International Journal of Bifurcation and Chaos

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

Chaos in physical systems governed by nonlinear PDEs have been amply observed and reported. However, rigorous proofs for their occurrences are challenging. In particular, for a second order PDE of hyperbolic type with a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed antidamping term in a linear governing equation, no proof for the onset of chaos was available even though chaos was expected to occur when the antidamping term becomes sufficiently strong. In this paper, we use an operator-factoring technique together with the analogy with the one-dimensional wave equation to prove that for the Klein–Gordon equation chaos occurs for a class of equations and boundary conditions when system parameters enter a certain regime. Chaotic and nonchaotic profiles of solutions are illustrated by computer graphics.

The dynamics of a low-dimensional ensemble consisting of a network of five discrete phase oscillators is considered. A two-parameter synchronization picture, which appears instead of the Arnol'd tongues with an increase of the system dimension, is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency–coupling" parameter plane. The cases of attractive and repulsive interactions are discussed.

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C⁰-normal form.

In this paper, the global exponential attractive sets of the T chaotic system are studied. The method for constructing Lyapunov functions that are applied to the former chaotic systems [Liao, 2004; Liao et al., 2008; Wang et al., 2011; Zhang et al., 2011a; Zhang et al., 2011b; Yu & Liao, 2008; Li et al., 2009; Yu et al., 2009] does not work for the T chaotic system. We overcome this difficulty by introducing a cross term xy. We get a perfect result that contains the case for the parameter values considered covering the most interesting case of the chaotic attractor system.

In the n-body problem, a collision singularity occurs when the position of two or more bodies coincide. By understanding the dynamics of collision motion in the regularized setting, a better understanding of the dynamics of near-collision motion is achieved. In this paper, we show that any double collision of the planar equilateral restricted four-body problem can be regularized by using a Birkhoff-type transformation. This transformation has the important property to provide a simultaneous regularization of three singularities due to binary collision.

We present some ejection–collision orbits after the regularization of the restricted four-body problem (RFBP) with equal masses, which were obtained by numerical integration.

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

Two methods are presented to encode and decode messages accurately with an expanded matrix representation of linear multivariable systems in which parameters are used to describe three-port chaotic secure communication networks. The matrix representation includes an optimal extended Kalman filter (EKF)-based observer, which is linear with time. The optimal linearization technique is used to find the exact linear models of the chaotic system at operating states of interest. Subsequently, the EKF algorithm is used to estimate the parameters and states in which a message is embedded. Using the EKF with the optimal linear model, the message can be adequately recovered at the receiver. However, the bit error rate is insufficiently small; therefore, presynchronizations and the carrier-digitalized sources are used to reduce the error rate to obtain robust communications. Numerical examples and simulation results demonstrate the effectiveness of the proposed methodology.

This paper considers observer-based chaos synchronization and chaos-based secure communication problems for a class of uncertain chaotic systems with both parameter uncertainties and channel noise. First, by introducing an augmented vector, the original system is transformed into a new system which has no channel noise. Second, based on the concept of relative degree, an auxiliary drive signal vector is constructed so that the observer matching condition is satisfied. Third, a reduced-order observer is designed to estimate the states of the new augmented system. Then, a high-order high-gain sliding mode observer is considered to estimate the auxiliary drive signals and their derivatives exactly in a finite time. After this, the combination of the reduced-order observer and the high-order high-gain sliding mode observer is used as the receiver in the secure communication mechanism. And a kind of secure information recovery method is developed. Finally, a four-dimensional hyperchaos system is used as the simulation example to illustrate the effectiveness of the proposed methods.

In the paper, we study the existence of periodic solutions in an invariant manifold for a predator–prey model with periodic harvesting rate. Without harvesting, the predator–prey system has a positive equilibrium of center type, which undergoes Hopf bifurcation. Under the periodic harvesting, we first reduce this predator–prey system to a complex normal form. Then, we define the Poincaré mapping of this system by using the polar coordinates, and prove that there exists an integral manifold and periodic solution with the period of the harvesting rate. An example is provided to illustrate the result.

We study the relationship between homoclinic orbits associated with repellers, usually called snap-back repellers, and expanding sets of smooth endomorphisms. We also consider critical homoclinic orbits and prove that the bifurcations they create constitute a codimension one phenomenon. This and other possible routes to the creation of homoclinic orbits are considered in low dimensions.

