International Journal of Bifurcation and Chaos

We apply the method of Lagrangian Descriptors (LDs) to a symmetric Caldera-type potential energy surface which has three index-1 saddles surrounding a relatively flat region that contains no minimum. Using this method we show the phase space transport mechanism that is responsible for the existence and nonexistence of the phenomenon of dynamical matching for this form of Caldera potential energy surface.

In this paper, by averaging the growth rate on each state, we analyze the dynamics of a discrete dynamical system coming from a system of ODEs. This differential system corresponds to a tritrophic Leslie type model which is formed by three populations (prey (P), mesopredator (MP) and superpredator (SP)), where the last two populations are generalist predators. We give sufficient conditions where the discrete model undergoes a Neimark–Sacker bifurcation at a coexistence point. This analysis is independent of the functional responses that govern the interactions. To illustrate our results, several applications are given, under the assumptions that the population P has logistic growth and that the relations MP–P and SP–MP are carried out through Holling type functional responses. From these applications, we conclude that there are sufficient conditions to guarantee that the three species coexist by means of a supercritical Neimark–Sacker bifurcation. Moreover, numerically we can detect that the discrete system exhibits a chaotic behavior.

An intrinsic similarity was noted between epidemic spreading models and diffusion operations in cryptography. In this paper, a new image encryption algorithm is proposed based on an epidemic spreading model. The notable feature of the model is the evolution of the network node state. We utilize the evolution of an epidemic spreading model, namely the susceptible-infectious-recovered (SIR) model, to drive the encryption progress by abstracting pixels to the network nodes. In the new cryptosystem, the pixels in plain-text are considered as susceptible individuals, permutation and diffusion are seamlessly integrated rather than having two independent stages, and the encryptions are performed in a two-dimensional and nonlinear manner. Simulation results demonstrate its high encryption speed and excellent encryption security.

A fractional-order three-species food chain ecosystem with prey refuge and Holling-II type functional response for predation is proposed and studied. Several sufficient conditions for the existence and uniqueness of the solution of the fractional-order system are obtained. The boundedness of the solution of the system is proven. We investigate the asymptotic behavior of the model by using eigenvalue analysis, and some sufficient conditions on local asymptotic stability of the equilibrium points are given. Furthermore, the conditions for the occurrence of bifurcation at some equilibrium points are presented. We find that the order of the proposed fractional-order ecosystem is one of the parameters for its bifurcation. Several numerical simulations are provided to show the effectiveness of our findings in this paper. Lastly, some new numerical simulations are given to discuss the influence of the half-saturation constant, prey refuge coefficient and the order of fractional-order derivative on the stability of the discussed fractional-order system.

A coupled model of a top tensioned riser (TTR) under combined excitations from vortices and time-varying tension is proposed. And the van der Pol oscillator is introduced to simulate the time-varying characteristics of vortex shedding. The first three mode approximations as a six degree-of-freedom (DOF) system of vibrations of the fluid-structure interaction system are derived. Two cases of resonances are analyzed by using the numerical method. One case is on the vortex induced vibration (VIV) of the six DOF system, where the 1-to-1 internal resonance (also named “lock-in” phenomenon) between frequencies of the first modes of structure and the wake oscillator are investigated. Bifurcation curves between dynamic responses in terms of the shedding frequency are derived. Comparing to the vibrations of the first order mode, the “lock-in” phenomenon for vibrations of the higher order modes can occur in the wider parameter range with large values. The second case is on the combined resonance between the 1-to-1 internal and 1-to-2 parametric resonance. Response curves versus the shedding frequency under different amplitude ratios are calculated by the Runge–Kutta method. Effects of system parameters including the amplitude ratio of the parametric excitation, shedding frequency on the response curves are investigated. Comparing to the case when only considering the 1-to-1 internal resonance in the VIV, results of the combined resonance show those large-amplitude responses of the structure and wake oscillator can be induced. Responses will increase rapidly as the amplitude of the varying tension increases. Unstable responses can occur within a large frequency band of nondimensional shedding frequency. Furthermore, the dynamic characteristics of the responses of the six DOF system including the bifurcation diagrams, time histories, phase-plane projection and Poincaré sections are computed for different amplitude ratios. Results indicate that complex dynamics such as the quasi-periodic and chaotic solutions can occur for different amplitude ratios. The analysis can be helpful for the parameter design of deep-water risers.

The main objective of this study is to explore the nonlinear dynamics and chaos detecting in the three-dimensional (3D) generalized Hamiltonian Yang–Mills system as coupled quartic oscillators. We investigate its Liouvillian integrability. The integrable system is obtained and its associated first integral of motion is explicitly given. Also, a variety of nonlinear phenomena could be expressed using various numerical analysis, such as the construction of the system’s Poincaré Surface of Section (PSS), the maximum Lyapunov exponent (mLE), and the modern numerical methods like the Smaller (SALI) and the Generalized (GALI) Alignment Indexes. In this context, a series of numerical simulations are appropriate in order to explore ordered and chaotic motions of the system both in two (2D) and three dimensions, and one can observe chaos–order–chaos transition of the system when any parameters on which the system depends vary. Finally, several numerical schemes are shown to demonstrate the effectiveness and rapidity of these proposed methods for distinguishing chaos and order states of the system.

