International Journal of Bifurcation and Chaos

The mean Poincaré recurrence time as well as the Lyapunov time are measured for the Fermi–Ulam model. It is confirmed that the mean recurrence time is dependent on the size of the window chosen in the phase space where particles are allowed to return. The fractal dimension of the region is determined by the slope of the recurrence time against the size of the window and two numerical values are measured: (i) $\mu =1$ confirming normal diffusion for chaotic regions far from periodic domains and (ii) $\mu =2$ leading to anomalous diffusion measured inside islands of stability and invariant curves corresponding to regular orbits, a signature of local trapping of an ensemble of particles. The Lyapunov time is the inverse of the Lyapunov exponent. Therefore, the Lyapunov time is measured over different domains in the phase space through a direct determination of the Lyapunov exponent.

In the last two years, many chaotic or hyperchaotic systems with megastability have been reported in the literature. The reported systems with megastability are mostly developed from their dynamic equations without any reference to the physical systems. In this paper, the dynamics of a single-link manipulator is considered to observe the existence of interesting dynamical behaviors. When the considered dynamical system is excited with (a) periodically forced input or (b) quasi-periodically forced input, it indicates the existence of megastability. This paper reports megastability in a physical dynamical system with infinitely many equilibria. The considered system has other dynamical behaviors like chaotic, quasi-periodic and periodic. These behaviors are analyzed using Lyapunov spectrum, bifurcation diagram and phase plots. The simulation results reveal that the objectives of the paper are achieved successfully.

We introduce the asymmetric wedge billiard as a generalization of the wedge billiard first studied by Lehtihet and Miller in 1986. This is a billiard system in which the billiard ball moves under the influence of a constant gravitational field, colliding elastically with two wedge walls with the collisions obeying the reflection law. Collision maps are given from which derivatives and area-preservation (or lack thereof) were determined. Expressions for the fixed points of the collision maps were also calculated and discussed. Long-term dynamics were determined computationally from which we observed integrable, quasi-periodic and chaotic behavior which were all dependent on the wedge angles.

In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

In this paper, the dynamics of a three-species food chain model with two predators infected by an infectious disease is investigated. The positivity and boundedness of the system, the existence of the equilibria and the basic reproductive number are given. Sufficient conditions for the local stability of all equilibria are obtained by analyzing the corresponding characteristic equations. By constructing suitable Lyapunov functions and taking the geometric approach, the global stability of all equilibria is proved. According to the center manifold theory, this model undergoes the phenomenon of backward and forward bifurcations in a certain range of the basic reproductive number $R_0$. By taking the disease transmission coefficient of predator as bifurcation parameter, Hopf bifurcation emerges in the neighborhood of the endemic equilibrium. Furthermore, the optimal control of the disease is discussed by the Pontryagin’s maximum principle. Various simulations are given to support the analytical results.

This study develops a general model of delayed p53 regulatory network in the DNA damage response by introducing microRNA 192-mediated positive feedback loop based on the existing research work. Through theoretical analysis and numerical simulation, we find that the delay as a bifurcation parameter can drive the p53-Mdm2 module to undergo a supercritical Hopf bifurcation, thereby producing oscillation behavior. Moreover, we demonstrate how the positive feedback loop formed by p53* and microRNA 192 (miR-192) with the feature of double-negative regulation produces oscillations. Further, a comparison is given to demonstrate that microRNA 192-mediated positive feedback loop affects the robustness of system oscillations. In addition, we show that ataxia telangiectasia mutated kinase (ATM), once activated by DNA damage, makes p53* undergo two Hopf bifurcations. These results reveal that both time delay and miR-192 play tumor suppressing roles by promoting p53 oscillation or high level expression, which will provide a perspective for promoting the development of anti-cancer drugs by targeting miR-192 and time delay.

