Hence, the circuit has only three circuit elements in series. We discuss this circuit topology, show several attractors and illustrate local activity via the memristor's DC v_M - i_M characteristic.