It is essential yet challenging to investigate heteroclinic trajectories in chaotic dynamics of high-dimensional systems. This paper presents an efficient approach for precise prediction of heteroclinic cycles in $n$-dimensional ($n\ge 3$) piecewise affine systems. Moreover, it systemically investigates different types of heteroclinic cycles in the considered systems, and proposes the corresponding existence conditions with rigorous proofs by analyzing the dynamics of stable and unstable manifolds. In addition, three numerical examples with both a heteroclinic cycle and a chaotic set are provided to verify the established results.

It is well known that dynamical systems are very useful tools to study viral diseases such as HIV, HBV, HCV, Ebola and Influenza. This paper focuses on a mathematical model of the cell-to-cell and the cell-free spread of HIV with both linear and nonlinear functional responses and logistic target cell growth. The reproduction number of each mode of transmission has been calculated and their sum has been considered as the basic reproduction number. Based on the values of the reproduction number, the local and global stabilities of the rest points are investigated. Choosing a suitable bifurcation parameter, some conditions for the occurrence of Hopf bifurcation are also obtained. Moreover, numerical simulations are presented to support the analytical results. Finally, to study the effect of the drug on the disease process, some control conditions are determined. Since two modes of transmission and both linear and nonlinear functional responses have been included in this manuscript, our obtained results are a generalization of those in the literature. Moreover, the results are obtained with weaker assumptions in comparison with the previous ones.

This paper studies the dynamics of a nonlinear network with many interacting neural populations and time-delayed couplings under electromagnetic radiation. The influence of the electromagnetic radiation is described by using flux-controlled memristor. The delay-induced instability and bifurcated periodic oscillations of the memristive multiplex network are determined by decomposing and analyzing the characteristic equations. The bifurcation diagrams are shown and complicated dynamical behaviors are explored, such as multi-periodic orbits and period doubling bifurcations. These interesting phenomena are also observed from the circuit realization. It is found that both time delay and electromagnetic radiation have great effects on the performance of the network, such as synchronization transitions and multistability coexistence.

This paper proposes a lossless encryption–compression algorithm for large-scale remote sensing images. Firstly, the red, green and blue components of color image are compressed by a lossless predictive encoding. Then, the lifting wavelet transform (LWT) is used to decompose the encoding results. And a new Rubik’s cube transformation is introduced to scramble the decomposed coefficients, which uses the chaotic sequence generated by 2D Cubic–Chebyshev map (2D-CCM). The initial values of 2D-CCM are obtained from the chi-square test values of the three components, which leads to the algorithm related to the plaintext image. After that, the scrambling coefficients are thresholding, and the position sequences generated in the process are encrypted and compressed by the proposed encrypted run-length encoding (E-RLE). The processed coefficients are further compressed by Huffman encoding. At the end, the final results are obtained by a novel helix diffusion which is related to the chaotic sequence. Experimental results show that, this algorithm achieves higher lossless compression ratio with lower time complexity, and the encryption scheme has higher security.

This paper focuses on the hybrid effects of memristor characteristics, coupling coefficient and time-delay on the recurrent neural network. We construct a novel time-delay recurrent neural network based on a passive hyperbolic tangent memristor, whose instability condition at equilibrium points is judged according to the Routh criterion. Under the changes of the memristor internal parameter, coupling coefficient and time-delay, some complex dynamic behaviors, including chaos and various coexisting attractors, of the time-delay memristive recurrent neural network are observed in detail through the bifurcation diagram, the largest Lyapunov exponent, and the basin of attraction. Finally, the theoretical results are verified by the circuit simulation through MULTISIM.

Recent studies have demonstrated that immune impairment is an essential factor in viral infection for disease development and treatment. In this paper, we formulate an age-structured viral infection model with a nonmonotonic immune response and perform dynamical analysis to explore the effects of both immune impairment and virus control. The basic infection reproduction number is derived for a general viral production rate, which determines the global stability of the infection-free equilibrium. For the immune intensity, we get two important thresholds, the post-treatment control threshold and the elite control threshold. The interval between the two thresholds is a bistable interval, where there are two immune-present infected equilibria. When the immune intensity is greater than the elite control threshold, only one immune-present infected equilibrium exists and it is stable. By assuming the death rate and virus production rate of infected cells to be constants, with the immune intensity as a bifurcation parameter, the system exhibits saddle-node bifurcation, transcritical bifurcation, and backward/forward bifurcation.

