International Journal of Bifurcation and Chaos

We study the dynamics in the neighborhood of fixed points in a 4D symplectic map by means of the color and rotation method. We compare the results with the corresponding cases encountered in galactic type potentials and we find that they are in good agreement. The fact that the 4D phase space close to fixed points is similar to the 4D representations of the surfaces of section close to periodic orbits, indicates an archetypical 4D pattern for each kind of (in)stability, not only in 3D autonomous Hamiltonian systems with galactic type potentials but for a larger class of dynamical systems. This pattern is successfully visualized with the method we use in the paper.

The approach to research the uncertainties and the regularities of nonlinear system behavior is developed in the paper. The peculiarity of the approach is connected with modifying the experimental bifurcation diagrams that allows to identify and to analyze certain aspects of the nonlinear dynamics evolution. The variety and the interrelation of the modified bifurcation diagrams are shown. Attention is focused on estimating the limits of the zone within which the nonlinear system behavior is characterized by the uncertainty and on making visible the particular tendencies of dynamics evolution within this zone. The results obtained on the experimental setup of PWM buck converter are used for illustrations. In particular, the uncertainty zone is assumed to be the unit of measurement to estimate the operating process stability margin. It is demonstrated that the properties corresponding to the practical experience appear as per this assumption.

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we verify that they are stable, and thus proper sinks. Regions of existence in parameter space of the found sinks are located using a continuation method; the basins of attraction are found numerically.

Finite time coherent sets [Froyland et al., 2010] have recently been defined by a measure-based objective function describing the degree that sets hold together, along with a Frobenius–Perron transfer operator method to produce optimally coherent sets. Here, we present an extension to generalize the concept to hierarchically define relatively coherent sets based on adjusting the finite time coherent sets to use relative measures restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the nonautonomous double gyre, the standard map, an idealized stratospheric flow, and empirical data from the Mexico Gulf during the 2010 oil spill. Also for the sake of analysis of computational complexity, we include an Appendix concerning the computational complexity of developing the Ulam–Galerkin matrix estimates of the Frobenius–Perron operator centrally used here.

This paper investigates nonlinear bifurcation behaviors in the input-series output-parallel (ISOP) connected DC–DC converters under peak current control operating in continuous conduction mode. Circuit simulation demonstrates typical nonlinear phenomena occurring in the ISOP connected DC–DC converters. A precise discrete-time mapping model is presented to analyze the stability and bifurcation phenomena in the ISOP connected DC–DC converters. Interaction of the outside voltage loop and inner loop leads to coexisting slow-scale and fast-scale bifurcation which is investigated under two different values of the bandwidth.

Vibratory behavior of two coupled oscillators is studied. The main system — Dahl type — is coupled to a very light system with a nonsmooth potential that can be endowed for passively controlling the main system. Invariant manifold of the system at the fast time scale is revealed and the system behavior at slow time scale around the infinity of the fast time scale is detected. This can give us the chance to forecast all possible attractors of the system during energy exchange between the two oscillators.

This paper further investigates a novel method, namely the mirror and double-mirror symmetry conversion both in z direction and in y direction, for generating complex grid multiwing chaotic attractors from a three-dimensional quadratic chaotic system. First, by designing a switching controller with even-symmetry multisegment quadratic function to extend the number of saddle-focus equilibrium points with index 2 in x direction, multiwing chaotic attractors are obtained. Based on this approach, then, different from the methods proposed in the previous literature, by the mirror and double-mirror symmetry conversion with respect to x-axis and y-axis respectively, various intended grid n × m-wing chaotic systems can be obtained. The principle and method for generating grid multiwing are also given. Numerical simulations and circuit realizations have demonstrated the feasibility and effectiveness of the proposed approaches.

In this paper, we consider Liénard systems of the form

This work introduces a FitzHugh–Nagumo (FHN) model of credit risk contagion based on the FHN system, which contains time-delay, Gaussian white noise, delayed feedback, weak periodic signal, and nonlinear resistance. The model depicts the dynamics behavior characteristics of evolution of credit risk contagion through simulation experiments. Meanwhile, numerical simulations show that, in a financial market, the dynamics system stability of credit risk contagion is positively related to the nonlinear resistance among participants of credit activities and to the inherent recovery capability attributed to the after-credit risk impact on economic subjects. However, the dynamics system stability of credit risk contagion is negatively related to the time-delay of credit risk contagion, the strength of Gaussian white noise, and the weak-signal cycle. Furthermore, the dynamics system of credit risk contagion introduces a series of Hopf bifurcation, inverse bifurcation and different degrees of chaotic oscillation phenomena with changes in these parameters.

Recently, simple dynamical systems such as the 1-d maps on the interval, gained significant attention in the context of statistical physics and complex systems. The decay of correlations in these systems, can be characterized and measured by correlation functions. In the context of symbolic dynamics of the nonchaotic multifractal attractors (i.e. Feigenbaum attractors), one observable, the symbol-to-symbol correlation function, for the generating partition of the logistic map, is rigorously introduced and checked by numerical experiments. Thanks to the Metropolis–Stein–Stein (MSS) algorithm, this observable can be calculated analytically, giving predictions in absolute accordance with numerical computations. The deep, algorithmic structure of the observable is revealed clearly reflecting the complexity of the multifractal attractor.

