International Journal of Bifurcation and Chaos

We investigate the effects of heterogeneous delays on the coupling of two excitable neural systems. Depending upon the coupling strengths and the time delays in the mutual and self-coupling systems, the compound system exhibits different types of synchronized oscillations of variable period. We analyze this synchronization based on the interplay of the different time delays and support the numerical results by analytical findings. In addition, we elaborate on bursting-like dynamics with two competing timescales on the basis of the autocorrelation function.

We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles having at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique.

Based on the detailed bifurcation analysis and the master stability function, bursting types and stable domains of the parameter space of the Rulkov map-based neuron network coupled by the mean field are taken into account. One of our main findings is that besides the square-wave bursting, there at least exist two kinds of triangle burstings after the mean field coupling, which can be determined by the crisis bifurcation, the flip bifurcation, and the saddle-node bifurcation. Under certain coupling conditions, there exists two kinds of striking transitions from the square-wave bursting (the spiking) to the triangle bursting (the square-wave bursting). Stable domains of fixed points, periodic solutions, quasiperiodic solutions and their corresponding firing regimes in the parameter space are presented in a rigorous mathematical way. In particular, as a function of the intrinsic control parameters of each single neuron and the external coupling strength, a stable coefficient of the Neimark–Sacker bifurcation is derived in a parameter plane. These results show that there exist complex dynamics and rich firing regimes in such a simple but thought-provoking neuron network.

The control of chaotic dynamics in a nonlinear mass-spring model with nonsmooth stiffness is analyzed here. This is carried out by applying the phase control technique, which uses a periodic perturbation of a suitable phase ϕ. For this purpose, we take as prototype model a system consisting of a double-well potential with an additional spring component, which acts into the system only for large enough displacements. The crucial role of the phase is evidenced by using numerical simulations and also by using analytical methods, such as the Melnikov analysis. We expect that our results might be fruitful with implications in some mechanical problems such as suspension of vehicles, among others, where similar models are extensively used.

In this paper, we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics; in particular, the dividing surface, the flux through the dividing surface (DS), and the gap time distribution. Our approach is to study these questions using simple, two degree-of-freedom Hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly. For our examples, we show that resonances within the normally hyperbolic invariant manifold may, or may not, lead to a "loss of normal hyperbolicity". Moreover, we show that the onset of such resonances results in a change in topology of the dividing surface, but does not affect our ability to define a DS. The flux through the DS varies continuously with energy, even as the energy is varied in such a way that normal hyperbolicity is lost. For our examples, the gap time distributions exhibit singularities at energies corresponding to the existence of homoclinic orbits in the DS, but these singularities are not associated with loss of normal hyperbolicity.

We consider the macroscopic, second order model of Kerner–Konhäuser for traffic flow given by a system of PDE. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We prove that under generic conditions on the so-called fundamental diagram, the surface of critical points has a fold or cusp catastrophe and each fold point gives rise to a Takens–Bogdanov bifurcation. In particular, limit cycles arising from a Hopf bifurcation give place to traveling wave solutions of the PDE.

We consider in-line and overlap geometry models of Josephson junctions with point or rectangular inhomogeneity and investigate the effect of their location on the Josephson vortices and the current. We analyze numerically the critical dependencies "current-magnetic field" caused by one- and two-point current injections. The obtained results elucidate the relation between these critical curves and the fractions of the injection current at the ends of the junction. We also find out similarities between the exponentially shaped junctions, and those with inhomogeneity at the end when a two-point current injection is present. We juxtapose the critical curves of the distinct junctions with inner inhomogeneity and discuss the similarity between them and the Josephson junctions with phase shifts.

The transitions of Josephson junctions from a superconducting mode to a resistive one as bifurcations of the static solutions of appropriately posed multiparametric compound boundary- and eigenvalue problems are interpreted and solved using the continuous analog of Newton method.

Classical electrical circuits consist of resistors and capacitors and are governed by integer-order model. Circuits may have so-called fractance which represents an electrical element with fractional order impedance. Therefore, fractional order derivative is important to study the dynamical behaviors of circuits. This paper extends the classical coupled Duffing circuits to cover a new fractional order Duffing system consisting of two identical periodic forced circuits coupled by a linear resistor. The fundamental resonance responses under various paired and unpaired fractional orders are investigated in detail using the harmonic balance in combination with polynomial homotopy continuation. The approximate solutions having a high degree of accuracy in the steady state response are sought. There exist different shapes of frequency versus response and excitation amplitude versus response curves under various fractional orders. Multiple-valued solutions and nonclassical bifurcations are observed analytically and verified numerically. The influence of coupling intensity on the fundamental resonance response is also examined. New contributions include the innovative introduction of fractional order to the coupled Duffing circuits, the explicit integration of fractional order and the linear consideration of higher harmonics to improve the nonlinear solutions of the lower harmonics without increasing the computational complexity.

In this paper, a three-species predator–prey system with diffusion and two delays is investigated. By taking the sum of two delays as a bifurcation parameter, it is found that the spatially homogeneous Hopf bifurcation can occur as the sum of two delays crosses a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are obtained by employing the center manifold theorem and the normal form theory. In addition, some numerical simulations are also given to illustrate the theoretical analysis.

