International Journal of Bifurcation and Chaos

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.

We prove analytically that 2-element memristive circuits consisting of a passive linear inductor in parallel with a passive memristor, or an active memristive device, can be described explicitly by a Hamiltonian equation, whose solutions can be periodic or damped, and can be represented analytically by the constants of the motion along the circuit Hamiltonian. Generalizations to 3-element and 2N-element memristive Hamiltonian circuits are also presented where complex bifurcation phenomena including chaos, abound.

In this paper, we study bifurcation scenarios characterized by period-adding cascades with alternating chaos in one class of piecewise-smooth maps (PWS). In this class, the state space is separated in three smooth zones defined by a saturation function. Some power converters controlled by Digital Pulse-Width Modulation (PWM) are physical applications of this class of PWS systems denoted by PWS₃. Chaos has virtually been detected and studied in all disciplines, however the characterization problem of chaos scenarios has many open problems, mainly in nonsmooth dynamical systems. Novel bifurcation scenarios have recently been reported such as bandcount adding and bandcount increment scenarios based on the numerical detection of bands (where bands are considered as strongly connected components). However, this approach known as Bandcounter cannot be applied to detect bifurcations in chaos scenarios without crisis bifurcations or to identify topological changes inside of one-band chaos. We have proposed a novel framework named Dynamic Linkcounter approach to characterize chaos and torus breakdown scenarios in PWS systems. In this paper, we report overlapping period-adding cascades interspersed with a dynamic linkcount adding cascade. Each complex dynamic link (CDL) structure is a fingered strange attractor increasing in an arithmetic progression the number of CDL or fingers when a bifurcation parameter is varied. Alternative point of view based on tent-map-like structures is given to illustrate the formation of fingered strange attractors.

A consumer behavior model is considered in the context of a network of interacting individuals in an energy market. We propose and analyze a simple dynamical model of an ensemble of coupled active elements mimicking consumers' behavior, where "word-of-mouth" interactions between individuals are important. A single element is modeled using the automatic control system framework. Assuming local (nearest neighbor) coupling we study the evolution of chains and lattices of the model consumers when varying the coupling strength and initial conditions. The results are interpreted as the dynamics of the decision-making process by the energy-market consumers. We demonstrate that a pitchfork bifurcation to the homogeneous solution leads to bistability of stationary regimes, while the autonomous system is always monostable. In the presence of inhomogeneities, this results in the formation of clusters of sharply positive and negative opinions. We also find that, depending on the coupling strength, the perturbations caused by inhomogeneities can be exponentially localized in space or delocalized. In the latter case, the coarse-graining of opinion clusters occurs.

The multiarrow formalism for coupled cell networks permits multiple arrows and self-loops. The Lifting Theorem states that any such network is a quotient of a network in which all arrows are single and self-loops do not occur. Previous proofs are inductive, and give no useful estimate of the minimal size of the lift. We give a noninductive proof of the Lifting Theorem, and identify the number of cells in the smallest possible lift. We interpret this construction in terms of the type matrix of the network, which encodes its topology and labeling.

This paper deals with the Shimizu–Morioka system, a special generalized Lorenz canonical form. Using techniques of elimination in the computation of algebraic varieties we obtain parameter-dependent normal forms on a center manifold. Our computation shows that the maximal number of limit cycles produced from Hopf bifurcations is four and only even number of limit cycles can be bifurcated near the two equilibria because of -symmetry. Our parameter-dependent normal forms enable us to give parameter conditions for the cases of none, two and four limit cycles separately. Furthermore, considering exterior perturbations, we give conditions under which one or three limit cycles can be produced from Hopf bifurcations. Moreover, we also give conditions for fold bifurcations, under which limit cycles coincide or disappear. Finally, our results are illustrated by numerical simulations.

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

This paper considers the synchronization problem for nonlinear systems with time-delay couplings. We assume that the error dynamics can be rewritten as a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for time-delay coupled systems by applying the multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain stability condition and avoids the use of the Lyapunov–Krasovskii approach. As a result, the condition based on the circle criterion does not contain the conservativeness caused by the Lyapunov–Krasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling.

Radar waveforms based on chaotic systems have occasionally been suggested for a variety of radar applications. In this paper, radar waveforms are constructed with solutions from a particular chaotic system, the Lorenz system, and are called Lorenz waveforms. Waveform properties, which include the peak autocorrelation function side-lobe and the transmit power level, are related to the system parameters of the Lorenz system. Additionally, scaling the system parameters is shown to correspond to an approximate time and amplitude scaling of Lorenz waveforms and also corresponds to scaling the waveform bandwidth. The Lorenz waveforms can be generated with arbitrary time lengths and bandwidths and each waveform can be represented with only a few system parameters. Furthermore, these waveforms can then be systematically improved to yield constant-envelope output waveforms with low autocorrelation function sidelobes and limited spectral leakage.

