International Journal of Bifurcation and Chaos

The orbital fates in a planar three-center problem of slightly negative energy are studied numerically, by classifying the dynamics into colliding with one of the centers, escaping to infinity and orbiting around the centers. We consider three energy levels, for each of them, the set of initial values leading to collision with the centers is determined. It presents an intriguing fractal structure. The complementary set, which corresponds to those initial values whose orbits are either regular or chaotic winding around the fixed centers, also exhibits a fractal structure. These fractal structures found here might lead to some observable physical feature in the future. The fractal structures imply the sensitivity to initial conditions, thus enable us to find initial values with which the orbits are chaotic, having a positive Lyapunov exponent.

We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.

Plastic deformation in a plane strain compression test of a dense sand specimen is studied using functional networks. Kinematical information for the deforming material is obtained using digital image correlation (DIC) and summarized by two types of complex network with different connectivity rules establishing links between the network nodes which represent the DIC observation sites. In the first, nodes are connected to a minimum fixed number of neighbors with similar kinematics such that the resulting network forms one connected component. In the second, nodes are connected to other nodes whose kinematical behavior lies within a fixed distance of each other in an observation space. The fixed radius is determined using optimization with a stopping criterion again with the resulting network forming one connected component. We find different network properties of each network provide useful information about plastic deformation and nonaffine kinematical processes emerging within the material. In particular, persistent shear bands and mesoscale structures within them (e.g. vortices) appear to be closely related to values of network properties including closeness centrality, clustering coefficients, k-cores and the boundaries of community structures determined using local modularity.

Using geometric inversion with respect to the origin, we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf–Takens bifurcation at infinity.

This article investigates the effect of the conventional and extended time-delayed feedback control techniques of chaos control on a first-order positive zero-crossing digital phase locked loop (ZC1-DPLL) using local stability analysis, two-parameter bifurcation studies and two-parameter Lyapunov exponent spectrum. Starting from the nonlinear dynamics of a ZC1-DPLL, we at first explore the time-delayed feedback control (TDFC) algorithm on a ZC1-DPLL in the parameter space. A condition for the optimum value of the system control parameter is derived analytically for a TDFC based ZC1-DPLL. Next, the extended time-delayed feedback control (ETDFC) technique on a ZC1-DPLL is described. It is observed that the application of the delayed feedback control (DFC) technique on the sampled values of the incoming signal inside the loop finally results in the nonlinear DFC of the phase error dynamics. We prove that, for some suitably chosen control parameters, an ETDFC based ZC1-DPLL has a broader stability zone in comparison with a ZC1-DPLL and its TDFC version.

In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhães, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis.

This paper presents an eight-wing chaotic attractor by replacing a constant parameter with a switch function in Qi four-wing 3-D chaotic system. The eight-wing chaotic attractor has more complicated topological structures and dynamics than the original one. Some basic dynamical behaviors and the compound structure of the proposed 3-D system are investigated. Poincaré-map analysis shows that the system has an extremely rich dynamics. The physical existence of the eight-wing chaotic attractor is verified by an electronic circuit FPGA.

We consider the dynamics of two pendulum clocks (with pendulums of the same length but different masses) suspended on the same beam. We give evidence that besides the complete and phase synchronizations, the considered system can exhibit long period synchronization and chaotic behavior. We argue that the observed phenomena are robust.

We consider the local bifurcations that can occur in a quasiperiodic orbit in a three-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at which the closed invariant curve intersects a "second Poincaré section". We show that these bifurcations can be classified depending on where the eigenvalues of this fixed point cross the unit circle.

Chaotic dynamical systems are increasingly considered for use in coding and transmission systems. This stems from their parameter sensitivity and spectral characteristics. The latter are relevant for channel estimation methods. In particular, the logistic map f_λ = λx(1 - x) has been employed in chaotic coding and spread spectrum transmission systems. For λ = 4, the statistical properties of sequences generated by f₄ are considered as ideal drive signals for channel estimation schemes. This assumption is proven in the present paper. To this end, the higher order statistical moments and the autocorrelation of time series generated by f₄ are derived. It is shown that for λ = 4 the zero mean time series is uncorrelated. The adaptation performance of finite impulse response (FIR) digital adaptive filters (DAF) used for channel estimation is analyzed. It is shown that using zero mean sequences of f₄ leads to the maximal possible FIR DAF performance. An optimal value for the damping parameter in the LMS scheme is derived that leads to the maximal performance and ensures stability. The analytic considerations are confirmed by simulation results.