Two different bifurcation scenarios of spontaneous neural firing patterns with decreasing extracellular calcium concentrations were observed in the biological experiment on identical pacemakers when potassium concentrations were fixed at two different levels. Six typical experimental scenarios manifesting dynamics closely matching those previously simulated using the Hindmarsh–Rose model and Chay model are provided as representative examples. Bifurcation scenarios from period-1 bursting to period-1 spiking via a complex process and via a simple process, period-doubling bifurcation to chaos, period-adding bifurcation with chaos, and period-adding bifurcation with stochastic burstings were identified. The results not only reveal that an experimental neural pacemaker is capable of generating different bifurcation scenarios but also provide a basic framework for bifurcations in neural firing patterns in a two-dimensional parameter space.

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

In this letter, we will show the existence of invariant sets called universal basins of attraction for typical nonlinear high-dimensional dynamical systems such as randomly sampled high-dimensional vector fields (ODEs) or maps. The method of analysis is based on the definition of an equivalence class between systems with the same number of neurons, the same number of time lags, and the same upper bound for one family of bifurcation parameters.

This paper proposes a chaos-based analog-to-information conversion system for the acquisition and reconstruction of sparse analog signals. The sparse signal acts as an excitation term of a continuous-time chaotic system and the compressive measurements are performed by sampling chaotic system outputs. The reconstruction is realized through the estimation of the sparse coefficients by the principle of chaotic parameter estimation. With the deterministic formulation, the analysis on the reconstructability is conducted via the sensitivity matrix from the parameter identifiability of chaotic systems. For the sparsity-regularized nonlinear least squares estimation, it is shown that the sparse signal is locally reconstructable if the columns of the sparsity-regularized sensitivity matrix are linearly independent. A Lorenz system excited by the sparse multitone signal is taken as an example to illustrate the principle and the performance.

A dynamical system with four quadratic nonlinearities is found to display a butterfly strange attractor. In a relatively large region of parameter space the system has coexisting point attractors and limit cycles. At some special parameter combinations, there are five coexisting attractors, where a limit cycle coexists with two equilibrium points and two strange attractors in different attractor basins. The basin boundaries have a symmetric fractal structure. In addition, the system has other multistable regimes where a pair of point attractors coexist with a single limit cycle or a symmetric pair of limit cycles and where a symmetric pair of limit cycles coexist without any stable equilibria.

In this paper, Lü system with a set of chaotic parameters is proved to be smoothly nonequivalent to Chen and Lorenz systems with any parameter. The analytical formula for the upper bound of Lyapunov dimension of attractors in Lorenz-type systems are presented under some suitable parameter conditions. These properties studied in this paper may contribute to a better understanding of the Lorenz-type systems.

The paper is devoted to the novel logic (SUC-logic) of the nonlinear dynamics forecasting. The SUC-logic is based on three main points: the special sections (S) to build the 2D projections of multidimensional spaces without the loss of useful information; the special units (U) of measurement to estimate the nonlinear dynamics evolution; the special consecutions (C) to realize the nonlinear dynamics forecasting step-by-step. The fractal approach to forecasting the nonlinear dynamics in real-time together with the approach to build the modified bifurcation diagrams to research the regularities and the uncertainties of the evolution scenarios are developed with the SUC-logic. The physical meaning of the uncertainty zone, the stability margin, the risk estimation, the farthest forecasting and the earliest forecasting are considered from the viewpoint of the nonlinear dynamics aspect. Reasonings and discussions are based on experimental and computational results.

In this paper, we consider the well-posedness and asymptotic behaviors of solutions of the fractional complex Ginzburg–Landau equation with the initial and periodic boundary conditions in two spatial dimensions. We explore the existence and uniqueness of global smooth solution by means of the Galerkin method and establish the existence of the global attractor. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor are presented.

In this paper, we study the coexistence of three types of attractors in an autonomous system in ℝ³: an equilibrium point, a limit cycle and a chaotic attractor. We give an analytical proof of the mechanism for the birth of two different types of these attractors.

This article reports a recent significant progress towards the implementation of the memristor. In the process, Sr-doped BaTiO₃ composite ceramic (thickness 1mm) was prepared by using the solid-state reaction method, whose basic properties (such as phases, microstructure and electric conductivity) were investigated by using XRD (X-ray Diffraction), SEM (Scanning Electron Microscopy) and an impedance analyzer. The electric conductivity of the Ba_0.77Sr_0.23TiO₃ (BST) ceramic shows that: (i) the conductivity of ceramic samples is increased with the increase of DC bias, demonstrating its nonlinear characteristics; (ii) the nonvolatile characteristics of ceramic samples are increased with the decrease of the temperature. Consequently, we conclude that the ceramic sample should have favorable nonvolatile behavior if the sample thickness is in nanoscale and the temperature is decreased to the room temperature. To that end, the current–voltage I–V curves of a BST nanometer film, prepared by sol–gel technique using the spin-coating method, are depicted. The experimental result illustrates its non-ohmic behavior and shows that the I–V curves are hysteretic. Both these indicate that the BST is a memristive system, and the BST monolayer nanometer film has important theoretical significance with a great potential in application to the memristor.