Three new routes to repetitive spiking, i.e. the delayed transcritical bifurcation, the delayed supercritical pitchfork bifurcation and the delayed subcritical pitchfork bifurcation, are revealed in this paper. We use bifurcation theory to classify bursting patterns related to three such delayed bifurcations. Then many new bursting patterns are obtained, including 24 new bursting patterns of point-point type, 27 new bursting patterns of point-cycle type and three new bursting patterns of point-torus type. Our study suggests that the classification of bursting remains to be further explored, since many new bursting patterns may be obtained based on new routes to repetitive spiking, even though we just consider codimension-1 bifurcations.

A scroll chaotic system containing a HP memristor model and triangular wave sequence is proposed in this article. Because the memristor is both a nonlinear element and a memory element intrinsically, it is considered a potential candidate to reduce system power consumption and circuit size. A reasonable mathematical structure of triangular wave sequence and the selection of appropriate amplitude, balance point and turning point reduce the dynamic range of signal input caused by the integrator. The proposed system produces a wealth of chaos, just by changing one parameter. Circuit simulations are conducted and the chaotic attractors can be observed. Theoretical analysis, computer simulation and calculation of maximum Lyapunov exponent have been used to research the basic dynamics of this system. The consistency of circuit implementation and computer simulations verifies the effectiveness of the system design.

A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Given 0 ≤ p ≤ q ≤ 1, a dynamical system is chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. It shows that, for any 0 ≤ p < q ≤ 1 or p = q = 0 or p = q = 1, a dynamical system which is null and chaotic can be realized.

The amplitude equation describes a reduced form of a reaction–diffusion system, yet still retains its essential dynamical features. By approximating the analytic solution, the amplitude equation allows the examination of mode instability when the system is near a bifurcation point. Multiple-scale expansion (MSE) offers a straightforward way to systematically derive the amplitude equations. The method expresses the single independent variable as an asymptotic power series consisting of newly introduced independent variables with differing time and space scales. The amplitude equations are then formulated under the solvability conditions which remove secular terms. To our knowledge, there is little information in the research literature that explains how the exhaustive workflow of MSE is applied to a reaction–diffusion system. In this paper, detailed mathematical operations underpinning the MSE are elucidated, and the practical ways of encoding these operations using MAPLE are discussed. A semi-automated MSE computer algorithm Amp_solving is presented for deriving the amplitude equations in this research. Amp_solving has been applied to the classical Brusselator model for the derivation of amplitude equations when the system is in the vicinity of a Turing codimension-1 and a Turing–Hopf codimension-2 bifurcation points. Full open-source Amp_solving codes for the derivation are comprehensively demonstrated and available to the public domain.

Recent years have witnessed a growing trend towards the development and deployment of distributed generation (DG). It is shown that electric distribution networks with DGs can encounter two types of local bifurcations: saddle-node bifurcation and structure-induced bifurcation. The structure-induced bifurcation occurs when a transition between two structures of the distribution network takes place due to limited amount of reactive power supports from renewable energies. The saddle-node bifurcation occurs when the underlying distribution network reaches the limit of its delivery capability. The consequence of structure-induced bifurcation is an immediate instability induced by reactive power limits of renewable energy. It is numerically shown that both types of local bifurcations can occur at both small distribution networks and large-scale distribution networks with DGs. Physical explanations of these two local bifurcations are provided. Studies of local bifurcations in distribution networks provide insights regarding how to design controls to enhance distribution networks with DGs.

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.

An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense.

In this paper, we develop a mathematical model to assess the strength of the effects of catastrophic anemia level on the dynamical transmission of malaria parasite within the body of a host. We first consider a temporal model. The important mathematical features of the model are thoroughly investigated. We found that the model exhibits forward bifurcation. We also consider a spatiotemporal model using reaction–diffusion equations. The model is numerically analyzed to assess the impact of anemia on the dynamical transmission of malaria parasite within the body of a host. Through numerical simulation, we found that malaria can lead to a catastrophic anemia level even if the parasite is nonpersistent within the body of a host. Numerical results also suggest that to reduce or control the anemia level, the strategy should be to accelerate innate cell reproduction rate or should have the ability to clean parasitized red blood cells (PRBCs) with a high mortality rate.