International Journal of Bifurcation and Chaos

The Boros–Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to their convergence. In the paper, we study the dynamics of a one-parameter family of maps which unfold the Boros–Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros–Moll map is a peculiar feature within the family. We relate this singularity with a specific property of the critical lines that occurs only for this special case. In particular, we explain how the unbounded chaotic region in the Boros–Moll map appears. We especially explain the main contact/homoclinic bifurcations that occur in the family. We also report some other bifurcation phenomena that appear in the considered unfolding.

Hyperbolic tangent function, a bounded monotone differentiable function, is usually taken as a neuron activation function, whose activation gradient, i.e. gain scaling parameter, can reflect the response speed in the neuronal electrical activities. However, the previously published literatures have not yet paid attention to the dynamical effects of the neuron activation gradient on Hopfield neural network (HNN). Taking the neuron activation gradient as an adjustable control parameter, dynamical behaviors with the variation of the control parameter are investigated through stability analyses of the equilibrium states, numerical analyses of the mathematical model, and experimental measurements on a hardware level. The results demonstrate that complex dynamical behaviors associated with the neuron activation gradient emerge in the HNN model, including coexisting limit cycle oscillations, coexisting chaotic spiral attractors, chaotic double scrolls, forward and reverse period-doubling cascades, and crisis scenarios, which are effectively confirmed by neuron activation gradient-dependent local attraction basins and parameter-space plots as well. Additionally, the experimentally measured results have nice consistency to numerical simulations.

In this paper, the Turing pattern formation in a diffusive toxin producing phytoplankton–zooplankton model with three-dimensional patch is investigated. Firstly, the conditions for the existence of Hopf bifurcation and stationary pattern are obtained based on the linear stability analysis. Then, the amplitude equations are derived by using nonlinear analysis and standard multiple-time-scale analysis. This study shows that there exists a pattern transition from spot pattern to stripe pattern with the decrease of the amount of toxicity, and the death rate of zooplankton also has a significant effect on the pattern structure. The obtained results suggest that the toxicity of phytoplankton and the death rate of zooplankton both play vital roles in the spatial distribution of population density.

The diffusive Holling–Tanner predator–prey model with no-flux boundary conditions and nonlocal prey competition is considered in this paper. We show the existence of spatially nonhomogeneous periodic solutions, which is induced by nonlocal prey competition. In particular, the constant positive steady state may lose the stability through Hopf bifurcation when the given parameter passes through some critical values, and the bifurcating periodic solutions near such values could be spatially nonhomogeneous and orbitally asymptotically stable.

The running-in attractor is a chaotic attractor with stability, which forms in the running-in process. Its “formation-stabilization-disappearance” stage can reflect the dynamic features of the nonlinear friction system. To reveal the dynamic features and evolution of the running-in attractor, in this work, wear experiments are implemented on a ring-on-disk tribometer. The friction coefficient signals are then extracted and reconstructed. Based on the reconstructed matrix of the friction coefficient signals, the convergence (CON) and standard deviation of the distance matrix (STD) are proposed. Results show that CON evolves from an increase to the stabilization and then to a decrease. STD presents an inverse variation tendency, evolving from a decrease to the stabilization and then to an increase. A larger CON indicates higher convergence and a more complex friction system. A smaller STD indicates a more stable friction system. These two new characteristic parameters effectively exhibit the dynamic behavior of the wear process.

We study the intersection of double-flip (period-doubling) bifurcations in a parameter plane. We derive normal forms for discrete-time and continuous-time systems. Using these normal forms, we clarify the bifurcation structure around the flip-flip bifurcation point. We apply these analytical results to a system of coupled ventricular cell models. We determine the coexistence of in-phase and anti-phase two-periodic solutions. We make the simplest model for generating discordant alternans and clarify that two parameters (free concentration of potassium ions in the extracellular compartment and the conductance of the gap junction) play key roles in generating discordant alternans.

Vibration-based energy harvesting is of increasing importance and there is a current challenge to improve energy harvesting capacity exploiting nonlinear and random effects. This article investigates random effects in a nonlinear energy harvesting system. The system is represented by a magnetoelastic structure with two piezoceramic layers attached to the root of a cantilever beam, obtaining a bimorph generator. The energy harvesting system is subjected to three excitation conditions: pure harmonic, pure random and a combination of harmonic and random excitations. Noise-to-Signal Ratio (NSR) is employed to quantify different combinations of the forcing terms, establishing a procedure to evaluate the system performance. This approach is based on Power Spectral Density (PSD) of input and output signals. Numerical simulations are carried out, identifying the better combinations of harmonic and random excitations for energy harvesting purposes. Discussions about the influence of the kind of response are carried out evaluating the differences between periodic and chaotic motions. Conclusions show that both random and nonlinear effects can be tuned in order to enhance energy harvesting capacity.

