International Journal of Bifurcation and Chaos

This paper proposes to use a state feedback method to control the Hopf bifurcation for a novel congestion control model, i.e. the exponential random early detection (RED) algorithm with a single link and a single source. The gain parameter of the congestion control model is chosen as the bifurcation parameter. The analysis shows that in the absence of the state feedback controller, the model loses stability via the Hopf bifurcation early, and can maintain a stationary sending rate only in a certain domain of the gain parameter. When applying the state feedback controller to the model, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the model possesses a stable sending rate in a larger parameter range. Furthermore, explicit formulae to determine the properties of the Hopf bifurcation are obtained. Numerical simulations are given to justify the validity of the state feedback controller in bifurcation control.

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

We report the theory and experiment of a new time-delayed chaotic (hyperchaotic) system with a single scalar time delay and a nonlinearity described by a closed form mathematical function. Detailed stability and bifurcation analyses establish that with the suitable delay and system parameters, the system shows a stable limit cycle through a supercritical Hopf bifurcation. Numerical simulations exemplify that the system depicts mono-scroll and double-scroll chaos and hyperchaos for a range of delay and other system parameters. Nonlinear behavior of the system is characterized by Lyapunov exponents and Kaplan–Yorke dimension. It is established that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate time delay. Finally, the system is implemented in an analogue electronic circuit using off-the-shelf circuit elements. It is shown that the behavior of the time delay chaotic electronic circuit qualitatively agrees well with our analytical and numerical results.

The dynamics of the map

It is shown that the dissipative chaotic dynamics of a charged particle in the field of a wave packet with an arbitrary but finite number of harmonics can be reliably suppressed by judiciously varying the constant phase of the main harmonic, ϕ₀, while keeping null the corresponding constant phases of the remaining harmonics. The dependence of the chaotic threshold on the wave packet parameters is predicted theoretically (Melnikov method) and confirmed numerically (Lyapunov exponents). In particular, it is shown that ϕ₀ is effective at suppressing the chaotic behavior existing when ϕ₀ = 0 over a wide range of values of the wave packet width, while the remaining parameters are kept constant.

Line spectrum of noise radiated from machinery vibrations of underwater vehicles is one of the most harmful signals that expose the characteristics of vehicles and locations. In order to distort the features and restrain the intensity of the line spectra, we attempt to chaotify the vibration system by time delay control. To avoid blindly numerical testing of the control parameters, stability of a two-dimensional vibration isolation floating raft system with two time-delayed feedback control is studied in this paper, aiming to provide guidance for chaotification. The system with dual equal time delay is investigated by the generalized strum method and the polynomial eigenvalues are adopted to analyze the stability of the controlled vibration isolation system with two unequal time delays. The critical control gains and delays for stability switches are obtained. By adjusting the control parameters beyond stable region, it is feasible to chaotify the system. Numerical simulations are conducted to compare the effect of two time delay with different control parameters and different control scheme to complicate the vibration isolation system.

We show that each sensitive group action on a dendrite contains an Auslander–Yorke chaotic subsystem. By this conclusion, we prove that each sensitive group action on a dendrite must have positive geometric entropy when the acting group is finitely generated, and no dendrite admits a sensitive nilpotent group action.

In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincaré Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.

The undesirable intermittency in periodically driven nonlinear systems such as switching power converters is a more common problem in power electronic circuit design. Such operations are usually avoided by modifying the circuit parameters which necessitates a proper understanding of the effects of various spurious signals. In the present work, the effect of different types of periodic interference signals like sinusoidal, triangular and saw-tooth waveforms have been considered and the influence of these signals in the input, control and reference voltages of the buck converter has been investigated. It is found that the presence of sinusoidal or triangular interference signal whose frequency is comparable with the switching frequency of the converter or its rational multiples manifests as symmetrical period-doubling bifurcation in intermittent periods. However, the presence of either sinusoidal or triangular interference signal having irrational frequency ratios with the switching frequency of the converter leads to quasi-periodic route to chaos. Also it is observed that the influence of saw-tooth interference signal results in intermittent chaos in the dynamics of the system. Further, the effect of simultaneous presence of more than one interference signal has been considered. The dynamics of this converter system has been mathematically described and analyzed with a simple discrete map. The ordered and the chaotic dynamics of this system have been investigated with suitable analytical, numerical and experimental means.

In this note, we study smooth classification of all germs of 0-resonant diffeomorphisms on ℝ³ having generic nonlinear parts. We prove that, for the Poincaré type diffeomorphisms, except for one germ, any two such germs are at least C³ conjugated if and only if their linear parts are similar; and for the non-Poincaré ones, any such two germs can be C^∞ conjugated provided that they have similar linear parts.

This paper presents a study of two-dimensional hexagonal cellular automata (CA) with periodic boundary. Although the basic construction of a cellular automaton is a discrete model, its global level behavior at large times and on large spatial scales can be a close approximation to a continuous system. Meanwhile CA is a model of dynamical phenomena that focuses on the local behavior which depends on the neighboring cells in order to express their global behavior. The mathematical structure of the model suggests the importance of the algebraic structure of cellular automata. After modeling the dynamical behaviors, it is sometimes an important problem to be able to move backwards on CAs in order to understand the behaviors better. This is only possible if cellular automaton is a reversible one. In the present paper, we study two-dimensional finite CA defined by hexagonal local rule with periodic boundary over the field ℤ₃ (i.e. 3-state). We construct the rule matrix corresponding to the hexagonal periodic cellular automata. For some given coefficients and the number of columns of hexagonal information matrix, we prove that the hexagonal periodic cellular automata are reversible. Moreover, we present general algorithms to determine the reversibility of 2D 3-state cellular automata with periodic boundary. A well known fact is that the determination of the reversibility of a two-dimensional CA is a very difficult problem, in general. In this study, the reversibility problem of two-dimensional hexagonal periodic CA is resolved completely. Since CA are sufficiently simple to allow detailed mathematical analysis, also sufficiently complex to produce chaos in dynamical systems, we believe that our construction will be applied many areas related to these CA using any other transition rules.

