International Journal of Bifurcation and Chaos

A general formalism describing a type of energy-conservative system is established. Some possible dynamic behaviors of such energy-conservative systems are analyzed from the perspective of geometric invariance. A specific 4D chaotic energy-conservative system with a line of equilibria is constructed and analyzed. Typically, an energy-conservative system is also conservative in preserving its phase volume. The constructed system however is conservative only in energy but is dissipative in phase volume. It produces energy-conservative attractors specifically exhibiting chaotic 2-torus and quasiperiodic behaviors including regular 2-torus and 3-torus. From the basin of attraction containing a line of equilibria, the hidden nature of chaotic attractors generated from the system is further discussed. The energy hypersurface on which the attractors lie is determined by the initial value, which generates complex dynamics and multistability, verified by energy-related bifurcation diagrams and Poincaré sections. A new type of coexistence of attractors on the equal-energy hypersurface is discovered by turning the system parameter values to their opposite. The basins of attraction under three sets of parameter values demonstrate that the Hamiltonian is the leading factor predominating the dynamic behaviors of the system with a closed energy hypersurface. Finally, an analog circuit is designed and implemented to demonstrate the consistent theoretical and simulation results.

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

In this work, we present a monoparametric family of piecewise linear systems to generate multiscroll attractors through a polynomial family defined by path curves that connect to the roots. The idea is to define path curves where the roots of a polynomial can take values by determining an initial and a final polynomial. As a consequence, structural stability and bifurcation of the system can be obtained. Structural stability is obtained by preserving the same stability of the initial and final polynomials. However, the system bifurcates by changing the stability of the final polynomial with respect to the initial polynomial. The aim is achieved by the design of a piecewise linear controller that is applied to affine linear systems. Our results are mathematically proved and numerical examples are also provided to illustrate the approach.

In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns.

The article is devoted to the study of the global behavior of the generalized Rabinovich system describing the process of interaction between waves in plasma. For this generalized system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using the approach where we transform the initial problem of finding the corresponding set to the conditional extremum problem and solve this problem. Furthermore, the rate of the trajectories going from the exterior of the attractive set to the interior of the attractive set is also obtained. Numerical localization of attractor is presented. Meanwhile, the volumes of the ultimate bound set and the global exponential attractive set are obtained, respectively. The main innovation of this article lays in considering the generalized form of the Rabinovich system with $v_3>0$ and obtaining the results on the ultimate bound set and globally exponential attractive set for this general case.

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

In this paper, a predator–prey model with prey-stage structure and prey-taxis is proposed and studied. Firstly, the local stability of non-negative constant equilibria is analyzed. It is shown that non-negative equilibria have the same stability between ODE system and self-diffusion system, and self-diffusion does not have a destabilization effect. We find that there exists a threshold value ${\xi }_0$ such that the positive equilibrium point of the model becomes unstable when the prey-taxis rate $\xi <{\xi }_0$, hence the taxis-driven Turing instability occurs. Furthermore, by applying Crandall–Rabinowitz bifurcation theory, the existence, the stability and instability, and the turning direction of bifurcating steady state are investigated in detail. Finally, numerical simulations are provided to support the mathematical analysis.

The importance of cooperation is self-evident to humans, yet the existence of corruption where law violators can avoid being punished by paying bribes to corrupt law enforcers may threaten the maintenance of cooperation. Although powerful monitoring has been used to resolve such matters, existing studies show that the effects of such measures are either transient or uncertain. Thus how to efficiently control the occurrence of corruption for the emergence of cooperation remains a challenge. Here, we introduce social exclusion into the public goods game, and respectively propose three measures to control corruption, namely, the exclusion of corrupt punishers, the exclusion of corrupt defectors, and the exclusion of both corrupt punishers and corrupt defectors. Our results show that the system dynamics driven by these three measures can exhibit many interesting dynamical outcomes including the dominance of defectors, rock-scissors-paper cycle, heteroclinic cycle, or interior attractor. We further demonstrate that excluding corrupt punishers can improve the situation of corruption more efficiently than excluding corrupt defectors. In addition, excluding both corrupt defectors and corrupt punishers can more effectively promote the emergence of cooperation for a broad parameter range.

In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic bifurcations of group-invariant solutions for a generalized stochastic Zakharov–Kuznetsov equation.

A singularity is described that creates a forward time loss of determinacy in a two-timescale system, in the limit where the timescale separation is large. We describe how the situation can arise in a dynamical system of two fast variables and three slow variables or parameters, with weakly coupling between the fast variables. A wide set of initial conditions enters the $𝜖$-neighborhood of the singularity, and explodes back out of it to fill a large region of phase space, all in finite time. The scenario has particular significance in the application to piecewise-smooth systems, where it arises in the blow up of dynamics at a discontinuity and is followed by abrupt recollapse of solutions to “hide” the loss of determinacy, and yet leave behind a remnant of it in the global dynamics. This constitutes a generalization of a “micro-slip” phenomenon found recently in spring-coupled blocks, whereby coupled oscillators undergo unpredictable stick-slip-stick sequences instigated by a higher codimension form of the singularity. The indeterminacy is localized to brief slips events, but remains evident in the indeterminate sequencing of near-simultaneous slips of multiple blocks.