A general method for formulating first-order time-delayed chaotic systems with simple linear time-delayed term is proposed. The formulated systems are realized with electronic circuit experiments. In order to determine the unknown coefficients in a general delayed differential equations for having chaotic solutions, we follow the route of period-doubling bifurcation to chaos. Firstly, the conditions for a time-delayed system having a stable periodic solution, generating from a destablized steady state, is analyzed with Hopf bifurcation theory. Then the delay time parameter is changed according to the bifurcation direction to search the chaotic state, which is identified by the Lyapunov exponents spectra. The theoretical analysis is well confirmed by numerical simulations and circuit experiments.

Firstly, the dynamics of the Lü system having an invariant algebraic surface are analyzed. Secondly, by using the Poincaré compactification in ℝ³, a global analysis of the system is presented, including the complete description of its dynamic behavior on the sphere at infinity. Lastly, combining analytical and numerical techniques, it is shown that for the parameter value b = 0, the system presents an infinite set of singularly degenerate heteroclinic cycles. The chaotic attractors for the Lü system in the case of small b > 0 are found numerically, hence the singularly degenerate heteroclinic cycles.

This paper proves that all the closed orbits (including limit cycles) of the general Lorenz family must be planar, but only to be curves in space based on the technique of classification and identification for the quadratic surface in three-dimensional space. This result indicates that the qualitative property of the closed orbit for the systems is very complicated. We hope that this study would be beneficial for further studies of the dynamically rich chaotic system.

We consider the problem of a classical Morse oscillator driven by an external periodic field of controllable characteristics and study systematically the effects of the initial phase of the external field on the probability of escape upon molecular dissociation at the end of the interaction time. First, it is shown that such effects indeed exist and may become important depending on the field amplitude and frequency. Then, an explanation is given in terms of the energy exchanges between the basic periodic orbits and the external field and of the associated modifications of the shape of regular islands in phase space. Finally, we exploit this explanation in order to make predictions regarding the behavior of the escape probability as a function of the initial phase and frequency of the external field.

Chaotic Ant Swarm (CAS) is an optimization algorithm based on swarm intelligence theory, which has been applied to find the global optimum solution in search space. However, it often loses its effectiveness and advantages when applied to large and complex problems, e.g. those with high dimensions. To resolve the problems of high computational complexity and low solution accuracy existing in CAS, we propose a Disturbance Chaotic Ant Swarm (DCAS) algorithm to significantly improve the performance of the original algorithm. The aim of this paper is achieved by three strategies which include modifying the method of updating ant's best position, neighbor selection method and establishing a self-adaptive disturbance strategy. The global convergence of the DCAS algorithm is proved in this paper. Extensive computational simulations and comparisons are carried out to validate the performance of the DCAS on two sets of benchmark functions with up to 1000 dimensions. The results show clearly that DCAS substantially enhances the performance of the CAS paradigm in terms of computational complexity, global optimality, solution accuracy and algorithm reliability for complex high-dimensional optimization problems.

Using the method of dynamical systems for the generalized Schrödinger equation, the bright soliton solution, dark soliton solution, uncountably infinite many periodic wave solutions and breaking bounded wave solutions are obtained. Exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.

A simple memristor-based chaotic circuit with an active flux-controlled memristor characterized by a smooth continuous cubic nonlinearity is designed. The proposed chaotic circuit can generate a 2-scroll chaotic attractor on a finite time scale and has an equilibrium set with its stability dependent on the initial state of the memristor. The complex dynamics of the proposed chaotic circuit under different initial state of the memristor are investigated both theoretically and numerically. In particular, some novel transient transition behaviors with different time scales are found in the memristor circuit. Experimental observations based on a universal circuit implementation platform are conducted to partially verify the numerical simulation results.

In this paper, a control method developed earlier is employed to consider controlling bifurcations in a class of Lorenz-like systems. Particular attention is focused on Hopf bifurcation control via linear and nonlinear stability analyses. The Lorenz system, Chen system and Lü system are studied in detail. Simple feedback controls are designed for controlling the stability of equilibrium solutions, limit cycles and chaotic motions. All formulas are derived in general forms including the system parameters. Computer simulation results are presented to confirm the analytical predictions.