The chaotic system has been exploited in metaheuristic methods of solving global optimization problems having a large number of local minima. In those methods, the selection of chaotic system is significantly important to search for solutions extensively. Recently, a novel chaotic system, the gradient model with perturbation methods (GP), was proposed, which can be regarded as the steepest descent method for minimizing an objective function with additional perturbation terms, and it is reported that chaotic metaheuristic method with the GP model has a good performance of solving some benchmark problems through numerical experiments. Moreover, a sufficient condition of parameter was theoretically shown for chaoticity in a simplified GP model where the descent term for the objective function is removed from the original model. However, the shown conditions does not provide enough information to select parameter values in the GP model for metaheuristic methods. Therefore, in this paper, we theoretically derive a sufficient condition under which the original GP model is chaotic, which can be usefully exploited for an appropriate selection of parameter values. In addition, we examine the derived sufficient condition by calculating the Lyapunov exponents of the GP model, and analyze its bifurcation structure through numerical experiments.

Since all kinds of noise exist in signals from real-world systems, it is very difficult to exactly estimate Lyapunov exponents from this time series. In this paper, a novel method for estimating the Lyapunov spectrum from a noisy chaotic time series is presented. We consider the higher-order mappings from neighbors into neighbors, rather than the mappings from small displacements into small displacements as usual. The influence of noise on the second-order mappings is researched, and an averaging method is proposed to cope with this noise. The mappings equations of the underlying deterministic system can be obtained from the noisy data via the method, and then the Lyapunov spectrum can be estimated. We demonstrate the performance of our algorithm for three familiar chaotic systems, Hénon map, the generalized Hénon map and Lorenz system. It is found that the proposed method provides a reasonable estimate of Lyapunov spectrum for these three systems when the noise level is less than 20%, 10% and 7%, respectively. Furthermore, our method is not sensitive to the distribution types of the noise, and the results of our method become more accurate with the increase of the length of time series.

An explicit, computationally efficient, recursive formula is presented for computing the normal form and center manifold of general n-dimensional systems associated with Hopf bifurcation. Maple program is developed based on the analytical formulas, and shown to be computationally efficient, using two examples.

New existence conditions of homoclinic orbits for some systems of ordinary quadratic differential equations with singular linear part are found. A realization of these conditions guarantees the existence of chaotic attractors at 3D autonomous quadratic systems. In addition, a chaotic behavior of solutions of these systems is determined by the 1D discrete map at some values of parameters r > 0, p ∈ ℝ, and γ ∈ (-d, ∞), where d > 0; n = 0, 1, 2, …. Examples of chaotic attractors are given.

This paper is concerned with the bifurcation of limit cycles from the period annulus of a quadratic reversible system. The outer boundary of the period annulus contains a degenerate critical point. The exact upper bound of the number of limit cycles is given. Our result shows that the conjecture on the cyclicity of (r4) system is correct.

The aim of this work is to study the existence and stability of traveling wave solutions for discrete-time model of delayed cellular neural networks distributed in the one-dimensional integer lattice ℤ¹. Since the dynamics of each given cell depends on its left and right neighboring cells, it is not easy to construct the traveling wave solutions. Using the method of step along with positive characteristic roots of the equations, we successfully prove the existence of traveling wave solutions. Moreover, we show that all the traveling wave solutions are unstable. We also provide some numerical results to support our results, and point out the different structures of traveling wave solutions between the continuous-time and discrete-time models.

This paper reports the finding of a five-dimensional (5D) new hyperchaotic system with three positive Lyapunov exponents, which is obtained by adding a nonlinear controller to the first equation of a 4D hyperchaotic system. The algebraical form of the hyperchaotic system is very similar to the 5D controlled Lorenz-like systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under unique equilibrium or three equilibria. To further analyze the new system, the corresponding hyperchaotic and chaotic attractor are firstly numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation, analysis of power spectrum and Poincaré projections. Moreover, some complex dynamical behaviors such as the stability of hyperbolic or nonhyperbolic equilibrium and two complete mathematical characterizations for 5D Hopf bifurcation are rigorously derived and studied.

We introduce the notion of (f, δ)-recurrence for the so-called (f, δ)-processes. We show that if a function of type 2^∞ has no attractive periodic points of periods greater than 2^N for some positive integer N and has an infinite ω-limit set, then each point belonging to its infinite ω-limit set is (f, δ)-recurrent provided that δ is small enough.

The existence of homoclinic orbit to a saddle focus point for third-order ODE systems is studied in this paper. Through a discussion on the absence of homoclinic orbit, we obtain the necessary condition for the coexistence of homoclinic orbit and saddle focus point for such systems. Some properties of the homoclinic orbit are also exposed. An example from Sprott systems is given to show the application of the results. Finally, a conclusion and a problem are proposed.

The memristor is a novel circuit element which is capable of maintaining an activity-dependent nonvolatile resistance and is therefore a candidate for use in next-generation storage and logic circuits. In this article, we present a model of the PEO-PANI memristor for use in the SPICE circuit simulation program which is especially suited to analog logic applications. Two variants are presented herein; accompanying each is a short description that explains any design decisions made, as well as elucidating on preferred simulation settings. It is shown that the model accurately replicates corresponding experimental results found in the literature. Simple simulations are used to show the suitability of each variant to specific experimental usage. Appendices contain verbatim implementations of the SPICE models.