Ising model at zero temperature leads to a ferromagnetic state asymptotically. There are two such possible states linked by symmetry, and Glauber–Ising dynamics are employed to reach them. In some stochastic or deterministic dynamical systems, the same absorbing state with ${{𝒵}}_2$ symmetry is reached. This transition often belongs to the directed Ising (DI) class where dynamic exponents and persistence exponent are different. In asymmetrically coupled sequentially updated logistic maps, the transition belongs to the DI class. We study changes in the nature of transition with an update scheme. Even with the synchronous update, the transition still belongs to the DI class. We also study a synchronous probabilistic update scheme in which each site is updated with the probability $p$. The order parameter decays with an exponent $0.5$ in this scheme. Nevertheless, the dynamic exponent $z$ is less than $2$ even for small values of $p$ indicating a very slow crossover to the Ising class. However, with a random asynchronous update, we recover $z=2$. In the presence of feedback, synchronous update leads to a transition in the DI universality class which changes to Ising class for synchronous probabilistic update.

Chaotic systems with hidden attractors have been widely studied in recent decades. In this field, systems without equilibrium are of special interest. Since multistable systems can alternate between several hidden attractors, it is difficult to detect their hidden oscillations using known analytical techniques. In this study, we propose to apply recurrence analysis methods as a possible solution to this problem. We consider the new quantity measure, called relative specific volume, which correlates with changes of the attractor. We choose the improved Sprott A system as an example of the multiscroll attractor to verify the proposed approach. We compare the new recurrence measure with other traditional quantitative methods for chaotic systems investigation. It is shown that the relative specific volume possesses the best correlation with the change of attractors among all considered metrics. The proposed approach can be used to study hidden oscillations in experimental data as well.

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

In this paper, a chaotic duplex H.264-codec-based secure video communication scheme is designed and its smartphone implementation is also carried out. First, an improved self-synchronous chaotic stream cipher algorithm equipped with a sinusoidal modulation, a multiplication, a modulo operation and a round down operation (SCSCA-SMMR) is developed. Using the sinusoidal modulation and multiplication, the improved algorithm can resist the divide-and-conquer attack by traversing multiple nonzero component initial conditions (DCA-TMNCIC). Meanwhile, also by means of the round down operation and modulo operation, on the premise that the DCA-TMNCIC does not work, the original keys cannot be further deciphered only by the known-plaintext attack, the chosen-plaintext attack and the chosen-ciphertext attack, respectively. Then, the Android low-level multimedia support infrastructure MediaCodec class is used to access low-level media encoder/decoder components and the H.264 hardware encoding/decoding is performed on real-time videos, so the chaotic video encryption and decryption can be realized in real-time by smartphones. Security analysis and smartphone experimental results verify the effectiveness of the proposed method.

The objective of this study is to investigate the complex dynamics of an eco-epidemic predator–prey system where disease is transmitted in prey species and predator population is being provided with alternative food. Holling type-II functional response is taken into consideration for interaction of predator and prey species. The half saturation constant for infected prey, the growth rate of susceptible prey and force of infection play a significant role to create complex dynamics in this predator–prey system where alternative food is present. It is seen that healthy disease-free system is possible here. The system shows some important dynamics viz. stable coexistence, Hopf bifurcation, period-doubling bifurcation and chaos. The analytical results obtained from the model are justified numerically.

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

The study is an attempt to explore the competitive effects on the top predators in the tri-trophic food chain model. Holling types II and IV functional responses are used to investigate the complex behavior of the tri-trophic prey–predator system. Reaction–diffusion systems have been used to represent temporal evolution and spatial interaction among the species. Consequently, local and global stabilities, and Hopf-bifurcation have been analyzed. The numerical simulation leads to spontaneous and interesting pattern formation in two dimensions, signifying diffusion interaction between mid- and top-level predators. The effect of diffusion on the chaotic dynamics of the model systems is also investigated. It is found that the chaos mechanism wave-like behavior of the chaotic attractor exists for a long time. The diffusion of this long-term behavior is controlled by using Hybrid Control Strategy.

This paper uncovers some striking and new complex phenomena in a memristive diode bridge-based Murali–Lakshmanan–Chua (MLC) circuit. These striking dynamical behaviors include the coexistence of multiple attractors and double-transient chaos. Also, period-doubling, chaos, crisis scenarios are observed in the system when varying the amplitude of the external excitation. Numerical simulation tools like phase portrait, cross-section basin of attraction, Lyapunov spectrum, bifurcation diagrams and time series are used to highlight the complex dynamical behaviors in the memristive system. Further, practical realizations of the circuit both in PSpice and real-laboratory measurements match well with the observed numerical simulations.

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.