International Journal of Bifurcation and Chaos

Turing instability and pattern formation in the Lengyel–Epstein (L–E) model with superdiffusion are investigated in this paper. The effects of superdiffusion on the stability of the homogeneous steady state are studied in detail. In the presence of superdiffusion, instability will occur in the stable homogeneous steady state and more complex dynamics will exist. As a result of Turing instability, some patterns are formed. Through stability analysis of the system at the equilibrium point, conditions ensuring Turing and Hopf bifurcations are obtained. To further explore pattern selection, the weakly nonlinear analysis and multiple scale analysis are employed to derive amplitude equations of the stationary patterns. Then complex dynamics of amplitude equations, such as the existence of homogeneous solutions, stripe and hexagon patterns, mixed structure patterns, their stability, interaction and transition between them, are analyzed. Then different patterns occur immediately. Finally, the numerical simulations are presented to show the effectiveness of theoretical analysis and patterns are identified numerically. Whereas in the existing results of such model with normal diffusion, no amplitude equations are derived and patterns are only identified through numerical simulations.

Strong-Allee effect and competition coefficients provide useful insight to the dynamics of eco-epidemiological systems. Basically, different competition coefficients in the prey population lead to the emergent carrying capacity. The study of population dynamics under the influence of strong-Allee parameter has attracted a lot attention from researchers. But the dynamics of such systems under the combined effect of disease, Allee and emergent carrying capacity is yet to receive attention from researchers. A simple eco-epidemiological model with strong-Allee effect in prey population is proposed and analyzed with the aim to observe the dynamics of such system under the combined influence of strong-Allee parameter and competition coefficients. The basic mathematical features of the model are analyzed with the help of equilibrium analysis, stability analysis, bistability and bifurcation theory. Our numerical simulations reveal that in the absence of strong-Allee effect, the three species eco-epidemiological system produces chaos by increasing the force of infection. However, we observe that chaotic dynamics thus obtained can be controlled by the Allee parameter as well as the competition coefficients.

Cellular automata locally define dynamical systems, discrete in space, time and in the state variables, capable of displaying arbitrarily complex global emergent behavior. One core question in the study of cellular automata refers to their limit behavior, that is, to the global dynamical features in an infinite time evolution. Previous works have shown that for finite time evolutions, the dynamics of one-dimensional cellular automata can be described by regular languages and, therefore, by finite automata. Such studies have shown the existence of growth patterns in the evolution of such finite automata for some elementary cellular automata rules and also inferred the limit behavior of such rules based upon the growth patterns; however, the results on the limit behavior were obtained manually, by direct inspection of the structures that arise during the time evolution. Here we present the formalization of an automatic method to compute such structures. Based on this, the rules of the elementary cellular automata space were classified according to the existence of a growth pattern in their finite automata. Also, we present a method to infer the limit graph of some elementary cellular automata rules, derived from the analysis of the regular expressions that describe their behavior in finite time. Finally, we analyze some attractors of two rules for which we could not compute the whole limit set.

In this report, by the numerical continuation method, we visualize and connect hidden chaotic sets in the Glukhovsky–Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system.

In this paper, we consider a family of systems with two predators feeding on one prey. We show how to construct a positively invariant set in which it is possible to define a Poincaré map for examining the behavior of the system, mainly in the case when both predators survive. We relate it to examples from earlier works.

In this paper, we study random dynamical systems generated by two Allee maps. Two models are considered — with and without small random perturbations. It is shown that the behavior of the systems is very similar to the behavior of the deterministic system if we use strictly increasing Allee maps. However, in the case of unimodal Allee maps, the behavior can dramatically change irrespective of the initial conditions.

Applying some transformation to an autonomous system, we obtain a new system, which might keep the dynamical behavior of the original system or generate different dynamics. But this is often accompanied by the appearance of discontinuous points, where the vector field for the new system is not continuous at these points. We discuss the effects of the discontinuous points, and provide two methods to construct systems with any preassigned number of chaotic attractors via some transformation. The first one does not change the geometric structure of the attractors, since the discontinuous points are out of the basin of attraction. The second one might make the new systems have different dynamics, like multiscroll chaotic attractors, or infinitely many chaotic attractors. These results illustrate that both the equilibria and the discontinuous points affect the global dynamics.

Let $(X,d)$ be a compact metric space and $F=\{f_1,f_2,\dots ,f_n\}$ be an $n$-tuple of continuous selfmaps on $X$. This paper investigates Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence from a set-valued view. On the basis of this research, we draw the main conclusions as follows: (i) If $F$ has a distributionally chaotic pair, especially $F$ is distributionally chaotic, the strongly nonwandering set $S\mathrm{\Omega }(F)$ contains at least two points. (ii) We give a sufficient condition for $F$ to be distributionally chaotic in a sequence and chaotic in the strong sense of Li–Yorke. Finally, an example to verify (ii) is given.

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case ${\rho }_1^1>{\lambda }_1^1$, ${\rho }_2^1<{\lambda }_2^1$, ${\rho }_1^1{\rho }_2^1<{\lambda }_1^1{\lambda }_2^1$, where $-{\rho }_i^1<0$ and ${\lambda }_i^1>0$ are the pair of principal eigenvalues of unperturbed system at the critical point $p_i$, $i=1,2$. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

Supercavity can increase the velocity of underwater vehicles greatly, however the launching state of vehicle and systematic parameters often lead to unstable motion. To solve the problem, the effect of parameters and initial conditions on the stability of vehicles is studied. With two variable parameters, namely cavitation number and feedback control gain of fin deflection angle, a simple dynamic model of supercavity system is studied. The multistability is verified through simulation. Robustness of the system is also analyzed based on its basins of attraction. There are various coexisting attractors in a relatively large region of parameter space of the supercavity system, namely coexistence of a stable equilibrium point and a periodic attractor, coexistence of various periodic attractors, coexistence of a periodic attractor with a chaotic attractor and so on, which explain the effect of parameters and initial values on stability of vehicles qualitatively. In addition, without major change in cavitation number, there is a negative correlation between the robustness of the vehicle and feedback control gain of fin deflection angle. The robustness can be improved through optimization of parameters.