In this paper, an image encryption algorithm based on compressive sensing (CS) and a hyperchaotic system is proposed. This new two-dimensional logical coupling sine chaotic map (2D-LCSCM) has a wider range of chaos, better ergodicity and hyperchaotic properties rather than for existing chaotic maps. By combining with the novel chaotic map and CS, we design an image compression encryption algorithm with good reconstruction accuracy and high security. In the substitution phase of the encryption process, the security performance of the algorithm is improved by the nonlinear transformation of the dynamic substitution box (S-box) and the linear transformation of the chaotic sequences. The performance analyses of the experimental results show that the proposed algorithm can resist various attacks and is highly secure.

This paper investigates the pullback attractor of Cohen–Grossberg neural networks with multiple time-varying delays. Compared with the existing references, the networks considered here are more general and cannot be expressed in the vector-matrix form due to multiple time-varying delays. After constructing a proper Lyapunov–Krasovskii functional and eliminating the terms involving multiple time-varying delays, two sets of new sufficient criteria on the existence of the pullback attractor are derived based on the theory of pullback attractors. In the end, two examples are given to demonstrate the effectiveness of our theoretical results.

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.

In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented.

We consider a nongeneric family of 3D Filippov linear systems with a discontinuity plane that have two parallel tangency lines, such that the region between them is the sliding region. We are interested in finding under what conditions the family has a crossing limit cycle, when the sliding region changes its stability. We call this phenomenon the pseudo-Hopf bifurcation. This class of systems is motivated by piecewise-linear control systems which have not yet been treated in the context of crossing limit cycles.

Our goal is to make a global study of the class $QsnS{N}_{11}$ of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite and infinite singularities. This class can be divided into two different families, being (A) possessing the finite saddle-node as the only finite singularity and (B) possessing the finite saddle-node and also a finite simple elemental singularity. In this paper we provide the complete study of the geometry of family (A). The family (A) modulo the action of the affine group and time homotheties are four-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the four-dimensional real projective space $ℝ{\mathbb{P}}^4$. As far as we know, this is the first time that a complete family is studied in the four-dimensional real projective space. The respective bifurcation diagram yields 36 topologically distinct phase portraits for systems in the closure $\overline{QsnS{N}_{11}(A)}$ within the representatives of $QsnS{N}_{11}(A)$ given by a specific normal form.

In [Chen et al., 2020], the third author and other coauthors studied global dynamics of the following system:

This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, which are not in the sense of Shil’nikov criteria. First, under some conditions, the system has infinitely many isolated equilibria. Moreover, all equilibria are nonhyperbolic and give the first Lyapunov coefficient. Furthermore, when all equilibria are weak saddle-foci, the system also has infinitely many chaotic attractors. Besides, the Lyapunov exponents spectrum and bifurcation diagram are given. Second, under another condition, all the equilibria constitute a curve and there exist infinitely many singular degenerated heteroclinic orbits. At the same time, the system can show infinitely many chaotic attractors.

Nonautonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but non-negligible rates before settling at a stable value, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explained by the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover a scaling law relating the duration of the transient to the rate of change of the parameter for a case where a chaotic attractor is involved. Additionally, we analyze the viability of recovering the transient dynamics by reversing the parameter to its original value, as an alternative to the control theory for systems with parameter drifts. We obtain the relationship between the paramater change rate and the number of trajectories that tip back to the initial attractor corresponding to the transient state.

A feedforward network is a unidirectionally coupled chain of dynamical systems in which the first cell is coupled to itself, and each successive cell is coupled to the next one. Feedforward networks have gained considerable interest because of their potential to enhance signal amplification and to manipulate the frequency of oscillations. Indeed, it has been shown that the growth rate of the bifurcation undergone by the final cell is much larger than the expected square root growth rate associated with the standard Hopf bifurcation. In this paper, we present a new approach to studying this growth rate phenomenon. We employ a two-time-scale analysis and asymptotic approximations to detect behavior associated with the growth rate phenomenon that has not been previously observed. In particular, we show that the Hopf bifurcation is not the only bifurcation capable of exhibiting this large growth rate behavior. Using asymptotic methods we show that it is not a special property of the Hopf bifurcation that allows for this accelerated growth rate; it is a combination of the unidirectional coupling and the higher-degree nonlinearities that cause this effect. Furthermore, we show that this large growth rate need not persist away from the bifurcation. In fact, the growth rate is asymptotic to the standard square root growth rate as the bifurcation parameter increases.

