International Journal of Bifurcation and Chaos

Recently it was shown that in the dynamical model of Chua circuit both the classical self-excited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of self-excited and hidden attractors is studied. A pitchfork bifurcation is shown in which a pair of symmetric attractors coexist and merge into one symmetric attractor through an attractor-merging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.

This paper is concerned with the study of the iteration of the quadratic coquaternionic map $f_{\mathrm{\text{c}}}(\mathrm{\text{q}})={\mathrm{\text{q}}}^2+\mathrm{\text{c}}$, where c is a fixed coquaternionic parameter. The fixed points and periodic points of period two are determined, revealing the existence of a type of sets of these points which do not occur in the classical complex case: sets of nonisolated points. This brings the need to consider a different concept of stability. The analysis of the stability, in this new sense, of the sets of fixed points and periodic points is performed and a discussion of certain type of bifurcations which occur, in the case of a real parameter c, is also presented.

We investigate the relationship between the switching-time length $T$ and the fractal-like feature that characterizes the behavior of dissipative dynamical systems excited by external temporal inputs for tracking movement. Seven healthy right-handed male participants were asked to continuously track light-emitting diodes that were located on the right and left sides in front of them. These movements were performed under two conditions: when the same input pattern was repeated (the periodic-input condition) and when two different input patterns were switched stochastically (the switching-input condition). The repeated time lengths of input patterns during these conditions were 2.00, 1.00, 0.75, 0.50, 0.35, and 0.25s. The movements of a lever held between a participant’s thumb and index finger were measured by a motion-capture system and were analyzed with respect to position and velocity. The condition in which the same input was repeated revealed that two different stable trajectories existed in a cylindrical state space, while the condition in which the inputs were switched induced transitions between these two trajectories. These two different trajectories were considered as excited attractors. The transitions between the two excited attractors produced eight trajectories; they were then characterized by a fractal-like feature as a third-order sequence effect. Moreover, correlation dimensions, which are typically used to evaluate fractal-like features, calculated from the set on the Poincaré section increased as the switching-time length $T$ decreased. These results suggest that an inverse proportional relationship exists between the switching-time length $T$ and the fractal-like feature of human movement.

Discontinuities are a common feature of physical models in engineering and biological systems, e.g. stick-slip due to friction, electrical relays or gene regulatory networks. The computation of basins of attraction of such nonsmooth systems is challenging and requires special treatments, especially regarding numerical integration. In this paper, we present a numerical routine for computing basins of attraction (BA) in nonsmooth systems with sliding, (so-called Filippov systems). In particular, we extend the Simple Cell Mapping (SCM) method to cope with the presence of sliding solutions by exploiting an event-driven numerical integration routine specifically written for Filippov systems. Our algorithm encompasses a method for dynamic construction of the cell state space so that a lower number of integration steps are required. Moreover, we incorporate an adaptive strategy of the simulation time to render more efficiently the computation of basins of attraction. We illustrate the effectiveness of our algorithm by computing basins of attraction of a sliding control problem and a dry-friction oscillator.

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

The Newton–Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter $𝜖$ varies. The multivariate Newton–Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the perturbation parameter on the geometry as well as of the basin entropy of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.

The competitive threshold linear networks have been recently developed and are typical examples of nonsmooth systems that can be easily constructed. Due to their flexibility for manipulation, they are used in several applications, but their dynamics (both local and global) are not completely understood. In this work, we take some recently developed threshold systems and by a simple modification in the parameter space, we obtain new global dynamic behavior. Heteroclinic cycles and other remarkable scenarios of global bifurcation are reported.

The Qi four-wing chaotic system is transformed into a Kolmogorov-type system, thereby building a bridge between a numerical chaotic system and a physical chaotic system that is convenient for analysis when finding common ground between the two. The vector field is decomposed into four types of torques: inertial, internal, dissipative and external. The angular momentum representing the physical analogue of the state variables of the chaotic system is identified. The cycling of energy among potential energy, kinetic energy, dissipation, and external energy is analyzed. The Casimir function is employed to identify the key factors producing chaos and other dynamical modes. The system is non-Rayleigh dissipative, which determines the extremal points of Casimir function to form a hyperboloid instead of ellipsoid.

In this paper, fractional-order recurrent neural network models with Caputo Derivative are investigated. Firstly, we mainly focus our attention on Hopf bifurcation conditions for commensurate fractional-order network with time delay to reveal the essence that fractional-order equation can simulate the activity of neuron oscillation. Secondly, for incommensurate fractional-order neural network model, we prove the stability of the zero equilibrium point to show that incommensurate fractional-order neural network still converges to zero point. Finally, Hopf bifurcation conditions for the incommensurate fractional-order neural network model are first obtained using bifurcation theory based on commensurate fractional-order system.

A popular biomathematics model of the Goodwin oscillator has been previously generalized to a more biologically plausible construct by introducing three time delays to portray the transport phenomena arising due to the spatial distribution of the model states. The present paper addresses a similar conversion of an impulsive version of the Goodwin oscillator that has found application in mathematical modeling, e.g. in endocrine systems with pulsatile hormone secretion. While the cascade structure of the linear continuous part pertinent to the Goodwin oscillator is preserved in the impulsive Goodwin oscillator, the static nonlinear feedback of the former is substituted with a pulse modulation mechanism thus resulting in hybrid dynamics of the closed-loop system. To facilitate the analysis of the mathematical model under investigation, a discrete mapping propagating the continuous state variables through the firing times of the impulsive feedback is derived. Due to the presence of multiple time delays in the considered model, previously developed mapping derivation approaches are not applicable here and a novel technique is proposed and applied. The mapping captures the dynamics of the original hybrid system and is instrumental in studying complex nonlinear phenomena arising in the impulsive Goodwin oscillator. A simulation example is presented to demonstrate the utility of the proposed approach in bifurcation analysis.