For a singular nonlinear traveling wave system of the first class, if there exist two node points of the associated regular system in the singular straight line, then the dynamics of the solutions of the singular system will be very complex. In this paper, two representative nonlinear traveling wave system models (namely, the traveling wave system of Green–Naghdi equations and the traveling wave system of the Raman soliton model for optical metamaterials) are investigated. It is shown that, if there exist two node points of the associated regular system in the singular straight line, then the singular system has no peakon, periodic peakon and compacton solutions, but rather, it has smooth periodic wave, solitary wave and kink wave solutions.

We generalize the chimney model by introducing nonlinear restoring and gravitational forces for the purpose of modeling swaying of trees at high wind speeds. We have derived the general equations governing the system using Lagrangian formulation. We studied the simplest case of a single element in more detail. The governing equation we arrive at for this case has not been studied so far. We study the chaotic properties of this simple building block and also the effect of directionality in the wind on the chaotic properties. We also consider the special case of two elements.

In this paper, we have investigated a diffusive predator–prey model with herd behavior. Also, we considered that the mortality of predators is linear as well as quadratic. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain non-Turing spatiotemporal patterns in the model with linear mortality of predators, and Turing pattern formation, namely, spotted pattern and mixed pattern (spots-stripes) in model with quadratic mortality of predators. The results focus on the effect of the changing mortality rates of predator in pattern dynamics of a diffusive predator–prey model and help us in the better understanding of the dynamics of the predator–prey interaction in real environment.

We present in this paper, two new chaotification methods: The first is about chaotifying one-dimensional discrete mappings using the notion of $S$-unimodality and the second one is about chaotifying one-dimensional discrete mappings using Collet–Eckmann condition. These two methods are valid only for one-dimensional discrete mappings.

We investigate the chaotic dynamics of a two-dimensional piecewise smooth map. The map represents the normal form of a discrete time representation of impact oscillators near grazing states. It is proved that, in certain region of the parameter space, the nonwandering set of the map is contained in a bounded region and that, restricted to the nonwandering set, the map is topologically conjugate to the two-sided shift map on two symbols.

The lemniscate sine and cosine are solutions of a $Z_4$-equivariant planar Hamiltonian system for all of which nontrivial solutions are nonhyperbolic periodic orbits. The forward Euler scheme is applied to this system and the one-parameter discrete-time $Z_4$-equivariant cubic dynamical system is obtained. The discrete-time system depending upon a parameter exhibits rich dynamics: numerical simulation shows that the system has attracting closed invariant curves, multiple periodic orbits and attracting sets exhibiting chaotic behavior. The approximating system of ordinary differential equations is constructed. We discuss the existence of closed invariant curves for the discrete-time system.

In this paper, we focus on clustering phenomena in a network composed of coupled chaotic circuits. In this investigation, the coupling strength is reflected by the distance information when the chaotic circuits are placed in a two-dimensional grid. We observe various clustering phenomena in the network of coupled chaotic circuits when we vary the scaling parameters, including the coupling strength, the distance between coupled chaotic circuits and the density of the chaotic circuits.

This paper presents an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multidimensional discrete dynamical systems based on a modified Marotto’s theorem developed by Li and Chen (called “Marotto–Li–Chen theorem”). Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and propose an algorithmic approach to solve this algebraic problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

This paper attempts to find some interesting and unique properties in the dynamics of a permanent magnet synchronous motor (PMSM). When compared with the existing literature on the dynamics of a PMSM, we find some interesting and unique behaviors which have not been reported so far. These are (i) occurrence of multistability, (ii) existence of hidden attractors and (iii) equilibrium point with two stable node-foci and one saddle point index-1. The above-said unique behaviors in the dynamics of a PMSM are not found in the literature to the best of authors’ knowledge. Three different cases with (a) $v/f$ (voltage/frequency) control, (b) constant load torque and (c) constant direct and quadrature-axis voltage, and load torque are considered to show the multistability in the dynamics of a PMSM. The multistability is confirmed by using the bifurcation analysis. In another case, when the load torque is selected as a feedback of quadrature-axis voltage, the system depicts hidden attractors (point, periodic and transient chaotic). An adaptive sliding mode control is designed to control the hidden transient chaotic behavior of the system. The simulation results confirm the suppression of the transient chaotic attractors with smaller stabilization time and chattering free control input.

In this paper, we continue our studies of the two-dimensional caldera potential energy surface in a parametrized family that allows for a study of the effect of symmetry on the phase space structures that govern how trajectories enter, cross, and exit the region of the caldera. As a particular form of trajectory crossing, we are able to determine the effect of symmetry and phase space structure on dynamical matching. We show that there is a critical value of the symmetry parameter which controls the phase space structures responsible for the manner of crossing, interacting with the central region (including trapping in this region) and exiting the caldera. We provide an explanation for the existence of this critical value in terms of the behavior of the Hénon stability parameter for the associated periodic orbits.