International Journal of Bifurcation and Chaos

In this work, we use the finite-time Lyapunov exponent (FTLE) for a version of the potential well model with an oscillating bottom. This model is described by a conservative two-dimensional mapping that exhibits a mixed phase space, namely it contains a sea of chaos and the regions of stability. Via FTLE, we show that throughout its orbit, an initial condition in the chaotic sea can become trapped in a certain region. We construct and analyze phase spaces for different ranges of Lyapunov exponents in finite time, highlighting these traps. For smaller and larger values of FTLE, we identified a very regular phase space and a phase space with more chaos. By means of low FTLE values, we observe some small regions in the phase space that present regularities. Therefore, we demonstrate that FTLE can be useful to identify the stickiness effect in a time-dependent potential well.

Using finite element simulation, we study the main features of rotating wave solutions of the cubic complex Ginzburg–Landau equation. To focus on the characteristics of the waves themselves, we have used a circular domain to avoid the effects of irregular boundaries; we have inhibited the formation of defects using an Archimedean wave centered at the origin as the initial condition, and we have chosen a domain size long enough to contain several spiral arms but not so long as to promote long-wave instabilities. This allows us to focus on the geometric features of the solutions often overlooked in traditional works with random initial conditions in large domains. We show with our simulations that the convective and absolute stabilities differ from those predicted for plane waves. Likewise, we show that the appearance of spirals and anti-spirals can occur in any of the quadrants of the parameter space and depends fundamentally on the initial conditions. Finally, regarding the stability of spiral waves against sideband disturbances, we show that the persistence of two-dimensional spiral waves, if fulfilled, does not correspond to the Eckhaus criterion. Our simulations allow us to corroborate the idea that two-dimensional spirals depend on the size of the domain and that they require new analytical results whose trends are identified in this work.

Chua’s circuit represents a paradigm for nonlinear dynamics and chaos in electronic circuits. In this paper, its discrete-time counterpart is considered showing the occurrence of nonlinear resonance when driven by an external sinusoidal input. Evidences of jump resonance are revealed from several regions of the parameter space. An approach based on quasi-analytic methods, numerical simulations and several experiments with digital microprocessors is presented, further confirming the richness of Chua’s circuit dynamical behavior. The high selectivity of the jump resonance observed from the discrete-time Chua’s circuit paves the way for the implementation of low-cost high-resolution frequency drift sensing devices.

This paper studies the transient dynamics in a fast–slow ecological model with a Monod–Haldane functional response. Geometric singular perturbation theory is utilized to examine the transient dynamics generated by intrinsic multiple time scales. An asymptotic series solution is derived for the normally hyperbolic sub-manifold in the slow flow. The paper proves the existence of relaxation oscillations near fold points and the existence of a smooth family of canard cycles arising from folded singularities. Furthermore, it investigates the resulting canard explosion and transient dynamics numerically. By incorporating the Monod–Haldane functional response, which represents the inhibitory effect on predation at high prey population levels, this study illustrates the occurrence of transient dynamics under this mechanism.

The spread of epidemic diseases is often considered to be bound up with human behavior. Both treatment and awareness-raising are seen as important preventive measures during an outbreak. In this paper, an SEIR model on heterogeneous network containing saturated treatment function with media information variable is constructed. Through mathematical analysis, the basic reproduction number and equilibriums of the system are derived. The global stability of disease-free equilibrium is further proved based on LaSalle’s invariant principle. It is also found that with a gradual increase of the delayed treatment parameter, the system exhibits a transition from forward bifurcation to backward bifurcation with the variation of the basic reproduction number. Finally, an optimal control problem is formulated with both treatment rate and media information dissemination rate as control variables and solved using the Pontryagin’s maximum principle. All the above results are verified by numerical experiments.