This paper presents results concerning bifurcations of 2D piecewise-smooth vector fields. In particular, the generic unfoldings of codimension-three fold–saddle singularities of Filippov systems, where a boundary-saddle and a fold coincide, are considered and the bifurcation diagrams exhibited.

We discuss the post-bifurcation dynamics of the general double Hopf normal form, which allows us to study two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including the classical Ruelle–Takens and quasiperiodic routes to chaos. However, it has not, to the best of our knowledge, been considered in the context of the double Hopf normal form, although it has been numerically observed and tracked in the post-double-Hopf regime.

We then apply the method of competitive modes to verify parameter regimes for which the double Hopf normal form exhibits chaotic behavior. Such an analysis is useful, as it allows us to potentially identify specific parameter regimes for which the system may exhibit strange or irregular behavior, something which would be extremely difficult otherwise in a system with so many parameters. Indeed, it is conjectured that for parameter sets with two of the square-mode frequencies competitive or nearly competitive, chaotic behavior is likely to be observed in the system. We apply the method of competitive modes to two representative cases where intermittent chaos is found, and the competitive mode analysis there seems to verify the occurrence of chaos in each of the two types of regimes.

We consider oscillations observed in the mathematical model describing the dynamics of transmembrane potential and changes of proton concentration outside and inside the cell of alga Chara corallina. This model shows sufficiently nonlinear regimes such as periods of birth/disappearance as well as parameter regions of chaotic motion. We perform bifurcation analysis of these complex oscillations via the Continuous Wavelet Transform with the Morlet wavelet. We show that this method, compared to classical bifurcation analysis allows to make numerical simulations at the continuous change of the key parameters and to obtain the spectral characteristics.

Bifurcation of heteroclinic cycle near 1:4 resonance in a self-excited parametrically forced oscillator with quadratic nonlinearity is investigated analytically in this paper. This bifurcation mechanism leads to the disappearance of a slow flow limit cycle giving rise to frequency-locking near the resonance. The analytical approach used to approximate the bifurcation is based on a collision criterion between the slow flow limit cycle and saddles involved in the bifurcation. The amplitudes of the 1:4-subharmonic solution and the slow flow limit cycle are approximated using a double perturbation procedure and the heteroclinic bifurcation is captured applying the collision criterion. For validation, the analytical results are compared to those obtained by numerical simulations.

Dynamical behavior of a nonsmooth master system which is coupled to a nonsmooth Nonlinear Energy Sink (NES) during free and forced oscillations is studied analytically and numerically. Invariant manifolds of the system and their stable zones at different time scales are revealed and finally application of coupled nonsmooth NES to the passive control process of the main nonsmooth system is highlighted.

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

Chaotification of dynamical systems is a hot topic in the research field of chaos in recent years. Previous studies showed that (even linear) discrete or continuous dynamical systems can be chaotified by designing appropriate controllers. Here, we study chaotification of linear impulsive differential systems. First, we propose a framework for chaotification of general linear impulsive differential systems that can be transformed into discrete maps. Then, we give technical details for how to chaotify several typical linear impulsive differential systems that are actually canonical forms, including how to design appropriate quadratic impulsive controllers, how to find snapback repellers in the Marotto theorem, etc. As one of the main theoretical results, we rigorously prove the existence of chaos in all the considered impulsive systems. In addition, numerical examples are used to verify the theoretical prediction in each case. We are expecting that our proposed approach can have practical applications in the engineering field.

By using the construction theorem of singular point values developed by [Liu & Li, 1990], the center conditions for three classes of Poincaré systems in the complex domain are obtained. Complete criteria are given. These general results show that in order to obtain the singular point values and simplify them, it is very important to analyze the construction of singular point values.

The neutral particle motion in a composite field (gravitoelectromagnetic field overlapping a constant external gravitomagnetic field) is analyzed. A detailed nonlinear dynamics description, using temporal series, Poincaré sections, phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis, is performed. New phenomena, e.g. the gravitational gun-type, gravitational chaotic gun-type and gravitational multigun-type effects are discovered, if the amplitude of the gravitoelectromagnetic field is sufficiently large. These effects induce a high acceleration of the neutral particles which lead to sudden jumps between different Larmor-type orbits. It results both in a chaotic behavior and patterns formation in gravitational systems.