Given a linear time-invariant control system, the purpose of this work is to define a four-parameter family of static state feedback such that the corresponding closed-loop control system exhibits controllable Hopf bifurcation of codimensions one and two.

This paper aims to propose a unified approach for the ultimate bound estimation of a class of High Dimensional Quadratic Autonomous Dynamical Systems (HDQADS). Using the proposed method and the optimization idea, a sufficient condition is then given for estimating the ultimate bounds of a class of HDQADS. To validate the above sufficient condition, this paper further investigates the ultimate bound estimation of a hyperchaotic system, a 6D and a 9D chaotic system, separately. Moreover, the ultimate bounds for a general Lorenz system, a low-order atmospheric circulation model, and a new 3D chaotic system are also discussed in detail. In particular, it should be pointed out that a unified and accurate ultimate bound estimation is attained for the generalized Lorenz system and it includes several well-known results as its special cases. Some numerical simulations are also given to verify and visualize the corresponding theoretical results.

In this paper, a new Lorenz-type system with chaotic attractor is formulated. The structure of the chaotic attractor in this new system is found to be completely different from that in the Lorenz system or the Chen system or the Lü system, etc., which motivates us to further study in detail its complicated dynamical behaviors, such as the number of its equilibrium, the stability of the hyperbolic and nonhyperbolic equilibrium, the degenerate pitchfork bifurcation, the Hopf bifurcation and the local manifold character, etc., when its parameters vary in their space. The existence or nonexistence of homoclinic and heteroclinic orbits of this system is also rigorously proved. Numerical simulation evidences are also presented to examine the corresponding theoretical analytical results.

Recently we have reported that a previously unidentified V-trough of sympathetic nerve activity (SNA) is a potential precursor of lethal cardiac events by examining 24-hour ambulatory electrocardiograms in which such an event was recorded by chance. The V-trough was marked by three consecutive compartments: a small variation lasting two hours, an abrupt descent lasting 30 min and a sharp ascent for 40 min. We reported that the hemodynamics consisting of heart rate, SNA and blood pressure (BP) is modeled excellently by the modification of a known chaotic electrical circuit, Chua circuit. A V-trough of SNA appears by increasing the resistive element between SNA and BP in the circuit, which corresponds to the impaired regulation of BP by SNA. This finding is consistent with an acknowledged finding that the depressed baroreflex (reflex of BP by SNA) may trigger a lethal arrhythmia.

An explicit upper bound Z(2, n) ≤ n + m - 1 is derived for the number of zeros of Abelian integrals M₁(h) = ∮_γ(h) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where γ(h) is an oval lying on the algebraic curve H_λ = (1/2)x² + (1/2)y² - (1/3)x³ - λy³ = h, P(x, y), Q(x, y) are polynomials of x and y, and max{deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M₁(h, λ) near λ = 0 and assume (∂^m/∂λ^m)M₁(h, λ)|_{λ = 0} not vanish identically.

We study the characteristics of stochastic resonance (SR) in the Duffing oscillator with three types of asymmetries in its double-well potential. The asymmetries controlled by a parameter α are introduced in the potential by varying (i) the depth of the left-well alone, (ii) the location of the minimum of the left-well alone and (iii) both depth and location of the minimum of the left-well alone. The characteristics of SR in the asymmetric cases are different from the symmetric case (α = 1). We find that asymmetry has a strong influence on the optimum noise intensity at which signal-to-noise ratio (SNR) is maximum, mean residence time at resonance and the probability distribution of residence time in the left- and right-wells. For a range of values of α, α ≠ 1, SNR is found to be relatively higher than for α = 1.

Network coding has been shown to be an effective coding technique to improve the throughput of a multicast session in a communication network. In general, an undirected network can be used to model a communication network in which nodes at both ends of a link can exchange messages. For a single multicast session in an undirected network with network coding, the maximum throughput can be estimated based on the minimum maxflow between the source node and the destination nodes. In this letter, we evaluate the impact of the network topology on the maximum multicast throughput of a network-coded communication network. In particular, the scalefree network, which represents the topology of most man-made networks including the Internet, will be compared with the random network. The maximum throughputs of the multicast sessions are evaluated for different choices of source and destination nodes. It is found that network coding can achieve better throughput of a scalefree network only if the source and destination nodes are chosen from high-degree nodes.

We perform an analytical study of the Hopf bifurcations and their degeneracies in Chua's equation. In the case of the equilibrium at the origin only codimension-two Hopf bifurcations appear. However, for the nontrivial equilibria we prove the existence of codimension-three Hopf bifurcations. Numerical results are in strong agreement with the analytical ones.