International Journal of Bifurcation and Chaos

We present a computational study of the impact of variation in cut location on finite-time mixing of a line segment by cutting and shuffling, which is a one-dimensional piecewise isometry (PWI), also known as an interval exchange transformation (IET). A line segment of unit length is repeatedly cut into subsegments and shuffled according to any one of a variety of permutations. To mimic practical process error, variations drawn from a normal distribution are used to perturb cut locations. Illustrative examples of the mixing behaviors and finite-time measures of mixing are used to analyze the effect of variation in cut location for different permutations of subsegment mixing order. Mixing is significantly improved under irreducible nonrotational permutations when the dynamics show a resonance-like structure without variation. Specifically, the requirement of an irrational subsegment length ratio for good mixing can be relaxed as the underlying periodic dynamics is perturbed by the stochastic variation in cut location. Thus, good mixing can occur even with only four subsegments of roughly the same length for most irreducible nonrotational permutations.

We introduce a novel method revealing hidden bifurcations in the multispiral Chua attractor in the case where the parameter of bifurcation $c$ which determines the number of spiral is discrete. This method is based on the core idea of the genuine Leonov and Kuznetsov method for searching hidden attractors (i.e. applying homotopy and numerical continuation) but used in a very different way. Such hidden bifurcations are governed by a homotopy parameter $\epsilon$ whereas $c$ is maintained constant. This additional parameter which is absent from the initial problem is perfectly fitted to unfold the actual structure of the multispiral attractor. We study completely the multispiral Chua attractor, generated via sine function, and check numerically our method for odd and even values of $c$ from 1 to 12. In addition, we compare the shape of the attractors obtained for the same value of parameter $\epsilon$ while varying the parameter $c$.

A dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despite the eigenvalues of the fixed points before and after the bifurcation being within the unit circle. In this paper, we show that similar bifurcation phenomena also occur when the stable orbit in question is of a higher periodicity or is chaotic. Accordingly, we propose a generalized definition of dangerous bifurcation suitable for any kind of attracting sets. We report two types of dangerous border collision bifurcations and show that, in addition to the originally reported mechanism typically involving singleton saddle cycles, there exists one more situation where the basin boundary is formed by a repelling closed invariant curve.

There are many methods for computing stable and unstable manifolds in autonomous flows. When the flow is nonautonomous, however, difficulties arise since the hyperbolic trajectory to which these manifolds are anchored, and the local manifold emanation directions, are changing with time. This article utilizes recent results which approximate the time-variation of both these quantities to design a numerical algorithm which can obtain high resolution in global nonautonomous stable and unstable manifolds. In particular, good numerical approximation is possible locally near the anchor trajectory. Nonautonomous manifolds are computed for two examples: a Rossby wave situation which is highly chaotic, and a nonautonomus (time-aperiodic) Duffing oscillator model in which the manifold emanation directions are rapidly changing. The numerical method is validated and analyzed in these cases using finite-time Lyapunov exponent fields and exactly known nonautonomous manifolds.

The classical theory of intermittency developed for return maps assumes uniform density of points reinjected from the chaotic to laminar region. Recently, we reported how the reinjection probability density (RPD) can be generalized. Estimation of the universal RPD is based on fitting a linear function to experimental or numerical data. Here we present an analytical approach to estimate the RPD. After this, we can get an analytic evaluation of the characteristic exponent traditionally used to characterize the intermittency type. The proposed theoretical method is general and very simple to use. It is compared with numerical computation, showing a good agreement between both. Our analytical results are compared with some celebrated classical numerical results on intermittency theory.

We consider a generalized pendulum equation depending on the scalar parameter $\mu$ having for $\mu =0$ a limit cycle $\mathrm{\Gamma }$ of the second kind and of multiplicity three. We study the bifurcation behavior of $\mathrm{\Gamma }$ for $-1\le \mu \le (\sqrt{5}+3)/2$ by means of a Dulac–Cherkas function.

In a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the $1:2:5$ resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in $3:2:1$ resonance.

Here we employ the Lyapunov–Schmidt procedure to investigate bifurcations in a general neutral functional differential equation (NFDE) when the infinitesimal generator has, for a critical value of the parameter, a pair of nonsemisimple purely imaginary eigenvalues with multiplicity $k$. We derive criteria, explicitly in terms of the system's parameter values, for the existence of two branches of bifurcating periodic solutions and for the description of the bifurcation direction of these branches. The general result is illustrated by a detailed case study of an oscillator.

Departing from the geodesic flow on a surface of negative curvature as a classic example of the hyperbolic chaotic dynamics, we propose an electronic circuit operating as a generator of rough chaos. Circuit simulation in NI Multisim software package and numerical integration of the model equations are provided. Results of computations (phase trajectories, time dependencies of variables, Lyapunov exponents and Fourier spectra) show good correspondence between the chaotic dynamics on the attractor of the proposed system and of the Anosov dynamics for the original geodesic flow.