To address the problem of dynamic degradation over a finite-precision platform of chaotic maps and the reversibility of linear chaotic maps, we propose an improved model over GF($2^n$) that is called the nondegenerate m-Dimensional $(m\ge 2)$ Integer-Domain Chaotic Maps (mD-IDCMs). This model incorporates modular exponentiation operation, and is capable of constructing nondegenerate IDCMs of any dimension. Moreover, we prove the irreversibility of mD-IDCM and analyze its chaotic behaviors in terms of positive Lyapunov Exponents (LEs). The results of theoretical analysis show that the proposed mD-IDCM model can obtain the desired positive LEs by appropriately configuring its coefficient matrix. Then, we present two instances, and analyze their LEs, Kolmogorov entropy, Sample entropy, Correlation dimension, and the dynamic analysis indicates that the chaotic map constructed by mD-IDCM has ergodicity within a sufficiently large chaotic range. Finally, we design a Pseudo-Random Number Generator (PRNG) with a key to verify the practicability of the mD-IDCM.

In dynamical systems, events that deviate significantly from usual or expected behavior are referred to as extreme events. This paper investigates the mechanism of extreme event generation in a 3D jerk system based on a generalized memristive device. In addition, regions of coexisting parallel bifurcation branches are explored as a way of investigating the multistability of the memristive system. The system is examined using bifurcation diagrams, Lyapunov exponents, time series, probability density functions of events, and inter-event intervals. It is found that extreme events occur via a period-doubling route and are due to an interior crisis that manifests itself as a sudden shift from low-amplitude to high-amplitude oscillations. Multistability is also identified when both control parameters and initial values are modified. Finally, an analog circuit based on the memristive jerk system is designed and experimentally realized. To our knowledge, this is the first time that extreme events have been reported in a memristive jerk system in particular and in jerk systems in general.

This paper investigates the impact of gene expression time delay and diffusion on the dynamic behavior of a class of Gierer–Meinhardt systems under Neumann boundary conditions. It provides necessary and sufficient conditions for the emergence of Hopf, Turing, and Turing–Hopf bifurcations. Utilizing the normal form of the Turing–Hopf bifurcation, the spatiotemporal dynamics close to the bifurcation point are categorized into six types, encompassing spatially inhomogeneous and homogeneous periodic solutions, as well as spatially homogeneous and inhomogeneous steady states, along with their transitions. Notably, it’s observed that the systems may lack stable spatially inhomogeneous periodic solutions under specific parameters, despite the emergence of Turing–Hopf bifurcation. These findings are supported by numerical simulations.

The nonlinear characteristic of activation function is a critical aspect that affects the performance of artificial neural networks (ANNs). Despite its importance, the impact of neuronal nonlinearity variations on the dynamics of ANNs has not been thoroughly studied in previous research. This paper aims to fill this gap by exploring the influence of nonlinearity changes in activation function due to memristive electromagnetic induction on the dynamical characteristics of ANNs. We first propose a memristor model with sinusoidal conductance function. Then, the effects on the biological neuron under the electromagnetic induction are simulated by replacing the gradient of hyperbolic tangent activation function with the conductance of the proposed memristor. Furthermore, a Hopfield neural network (HNN) is constructed by taking the nonlinearity changes of activation function into consideration and its dynamical behaviors are analyzed through Lyapunov exponent spectra, bifurcation diagram, biparameter-based dynamic maps and phase portraits. The numerical simulation results show complex dynamical behaviors such as single-scroll chaotic spiral attractor, double-scroll chaotic attractors and initial-value-dependent offset boosted attractors, demonstrating that nonlinearity changes of activation function induced by the electromagnetic induction have a significant impact on the dynamics of HNN. To further verify our findings, we implement the proposed HNN in hardware and find that the results are in line with our numerical simulations. Finally, we apply the chaotic HNN in the mobile robot path planning task, which achieves higher coverage rate than the random number-based random walk method.

Chaos and sub-harmonic bifurcations of a simple SIR model with periodically and parametrically nonlinear incidence rate are investigated analytically with the Melnikov method and sub-harmonic Melnikov method, respectively. The critical curves of chaos for the model with the periodically and parametrically nonlinear incidence rate are studied analytically for the first time. Some new interesting dynamic phenomena are presented and proved rigorously, including the nonchaotic band, controllable frequencies and even controllable frequency interval, which may exist for some system parameters in this model. The sub-harmonic periodic orbit family is derived. It is presented that sub-harmonic bifurcations with even order and integer order may exist in the system. The sub-harmonic bifurcation conditions are obtained rigorously for two different starting points. Numerical simulations including phase portraits, Poincaré sections, basins of attraction and Lyapunov exponents are given to verify the chaos threshold obtained by the analytical results. It is also shown that the system can undergo chaos through even-order sub-harmonic bifurcations.