We disprove the conjecture that the existence of a DC3-scrambled pair is preserved under iteration and show that a slightly strengthened definition of distributional chaos of type 3, denoted by DC2$\frac{1}{2}$, is iteration invariant, i.e. that $f^n$ is DC2$\frac{1}{2}$ if and only if $f$ is. Unlike DC3, DC2$\frac{1}{2}$ is also conjugacy invariant and implies Li–Yorke chaos. The definition of DC2$\frac{1}{2}$ is the following: a pair $〈x,y〉$ is DC2$\frac{1}{2}$-scrambled iff ${\mathrm{\Phi }}_{〈x,y〉}(0)<{\mathrm{\Phi }}_{〈x,y〉}^{\ast }(0)$, where ${\mathrm{\Phi }}_{〈x,y〉}(\delta )$ (resp., ${\mathrm{\Phi }}_{〈x,y〉}^{\ast }(\delta )$) is lower (resp., upper) asymptotic density of the set of times $k$ when $d(f^k(x),f^k(y))<\delta$, and both densities are defined at 0 as limits of their values for $\delta \to 0^{+}$. DC$2\frac{1}{2}$ shares similar properties with DC1 and DC2 but it is essentially weaker than DC2.

We have recently developed a chaos-based stream cipher based on augmented Lorenz equations as a star network of Lorenz subsystems. In our method, the augmented Lorenz equations are used as a pseudorandom number generator. In this study, we propose a new method based on the augmented Lorenz equations for generating binary pseudorandom numbers and evaluate its security using the statistical tests of SP800-22 published by the National Institute for Standards and Technology in comparison with the performances of other chaotic dynamical models used as binary pseudorandom number generators. We further propose a faster version of the proposed method and evaluate its security using the statistical tests of TestU01 published by L’Ecuyer and Simard.

We develop a financial market model with interacting chartists and fundamentalists and chase sellers, the model dynamics is driven by a two-dimensional discontinuous piecewise linear map. Assume that the fixed point on the left side of border is restricted to regular saddle, we provide a more or less complete analytical treatment of the model dynamics by characterizing its possible outcomes in parameter space. The interpretation of structure for basin boundary and chaotic attractor is given by using contact bifurcation resulting from the contact between invariant set and the border. The critical value of occurring boundary crisis is given. In addition, we show that quite different scenarios can trigger real world phenomena such as bull and bear market dynamics and excess volatility.

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov–Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.

Neuron models are available for computational neurodynamics and the main dynamical properties can be reproduced in the numerical scheme for further dynamical analysis. During model setting, some important biophysical factors should be considered and thus reliable neuron models can be approached. In this paper, a neuron model driven by autapse connection is investigated with the effect of electromagnetic induction being considered as well. A digital neuronal circuit is designed by using FPGA, the dynamical response and biological function of autapse connection. It is found that positive feedback in autapse can modulate the oscillating behaviors in the digital circuit, which could be effective for further investigation on digital neuronal network.

Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law nonlinearity is investigated by the method of dynamical systems. The functions $\varphi (\xi )$ in the solutions $q(x,t)=\varphi (\xi )\mathrm{\text{exp}}(i(-kx+\omega t)),$ $(\xi =x-vt)$ satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of $\varphi (\xi )$, under 23 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 92 exact explicit solutions of system (6) are derived.

Existence and number of invariant cones in general 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane are investigated. Implicit parametric expressions of two proper half slope maps whose intersections determine the existence and number of invariant cones are obtained. Based on these expressions, some sufficient conditions for the existence of at most three invariant cones are provided, and it is proved that the maximum number of invariant cones for some special cases is equal to 1 plus the maximum number of limit cycles in planar piecewise linear systems with a straight line separation. Moreover, it is illustrated by a numerical example with four invariant cones that the maximum number of invariant cones is not less than four. Specially, the main results provide a method to completely solve the existence and number of invariant cones in any specific 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane by using numerical method.

We describe the dynamics of the logistic map $x_{T+1}=\lambda m_T(1-m_T)$ endowed with memory $m_T$ of past iterations with geometric decay. We find that most of the relevant properties of the dynamics can be explained by means of a scaling of the standard logistic map without memory. The bifurcation diagram is therefore a scaled version of the standard map, with the extreme values of the logistic parameter $\lambda$ for confined trajectories following also the scaling law. However, contrary to the ahistoric model, the trajectories may diverge for smaller values of $\lambda$, depending on the initial condition $x_1$. Simple conditions for divergence with given logistic map and memory are provided.

This paper provides a general framework for modifying chaotic systems on the premise of preserving chaos. It formulates effective techniques to construct some derivative systems from an existing chaotic system by replacing a parameter or a term in the dynamical system with certain time-dependent or state-dependent functions. This study derives an alternative for chaotic system construction and also provides more flexibility for chaotic synthesis. Numerical simulations demonstrate the effectiveness and validity of the proposed techniques. As a consequence, some new chaotic systems are also obtained.

We report a new result on the traveling wave solutions of a biological invasion model with density-dependent migrations and Allee effect. It has been shown in the literature that such a model can exhibit one periodic wave solution by using Hopf bifurcation theory. In this paper, global bifurcation theory is applied to prove that there exists maximal one periodic solution which can be reached in a large feasible parameter regime. The basic idea used in our technique is to examine the monotonicity of the ratio of related Abelian integrals. Especially, the existence condition for the solution near a homoclinic loop is obtained.