A crisis of amplitude control can occur when a system is multistable. This paper proposes a new chaotic system with a line of equilibria to demonstrate the threat to amplitude control from multistability. The new symmetric system has two coefficients for amplitude control, one of which is a partial amplitude controller, while the other is a total amplitude controller that simultaneously controls the frequency. The amplitude parameter rescales the basins of attraction and triggers a state switch among different states resulting in a failure of amplitude control to the desired state.

In this paper, we investigate the dynamics of a five-dimensional virus model incorporating saturation incidence rate, CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model by proving the positivity and boundedness of the solutions. Our model admits three possible equilibrium solutions, namely the infection-free equilibrium $E_0$, the infectious equilibrium without immune response $E_1$ and the infectious equilibrium with immune response $E_2$. Moreover, by analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcation at the equilibrium point $E_2$ are established, respectively. Further, by using fluctuation lemma and suitable Lyapunov functionals, it is shown that $E_0$ is globally asymptotically stable when the basic reproductive numbers for viral infection $R_0$ is less than unity. When the basic reproductive numbers for immune response $R_1$ is less than unity and $R_0$ is greater than unity, the equilibrium point $E_1$ is globally asymptotically stable. Finally, some numerical simulations are carried out for illustrating the theoretical results.

The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li–Yorke pair. However, the definition can be strengthened to get DC$2\frac{1}{2}$ which is a topological invariant and implies Li–Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC$2\frac{1}{2}$ systems must have zero topological entropy.

Lowered excitability leads to unstable meandering of spiral tip, which result in breakup of spiral waves into chaotic states induced by Doppler effects. This phenomenon is responsible for the transition from tachycardia to ventricular fibrillation in cardiac tissues. Numerical simulations show that low-energy local periodic forcing (LPF) applied around spiral tip can efficiently suppress the meandering behavior and consequently prevent spiral breakup. The controllable phase diagrams that describe the amplitude and period of LPF against excitability parameter are presented to illustrate the control region. The underlying mechanism of suppressing spiral meandering behavior is explored by greatly decreasing the radius of the meandering tip. The proposed scheme can potentially contribute to controlling cardiac arrhythmia.

The Qi chaotic system is transformed into Kolmogorov type of system. The vector field of the Qi chaotic system is decomposed into four types of torques: inertial torque, internal torque, dissipation and external torque. Angular momentum representing the physical analogue of the state variables of the chaotic system is identified. The Casimir energy law relating to the orbital behavior is identified and the bound of Qi chaotic attractor is given. Five cases of study have been conducted to discover the insights and functions of different types of torques of the chaotic attractor and also the key factors of producing different types of modes of dynamics.

In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs.

Considerable evidence suggests that in complex diseases the deteriorations are often abrupt, and can be viewed as a bifurcation or a critical transition from one asymptotically stable equilibrium to another one at a tipping point. Here, we propose some new ideas to detect early-warning signals of such critical transitions from the perspective of qualitative theory of ordinary differential equations. Specifically, we theoretically derive three indicators that serve as a general early-warning signal indicating an imminent bifurcation or sudden deterioration before the critical transition occurs. Then, we verify our theoretical results by numerical simulations for three examples. Our work forms a starting point to motivate new mathematical insights into predictability for critical transitions of dynamical systems from a new perspective.

New classes of Lyapunov-type functions are suggested for the estimation of Lyapunov dimension of Lorenz-like systems. Lyapunov dimension formulas for Lorenz and Tigan systems are obtained.

The study of the strange nonchaotic attractors is an interesting topic, where the dynamics are neither regular nor chaotic (the word chaotic means the positive Lyapunov exponents), and the shape of the attractors has complicated geometry structure, or fractal structure. It is found that in a class of planar first-order nonautonomous systems, it is possible that there exist attractors, where the shape of the attractors is regular, the orbits are transitive on the attractors, and the dynamics are not chaotic. We call this type of attractors as regular nonchaotic attractors with positive plural, which are different from the strange nonchaotic attractors, attracting fixed points, or attracting periodic orbits. Several examples with computer simulations are given. The first two examples have annulus-shaped attractors. Another two examples have disk-shaped attractors. The last two examples with externally driven terms at two incommensurate frequencies have regular nonchaotic attractors with positive plural, implying that the existence of externally driven terms at two incommensurate frequencies might not be the sufficient condition to guarantee that the system has strange nonchaotic attractors.

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

The phase coherence phenomenon of chaotic motion is unique to continuous time dynamical systems and of significance in many disciplines such as nonlinear physics and biology. In this paper, we present a geometric description of phase coherence of chaotic motion and show that chaotic sets near a homoclinic orbit or heteroclinic cycle are phase coherent in this geometric setting under a mild condition.