In this paper, a disease transmission model of SEIR type with stage structure is proposed and studied. Two kinds of time delays are considered: the first one is the mature delay which divides the population into two stages; the second one is the time lag between birth and being able to move freely, which we call the freely-moving delay. Our mathematical analysis establishes that the global dynamics are determined by the basic reproduction number $R_0$. If $R_0<1$, then the disease free equilibrium $E_1$ is globally asymptotically stable, and the disease will die out. If $R_0>1$, then a unique positive equilibrium $E_2$ exists, and $E_2$ is locally asymptotically stable when the freely-moving delay is less than the critical value. We show that increasing this delay can destabilize $E_2$ and lead to Hopf bifurcations and stable periodic solutions. By using the normal form theory and the center manifold theory, we derive the formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to verify the theoretical analysis and some biological implications are discussed.

Characterizing complicated behavior from time series constitutes a fundamental problem of continuing interest and it has attracted a great deal of attention from a wide variety of fields on account of its significant importance. We in this paper propose a novel wavelet multiresolution complex network (WMCN) for analyzing multivariate nonlinear time series. In particular, we first employ wavelet multiresolution decomposition to obtain the wavelet coefficients series at different resolutions for each time series. We then infer the complex network by regarding each time series as a node and determining the connections in terms of the distance among the feature vectors extracted from wavelet coefficients series. We apply our method to analyze the multivariate nonlinear time series from our oil–water two-phase flow experiment. We construct various wavelet multiresolution complex networks and use the weighted average clustering coefficient and the weighted average shortest path length to characterize the nonlinear dynamical behavior underlying the derived networks. In addition, we calculate the permutation entropy to support the findings from our network analysis. Our results suggest that our method allows characterizing the nonlinear flow behavior underlying the transitions of oil–water flows.

Predicting the nonlinear vibration responses of a Z-shaped folding wing during the morphing process is a prerequisite for structural design analysis. Therefore, the present study focuses on the nonlinear dynamical characteristics of a Z-shaped folding wing. The folding wing is divided into three carbon fiber composite plates connected by rigid hinges. The nonlinear dynamic equations of the system are derived using Hamilton’s principle based on the von Kármán equations and classical laminate plate theory. The mode shape functions of the system are then obtained using finite element analysis. Galerkin’s approach is employed to discretize the partial differential governing equations into a two-degree-of-freedom nonlinear system. The case of 1:1 inner resonance is considered. The method of multiple scales is employed to obtain the averaged equations of the system. Finally, numerical simulation is performed to investigate the nonlinear dynamical characteristics of the system. Bifurcation diagrams and wave-form diagrams illustrate the different motions of the Z-shaped folding wing, including periodic and chaotic motions under given conditions. The influence of transverse excitations on the bifurcations and chaotic motion of the Z-shaped folding wing is investigated numerically.

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation $\mathrm{\Omega }=n\omega +𝜖\nu$ (where $\nu$ is not rational relative to $\omega$) and $n=1,2,4,6$ is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when $n=3,5$,7–15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincaré maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.

We study the maximal number of limit cycles of piecewise linear near-Hamiltonian systems, whose unperturbed system has two saddles. Using the method developed in [Xiong, 2015], we prove that two limit cycles can be bifurcated from the periodic orbits.

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e. type I, type I${}^{\prime }$, type II, type II${}^{\prime }$, type III, and type III${}^{\prime }$, and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large deviations theorem of various types. Some characteristics of the ergodic measure are obtained and then applied to prove that every dynamical system satisfying the large deviations theorem of type I${}^{\prime }$ is ergodic, which is equivalent to the large deviations theorem of type II${}^{\prime }$ in this regard, and that every uniquely ergodic dynamical system restricted on its support satisfies the large deviations theorem. Moreover, we prove that every dynamical system satisfying the large deviations theorem of type III is an $E$-system.

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

In this paper, we study a model of generalized Dullin–Gottwald–Holm equation, depending on the power law nonlinearity, that derives a series of planar dynamical systems. The study of the traveling wave solutions for this model derives a planar Hamiltonian system. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions, under different parametric conditions. These results completely improve the study of traveling wave solutions to the mentioned model stated in [Biswas & Kara, 2010].

Recent evidence suggests that a system with only stable equilibria can generate chaotic behavior. In this work, we study a chaotic system with two stable equilibrium points. The dynamics of the system is investigated via phase portrait, bifurcation diagram and Lyapunov exponents. The feasibility of the system is introducing its electronic realization. Moreover, the chaotic system is used in Symmetric Chaos Shift Keying (SCSK) and Chaotic ON-OFF Keying (COOK) modulated communication designs for secure communication. It is determined that the SCSK modulated communication system implemented with the chaotic system is more successful than COOK modulation for secure communication.

In this paper, the unique normal form for a class of three-dimensional nilpotent vector fields with symmetry is investigated. The key technique used is a combination of multiple Lie brackets, linear grading function, new notations of block matrices and first integral of the linear part of the given vector field which avoids complicated calculations. The first and the second order normal forms are computed successively, and the uniqueness of the second order normal form is proved under certain conditions.