The problem of control synthesis for master–slave synchronization of continuous time chaotic systems of Lur'e type using sampled feedback control subject to sampling time random fluctuation and data packet dropouts is investigated. New stability and stabilization conditions are proposed based on Linear Matrix Inequalities (LMIs). The idea is to connect two very efficient approaches to deal with delayed systems: the discretized Lyapunov functional for systems with pointwise delay and the convex analysis for systems with time-varying delay. Simulation examples based on synchronizing coupled Chua's circuits are used to illustrate the effectiveness of the proposed methodology.

First, for the family P_n,c(z) = zⁿ + c, we show that the geometric limit of the Mandelbrot sets M_n(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(P_n,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ z^t + c for real t ≥ 2 and the rational maps z ↦ zⁿ + c + a/zⁿ.

In this paper, based on the method of upper and lower solutions, delayed bifurcation in first-order singularly perturbed problems with a nongeneric turning point is studied. The asymptotic behavior of the solutions is understood by constructing the upper and lower solutions with the desired dynamical properties. As an application of the obtained results, delayed phenomenon of degenerate Hopf bifurcation in a planar polynomial differential system with a slowly varying parameter is discussed in detail and the maximal delay is calculated. Numerical simulations are carried out to verify the theoretical results.

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

A saddle-center transition of eigenvalues in the linearization about Hamiltonian equilibria, and the attendant planar homoclinic bifurcation, is one of the simplest and most well-known bifurcations in dynamical systems theory. It is therefore surprising that anything new can be said about this bifurcation. In this tutorial, the classical view of this bifurcation is reviewed and the lifting of the planar system to four dimensions gives a new view. The principal practical outcome is a new formula for the nonlinear coefficient in the normal form which generates the homoclinic orbit. The new formula is based on the intrinsic curvature of the lifted path of equilibria.

In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.

The van der Pol circuit with neon glow lamp operates as a relaxation oscillator driven by a sinusoidal voltage source. Its dynamics is described by the function which maps an arc of circle into itself, preserves orientation on this arc, and has at most one jump-discontinuity point on the circle. Properties of this mapping are discussed with emphasis on rotation numbers, types of periodic orbits, saddle-node and border-collision bifurcations. The so-called period-adding phenomenon is explained. Some remarks are given about dynamics, where the mapping does not preserve the orientation because of too long neon lamp lighting.

Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.

By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.

In this study, we investigate the bifurcation of the quasi-periodic solution in a ring of three-coupled, bistable oscillators. The quasi-periodic solution under study is a propagating wave solution with three-phase synchronized envelope, and we name it "ICC3ϕ" which means invariant closed curve for which time waveforms have three-phase synchronized envelope. We obtain the two-parameter bifurcation diagram of ICC3ϕ with respect to the coupling factor α versus the nonlinear strength ϵ. Consequently, we detect several Arnold tongues showing various periodic solutions of ICC3ϕ. By investigating the bifurcation on the boundary of these Arnold tongues, we clarify the transition from periodic to quasi-periodic solution of ICC3ϕ or the inverse. Specifically, the transition of the period-4, period-5, and period-7 solutions to ICC3ϕ is due to a saddle-node bifurcation, which has no hysteresis. In contrast, the transition from the period-3 solution to ICC3ϕ is a combination of subcritical pitchfork and heteroclinic bifurcations; hence, it has hysteresis. We draw diagrams showing the connection of unstable manifolds of heteroclinic bifurcation.

In our preceding study, we investigated the appearance and disappearance scenario of (quasi-periodic) switching solution in a ring of six-coupled, bistable oscillators. This system was expressed as twelve-dimensional phase space, which was divided into linear subspace H and its complimentary linear subspace H^⊥. We investigated the stability of switching solution in H by calculating the conditional Lyapunov exponent to the direction of H^⊥. As a result, we found a critical coupling factor α_crt at which switching solution lost its stability in the direction of H^⊥. However, we did not know what type of bifurcation occurred at this point. In this study, we elucidate that switching solution causes a supercritical pitchfork bifurcation at α_crt.