The concept of conformal transformation is proposed through the study of the spatial structure of $n$-dimensional hypercubes. Based on conformal transformation, a novel algorithm, called topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the topological equivalence classes of all balanced linearly separable Boolean functions and the number of Boolean functions in each of the topological equivalence classes are obtained. In addition, the properties of conformal transformation also show an application prospect for decomposing nonlinearly separable Boolean functions.

In this paper, the existence of periodic motions of a discontinuous delayed system with a hyperbolic switching boundary is investigated. From the delay-related $G$-function, the crossing, sliding and grazing conditions of a flow to the switching boundary are first developed. For this time-delayed discontinuous dynamical system, there are 17 classes of generic mappings in phase plane and 66 types of local mappings in a delay duration. The generic mappings are determined by subsystems in three domains and two switching boundaries. Periodic motions in such a delay discontinuous system are constructed and predicted analytically from specific mapping structures. Three examples are given for the illustration of periodic motions with or without sliding motion on the switching boundary. This paper shows how to develop switchability conditions of motions at the switching boundary in the time-delayed discontinuous systems and how to construct the specific periodic solutions for the time-delayed discontinuous systems. This study can help us understand complex dynamics in time-delayed discontinuous dynamical systems, and one can use such analysis to control the time-delayed discontinuous dynamical systems.

In this paper, we analyze a class of three-dimensional eco-epidemiological models where prey is subject to Allee effects and infection. We first establish the existence, uniqueness, positivity and uniform ultimate boundedness of the solutions for the proposed system in the positive octant. For three subsystems, we investigate the existence of their respective trivial and positive equilibria and determine the conditions for some bifurcations (Hopf bifurcation, Bogdanov–Takens bifurcation of codimension-2 and saddle-node bifurcation) to occur. We find that the Allee effect, nonmonotonic functional response and intra-class competition in susceptible preys enable the S–I and S–P subsystems to have richer dynamics. For example, the S–I subsystem can have up to three positive equilibria, the S–P subsystem with nonmonotonic functional response can have two positive equilibria while it is impossible in monotonic situation, and high intra-class competition in susceptible preys may lead to the extinction of the predator population, etc. We show that the strong Allee effect can create a separatrix curve (or surface), leading to multistability. Then, we study the uniform persistence of the full system and identify an interior periodic orbit by applying Poincaré map and bifurcation theory. Our analysis reveals that the introduction of the infection or predation may act as a biological control to save the population from extinction and the interaction between these two factors yields a diverse array of biologically relevant behaviors. Finally, some numerical simulations are performed to support and supplement our analytical findings.

Sliding bifurcations of different types are defined to characterize the topological features of trajectories around the switching manifolds of nonsmooth dynamical systems. In this paper, the stick-slip transitions, which are related to the dynamical scenarios of sliding bifurcations, of the self-excited dry friction backward whirls of a general rotor/stator rubbing system, are investigated. The four-degree-of-freedom piecewise smooth rotor/stator rubbing model is said to be general because it takes into account main factors in the rotor/stator rubbing systems, including both the dynamics of the rotor and the stator as well as the dry friction and the flexibility on the contact surfaces. The switching manifold that separates two discontinuous vector fields is defined as the curved hypersurface in a nine-dimensional extended state space where the relative velocity at the contact points equals zero. After deriving the formulae defining the sliding regions and their boundaries for the piecewise smooth system, two extreme cases with rigid and soft contact surfaces are theoretically analyzed and confirmed to correspond respectively to a continuous pure rolling with full sliding region and a continuous crossing without sliding region on the switching manifold. Furthermore, three types of sliding bifurcations, namely, crossing-sliding, grazing-sliding and switching-sliding, are observed in the dry friction backward whirls of the present model in a semi-analytical way. Moreover, hybrids of the three kinds of the sliding solutions in one period of oscillation are also identified with the variation of system parameters and initial conditions. The main scenarios of the switching transition of sliding bifurcations in the self-excited dry friction backward whirl of the general rotor/stator rubbing system with the variation of system parameters are also summarized.