International Journal of Bifurcation and Chaos

In this paper, we consider the family of rational maps

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits, we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four-dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied, we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed. The sticky chaotic orbits close to the bifurcation point of the simple unstable x1v2 orbit and close to the double unstable z-axis orbit that we have examined, have comparable diffusion speeds. These speeds are much slower than of the orbits in the neighborhood of x1v2 simple unstable periodic orbits away from the bifurcating point, or of the double unstable orbits of the same family having very different eigenvalues along the corresponding unstable eigendirections.

The paper presents a complete description of the periodic solutions of the Duffing equation:

These results are obtained by showing that the half-period Poincaré map offered by the Duffing equation is asymptotically equivalent to a map of a circle into itself, according to:

Chaos generation is an interesting research topic in the study of coupled complex dynamical networks. In this paper, based on mathematical analysis of Lyapunov exponent and boundedness of networks, the emergence of chaos for a class of nonlinear complex networks is investigated and some new criteria of chaos generation are derived. The effectiveness of theoretical results is verified by a numerical example.

In this paper, we propose a new asynchronous multiuser communication system based on spread spectrum and chaotic symbolic dynamics modulation. By combining spread spectrum and chaotic modulation, the proposed system provides increased security by reducing the probability of detection while allowing multiuser transmissions. The sequence synchronization of chaos communication system is studied. A time acquisition technique based on serial search is proposed to achieve synchronization. An analysis is carried out to determine the probability of detection, the probability of false alarm and the bit error rate of the system. Simulation results show that the proposed system can achieve sequence synchronization under low signal-to-noise ratios, and also confirm our analytically computed expressions.

In this paper, we examine the global dynamics of the complete Owen–Sherratt model describing the tumor–macrophage interactions. We show for this dynamics that there is a positively invariant polytope. We give upper and lower ultimate bounds for densities of cell populations involved in this model. Besides, we derive sufficient conditions under which each trajectory in tends to the mutant cells-free equilibrium point or to the equilibrium point of macrophages in isolation or to the coordinate plane corresponding to the absence of normal tissue cells depending on initial conditions. The biological sense of our results is discussed as well.

In this paper, we will analyze the bifurcation dynamics of an in vivo model of Plasmodium falciparum. The main attention of this model is focused on the dynamics of cross-reactivity from antigenic variation. We apply the techniques of coupled cell systems to study this model. It is shown that synchrony-breaking Hopf bifurcation occurs from a nontrivial synchronous equilibrium. In proving the existence of a Hopf bifurcation, we also discover the condition under which possible 2-color synchrony patterns arise from the bifurcation. The dynamics resulting from the bifurcation are qualitatively similar to known behavior of antigenic variation. These results are discussed and illustrated with specific examples and numerical simulations.

A novel approach for constructing a high-order Chua's circuit is proposed. Based on a dual-port RCL network, a high-order Chua's circuit which can generate multi-directional multi-scroll (MDMS) chaotic attractors can be realized by introducing a RC structure and suitable nonlinear functions. First, a fourth-order Chua's circuit is achieved by adding a capacitor, a resistance and a controlled-source constituted by stair functions in the third-order Chua's circuit. And then, this recursive method can be adopted in higher-order Chua's circuit which can generate multi-scroll chaotic attractors in more directions. Finally, a sixth-order Chua's circuit is designed and its experimental results demonstrate the feasibility of this method.

A delayed predator–prey system with diffusion and Dirichlet boundary conditions is considered. By regarding the growth rate a of prey as a main bifurcation parameter, we show that Hopf bifurcation occurs when the parameter a is varied. Then, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived.

We study the number of limit cycles of the discontinuous piecewise linear differential systems in ℝ²ⁿ with two zones separated by a hyperplane. Our main result shows that at most (8n - 6)^n-1 limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result, we use the averaging theory in a form where the differentiability of the system is not necessary.

Let T = {τ₁, τ₂, …, τ_K; p₁, p₂, …, p_K} be a position dependent random map on [0, 1], where {τ₁, τ₂, …, τ_K} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p₁, p₂, …, p_K} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.

This work deals with the nonlinear dynamics of a microelectromechanical system constituted by an imperfect microbeam under an axial load and an electric excitation. The device is characterized by a bistable static configuration. We analyze the single-mode dynamics and describe the overall scenario of the response, up to the inevitable escape, when both the frequency and the electrodynamic voltage are considered as driving parameters. We observe the presence of several competing attractors leading to a considerable versatility of behavior, which may have many feasible applications. Extensive numerical simulations are performed. The frequency-dynamic voltage behavior chart is obtained, which detects the theoretical boundaries of appearance and disappearance of the main attractors. Taking into account the erosion of the double well, we investigate the final response when each attractor vanishes. All these results represent the limit when disturbances are absent, which never occurs in practice. To extend them to the practical case where disturbances exist, we develop a dynamical integrity analysis. This is performed via curves of constant percentage of local integrity measure, which give quantitative information about the changes in the structural safety. For each attractor, we examine both the practical disappearance, by analyzing the robustness of its basin along the range of existence, and the practical final response, by detecting where safe jump to another attractor may be ensured and where instead dynamic pull-in may arise. These curves may be used to establish safety factors in order to operate the device according to the desired outcome, depending on the expected disturbances.