A megastable oscillator with various types of attractors is proposed. The oscillator shows interesting dynamics like cloud, kite, and arrow-like attractors. As we know, such a megastable oscillator with the dynamic shapes was not previously reported. There is an infinite number of arrow-like attractors in this oscillator. Investigating equilibrium points of the oscillator and their stabilities shows that two of the attractors, the largest cloud-like and smallest kite-like, are self-excited, while the others are hidden. The basin of attraction for the oscillator is studied for various attractors. Then, a forcing term is added to the oscillator, and its dynamics are investigated. The bifurcations and Lyapunov exponents of each type of attractor are investigated by changing the coefficient of the sinusoidal term $K$. Each of the attractors shows various periodic, quasi-periodic, and chaotic dynamics for various $K$. Also, the basin of attraction of the forced oscillator is investigated.

In this paper, we study the limit cycles in the discontinuous piecewise linear planar systems separated by a nonregular line and formed by linear Hamiltonian vector fields without equilibria. Motivated by [Llibre & Teixeira, 2017], where an open problem was posed: Can piecewise linear differential systems without equilibria produce limit cycles? We prove that such systems have at most two limit cycles, and the limit cycles must intersect the nonregular separation line in two or four points. More precisely, the exact upper bound of crossing limit cycles is two, and this upper bound can indeed be reached: either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points. Based on Poincaré map, the stability of various limit cycles is also proved. In addition, we give some concrete examples to illustrate our main results.

At present, stochastic resonance (SR) based on Duffing and Langevin equations has been widely studied, but there is little research on SR phenomenon in higher-dimensional fractional-order systems. Based on the fractional-order Lorenz-like system, the SR phenomenon is studied from the perspective of equilibrium point in this paper. The dynamic process of SR phenomenon is presented, and several key parameters are derived. The results of numerical simulation show that the boundary of SR and chaos system will be broken if the order of the fractional-order system or internal parameters is properly adjusted, and the effect of SR phenomenon will also be influenced. This paper connects two important nonlinear phenomena (chaos and SR) based on the fractional-order system, which provides a new idea for the conversion between chaos and SR systems.

Impulsive control strategies have been widely used in cancer treatment and linear impulsive control has always been considered in previous studies. We propose a novel tumour-immune model with nonlinear killing rate as state-dependent feedback control, which can better reflect the saturation effects of the tumour and immune cell mortalities due to chemotherapy, and its dynamic behaviors are investigated. The paper aims to discuss the transcritical and subcritical bifurcations of the model. To begin with, the threshold conditions for tumour eradication and tumour persistence in the model without pulse interventions are provided. We define the Poincaré map of the proposed model and then address the existence and orbital asymptotically stability of the model’s tumour-free periodic solution. Furthermore, by using the bifurcation theory of the discrete one-parameter family of maps, which is determined by the Poincaré mapping, we investigate the model’s transcritical and subcritical pitchfork bifurcations with respect to the key parameter.

Phytoplankton patterns have been observed widely in aquatic systems. Although pattern formation has been investigated based on many PDEs, discrete models on aquatic systems can provide more complex dynamics. A discrete toxic-phytoplankton–zooplankton model is studied in this paper, with the consideration of Allee effect and cross-diffusion. Focusing on Allee effect coefficient, flip and Neimark–Sacker bifurcation analyses are carried out. And focusing on cross-diffusion coefficient, Turing bifurcation analyses are carried out. Parameter conditions and bifurcation diagram of these bifurcations are obtained correspondingly. Numerical simulations are then performed which are consistent with results of theoretical analysis. Irregular patterns can be formed by flip bifurcation. Spirals can be formed by Neimark–Sacker bifurcation. Spots and stripes can be formed by Turing bifurcation. When Turing and flip, or Turing and Neimark–Sacker bifurcations both occur, special patterns can be obtained.

Motivated by an attractive problem of elucidating the causes for the occurrence of complex oscillatory regimes in chemical reactions, we study the constructive role of noise in a three-dimensional enzymatic “substrate-two products” model. Stochastic effects are investigated in the parameter zone of birhythmicity where regular oscillatory attractors coexist with chaotic ones. Chaos–order transformations caused by stochastic transitions between these attractors are studied parametrically using the Lyapunov exponents. A special form of stochastic excitement of large-amplitude bursts consisting of the high-frequency spikes is revealed and investigated by statistical analysis.

In this paper, we obtain a lower bound for the maximum number of crossing limit cycles for cubic planar switching systems with two regions separated by a straight line. By pseudo Hopf bifurcation and $𝜖$-order Lyapunov constants, we prove that there exist cubic near-integrable switching systems with at least 26 small-amplitude limit cycles bifurcating from an elementary center after suitable perturbations. To the best of our knowledge, this is the best lower bound so far for the cubic class by using a first order analysis.