In this paper, a general and systematic scheme is provided to research strong and weak resonances derived from delay-induced various double Hopf bifurcations in delayed differential systems. The method of multiple scales is extended to obtain a common complex amplitude equation when the double Hopf bifurcation with frequency ratio k₁:k₂ occurs in the systems under consideration. By analyzing the complex amplitude equation, we give the conditions of the strong and weak resonances respectively in the analytical expressions. The weak resonances correspond to the codimension-two double Hopf bifurcations since the amplitudes and the phases may be decoupled, but the strong resonances to the codimension-three double Hopf bifurcations in the system. It is seen that the weak resonances happen in the system even for a lower-order ratio, i.e. k₁ + k₂ ≤ 4. As applications, two examples are displayed. Three cases of the delay-induced resonance with 1:2, 1:3, 1:5 and are discussed in detail and the corresponding normals are represented. Thus, the relative dynamical behaviors can be easily classified in the physical parameter space in terms of nonlinear dynamics. The results show the provided conditions may be used to determine that a resonance is strong or weak.

In this paper, we present a simple approach to produce n-scroll chaotic attractors in 3D quadratic continuous time systems. The method of analysis is based on the number of equilibrium points. Some numerical results are also given and discussed.

In this paper, we study two classes of three-dimensional differential systems. By using the Poincaré map and the averaging method, we give sufficient conditions for the existence of invariant surfaces consisting of periodic orbits for the systems. Illustrative examples are given for our main results.

In the present paper, we study the effect of time delays in evolutionary games with one population of users and two strategies. The case where the delays, corresponding to different strategies, are not the same is considered. The local stability of the stationary state for the replicator dynamics is analyzed. We show that there is Hopf bifurcation. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and normal form theory. The stochastic evolutionary game with delay is taken into consideration. We also study the behavior of the first and second solution moments for linear stochastic differential delay equation in the presence of white and colored noise. The last part of the paper includes numerical simulations and conclusions.

In this paper, two notes about the ergodicity of partially hyperbolic systems are given. First one is the ergodicity for a C² volume preserving partially hyperbolic diffeomorphism of a smooth compact Riemannian manifold which is essentially accessible and weak central exponentially bunched. Second one is that for a C² partially hyperbolic diffeomorphism, if both forward and backward center bunched are a full probability set, then it is center bunched in the sense of [Burns & Wilkinson, 2010].

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

In this paper, by using the topological degree theory for multivalued maps and the method of guiding functions, the existence and global bifurcation for periodic solutions of a class of differential variational inequalities are studied.

In this paper, a unified "scalar" discrete-time model is proposed for enhanced bifurcation prediction in digitally controlled H-bridge grid-connected inverter system. The model is established by analytically expressing the elements of the matrix exponential involved in the exact discrete-time model. In high-order system, different from the discrete-time model using the state-transition matrix directly, obtaining the discrete-time model of the state-variable vector only, in this system, the discrete-time model of each state variable can be obtained directly and separately. In addition, the stability boundary of Hopf bifurcation, period-doubling bifurcation and saddle-node bifurcation can be obtained by analytical expressions. The modeling method is validated by an LCL-filter system as an example using practical circuit parameters. The results of the analysis can simplify the modeling process and enhance the bifurcation prediction of the digitally controlled inverter system.

We construct examples proving that there exist C¹-dynamical systems in [0, 1) having forward orbits unstable in the sense of Lyapunov with negative Lyapunov exponent and forward orbits stable in the same sense, but having positive Lyapunov exponents. Such examples complete some results from Demir and Koçak on one-dimensional systems and prove the appearance of Perron effects in such systems. Additionally we consider similar problems in two-dimensional dynamical systems and also in the setting of nonautonomous systems.

This paper analyzes the dynamics of the spread of tuberculosis (TB) on complex metapopulation, that is, networks of populations connected by migratory flows whose configurations are described in terms of connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. The migration and transmission processes occur simultaneously. For uncorrelated networks, we give a necessary and sufficient condition for the instability of the disease-free equilibrium. The existence of endemic equilibria is also discussed. Finally, the prevalence of the TB infection across the metapopulation as a function of the path connectivity is studied using numerical simulations.

We consider a nonlinear system perturbed by two harmonic forcings of different frequencies. The slow forcing drives the system into an oscillatory regime while the fast perturbation enhances the effect of the slow periodic drive. The vibrational resonance occurs when this enhancement is optimal, usually when the fast perturbation has an amplitude much higher than the slow periodic forcing. We show that this resonance can also happen when the amplitude of the fast perturbation is far below the amplitude of the slow periodic forcing due to a peculiar condition of the phase space. Moreover, this resonance presents an extreme sensitivity to small variations of the fast perturbation. We explore here this phenomenon that we call ultrasensitive vibrational resonance.