In this paper, we present some kinds of generalized homoclinic loops in planar piecewise smooth systems. By establishing Poincaré map, we study the stability of generalized homoclinic loops for each case. Some criteria are given for their stability, respectively. As applications, several concrete piecewise systems are considered.

Atmospheric flows exhibit fractal fluctuations and inverse power law for power spectra indicates an eddy continuum structure for the self-similar fluctuations. A general systems theory for aerosol size distribution based on fractal fluctuations is proposed. The model predicts universal (scale-free) inverse power law for fractal fluctuations expressed in terms of the golden mean. Atmospheric particulates are held in suspension in the fractal fluctuations of vertical wind velocity. The mass or radius (size) distribution for homogeneous suspended atmospheric particulates is expressed as a universal scale-independent function of the golden mean, the total number concentration and the mean volume radius. Model predicted spectrum is compared with the total averaged radius size spectra for the AERONET (aerosol inversions) stations Davos and Mauna Loa for the year 2010 and Izana for the year 2009. There is close agreement between the model predicted and the observed aerosol spectra. The proposed model for universal aerosol size spectrum will have applications in computations of radiation balance of earth–atmosphere system in climate models.

We provide a constructive method to obtain polynomial vector fields in ℝ³ having infinitely many limit cycles starting from polynomial vector fields in ℝ² with a period annulus. We present two examples of polynomial vector fields in ℝ³ having infinitely many limit cycles, one of them of degree 2 and the other one of degree 12. The main tools of our method are the Melnikov integral and the Hamiltonian structure.

In recent years, there has been expanding research on the applications of fractional calculus to the areas of signal processing, modeling and controls. Analog circuit implementation of chaotic systems is used in studying nonlinear dynamical phenomena, which is also applied in realizing the controller development. In this paper, chain fractance and tree fractance circuits are constructed to realize the fractional-order Chen–Lee system. The results are in good agreement with those obtained from numerical simulation. This study shows that not only is this system related to gyro motion but can also be applied to electronic circuits for secure communication.

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is proposed. We investigate bifurcations and basins in the parametrically forced system when logistic map is used for the one-dimensional subsystem. On a parameter plane, crossroad areas centered at fold cusp points for several orders are detected. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling intensity is analyzed. Basin bifurcations and preimages with respect to critical curves are described. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity regarding parameters.

For the 𝔰𝔬(4) free rigid body the stability problem for the isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found, we construct a Lyapunov function. These Lyapunov functions are linear combinations of Mishchenko's constants of motion.

Oscillatory patterns in a one-pool model with Ca²⁺-activated IP₃ degradation are presented and classified, based on fast/slow dynamical analysis and two-parameter bifurcations. The whole parameter space can be partitioned into two parts, that is, the oscillatory and rest regions. For this purpose, we perform the continuation of fold bifurcation, Hopf bifurcation and saddle-node bifurcation of limit cycles, respectively. Several distinct topological types of complex intracellular Ca²⁺ oscillations (especially bursting) are investigated in the oscillatory area. The classification and transition of different types of Ca²⁺ oscillations may offer a possible explanation for the differences in dynamic behavior observed in real cells in response to different levels of stimulation.

The existence of transport barriers in 3/2 degrees of freedom degenerate Hamiltonian systems is studied using the associated stroboscopic maps. For small enough amplitudes of the perturbations, a transport barrier (formed by infinitely many invariant rotational circles) is proven to exist and to form in the degenerate annulus. These results are applied to Hamiltonian models which describe some magnetic configurations in tokamak plasmas.

This paper discusses the planar closed orbits of the three-dimensional autonomous system

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, namely the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and long-term stability times, which are established under smallness conditions of the parameters. We also provide some applications to concrete examples, which exhibit a linear or long-term stability behavior.

We study the mechanism of experimentally observed phase waves and clusters in yeast extracts (cells with destroyed membranes) placed into the unstirred medium (gel). As a mathematical model, the distributed Selkov system is used, since it describes the key step of glycolytic reaction cascade — the phosphofructokinase-catalyzed reaction. We argue that the emergence of spatial phase clusters does not correspond to the Turing mechanism because diffusion coefficients used for two considered reagents are taken as equal. We show that the actual background of this phenomenon is connected with various local rotation velocities in phase space. In this case, large diffusion coefficients stabilize spatial patterns and small diffusion provides an asynchronous regime only.