This paper is mainly concerned with the existence of chaos in a two-dimensional spatiotemporal discrete system with delayed perturbations. By applying the snap-back repeller theory, two criteria of chaos in the sense of both Li–Yorke and Devaney are established. Moreover, the lower bounds of the parameters that make the system chaotic are obtained. Finally, an example and its computer simulations of the dynamical behavior, spatiotemporal graphs, and the trends of the largest Lyapunov exponents with respect to the initial values and parameters are given to demonstrate the correctness of the theoretical results.

In previous studies, we developed two techniques aimed at expanding the scope of constructing a periodic orbit dividing surface within a Hamiltonian system with three or more degrees of freedom. Our approach involved extending a periodic orbit into a torus or cylinder, thereby elevating it into a higher-dimensional entity within the energy surface (see [, , , , ]). Recently, we introduced two alternative methods for creating dividing surfaces, distinct from the utilization of periodic orbits, by employing 2D surfaces (geometric entities) or 3D surfaces within a Hamiltonian system with three degrees of freedom (refer to [, , , ]). In these studies, we applied these surfaces in a quadratic normal form Hamiltonian system with three degrees of freedom. In this series of two papers, we extend our results to 2D generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper presents the first method of constructing 2D generating surfaces.

In this work, we present a method that determines optimal multistep Dynamic Mode Decomposition (DMD) models via Entropic Regression (ER), which is a nonlinear information flow detection algorithm. Motivated by the Higher-Order DMD (HODMD) method of [, ], and the ER technique for network detection and model construction found in [, ; , ], we develop a method that we call ERDMD, which produces high fidelity time-delay DMD models that allow for nonuniformity in the delays. This optimal choice of delays is discovered by maximizing informativity as measured through ER. These models are shown to be highly efficient and robust. We test our method over several data sets generated by chaotic attractors and show that we are able to build excellent reconstructions using relatively minimal models. We likewise are able to better identify multiscale features via our models which enhances the utility of DMD.

We review a number of methods to prove nonintegrability of Hamiltonian systems and focus on 3 Degrees-of-Freedom (DoF) systems listing the known results for the prominent resonances. Associated with the Hamiltonian systems are the averaged-normal forms that provide us with geometric insight, approximations of orbits and measures of chaos. Symmetries do change the qualitative and quantitative pictures; we illustrate this for the 1:2:1 resonance with discrete symmetry in the 1st and 3rd DoF. In this case, the averaged-normal form is still nonintegrable, but it becomes integrable when adding discrete symmetry in all DoF. Apart from the short-periodic solutions obtained by averaging, we find many periodic solutions. There is numerical evidence of the presence of Šilnikov bifurcation which clarifies the presence of nonintegrability phenomena qualitatively and quantitatively.

The classical Hastings–Powell model is well known to exhibit chaotic dynamics in a three-species food chain. Chaotic dynamics appear through period-doubling bifurcation of stable coexistence limit cycle around an unstable coexisting equilibrium point. A specific choice of parameter value leads to a situation, where the chaotic attractor disappears through a collision with an unstable limit cycle originated due to subcritical Hopf-bifurcation around the coexistence equilibrium. As a consequence, the top predator goes to extinction. The main objective of this work is to explore the structural sensitivity of this phenomenon by replacing the Holling-type II functional responses with Ivlev functional responses. In this work, we have shown the existence of two Hopf-bifurcation thresholds and numerically verified the existence of an unstable limit cycle. The model with Ivlev functional responses does not indicate any possibility of extinction of the top predator due to any collision of chaotic attractor with the unstable limit cycle for the chosen range of parameter values. Moreover, the model with Ivlev functional responses depicts an interesting scenario of bistable oscillatory coexistence.