International Journal of Bifurcation and Chaos

Based on the Hodgkin–Huxley and Hindmarsh–Rose models, this paper proposes a geometric phenomenological model of bursting neuron in its simplest form, describing the dynamic motion on a mug-shaped branched manifold, which is a cylinder tied to a ribbon. Rigorous mathematical analysis is performed on the nature of the bursting neuron solutions: the number of spikes in a burst, the periodicity or chaoticity of the bursts, etc. The model is then generalized to obtain mixing burst of any number of spikes. Finally, an example is presented to verify the theoretical results.

In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations.

The nonlinear dynamics of an underdamped sinusoidal potential system is experimentally and numerically studied. The system shows regular (nonchaotic) periodic motion when driven by a small amplitude ($F_1$) sinusoidal force (frequency $\omega =\frac{2\pi }{\tau }$). However, when the system is driven by a similarly small amplitude biharmonic force (frequencies $\omega$ and $\frac{\omega }{2}$ with amplitudes $F_1$ and $F_2$, respectively) chaotic motion appear as a function of amplitude ($F_2$) of the $\frac{\omega }{2}$-frequency component for a fixed $F_1\approx 0.22$. We investigate the effect of an additional constant force $\mathrm{\Delta }F$ on the dynamics of the system in the ($F_2-\mathrm{\Delta }F$) space. We find that $\mathrm{\Delta }F$ can cause chaotic motion to move to regular motion and regular motion can also become chaotic in certain ($F_2-\mathrm{\Delta }F$) domains.

We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a “graphene-type” force law and one inspired by Hollomon’s law describing “work-hardening” effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of nonanalytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi–Pasta–Ulam–Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles $N$ and the total energy $E$ are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large $N$, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.

The nanoscale implementations of ternary logic circuits are particularly attractive because of high information density and operation speed that can be achieved by using emerging memristor technologies. Memristor is a nanoscale device with nonvolatility and adjustable multilevel states, which creates an intriguing opportunity for the implementation of ternary logic operations. This paper proposes a novel memristor-based design for stateful ternary logic, including AND, OR, NOT, NAND, NOR, and COPY operations. In the proposed memristor ternary logic (MTL) design, the resistance of memristor is the only logic state variable for representing the input and output. By sensing the value of the input memristors, the resistance of the output memristor changes accordingly. Furthermore, the MTL gates are not only capable of performing logic operations, but also storing logic values. To illustrate the potential of the methodology, a single-input-three-output ternary decoder is designed by using the proposed ternary logic circuits. Simulation results verify the effectiveness of the presented design.

The security of chaotic cryptographic system can be theoretically evaluated by using conventional statistical tests and numerical simulations, such as the character frequency test, entropy test, avalanche test and SP 800-22 tests. However, when the cryptographic algorithm operates on a cryptosystem, the leakage information such as power dissipation, electromagnetic emission and time-consuming can be used by attackers to analyze the secret keys, namely the Side Channel Analysis (SCA) attack. In this paper, a cryptanalysis method is proposed for evaluating the security of a chaotic block cryptographic system from a hardware perspective by utilizing the Template Attacks (TAs). Firstly, a chaotic block cryptographic system is described briefly and implemented based on an Atmel XMEGA microcontroller. Then the TA using a multivariate Gaussian model is introduced. In order to reduce computational complexity and improve the efficiency of TA, the Hamming weight is used in this work to model power consumption traces. The proposed TA method has the following advantages including (a) using the sum of difference to select points of interest of traces, (b) using a data processing method to minimize the influences on power information modeling from the redundant sampling points, and (c) all the traces are aligned precisely before establishing the templates. Experimental results show that the TA can be used to attack the chaotic cryptographic systems and is more efficient, i.e. $\sim$32% less attack traces than correlation power analysis, when the templates are properly built.

This paper is dedicated to a study of a diffusive one-prey and two-cooperative-predators model with C–M functional response subject to Dirichlet boundary conditions. We first discuss the existence of positive steady states by the fixed point index theory and the degree theory. In the meantime, we analyze the uniqueness and stability of coexistence states under conditions that one predator’s consumer rate is small and the effect of interference intensity of another predator is large. Then, steady-state bifurcations from two strong semi-trivial steady states (provided that they uniquely exist under some conditions) and from one weak semi-trivial steady state are investigated in detail by the Crandall–Rabinowitz bifurcation theorem, the technique of space decomposition and the implicit function theorem. In addition, we study the asymptotic behaviors including the extinction and permanence of the time-dependent system by the comparison principle, upper-lower solution method and monotone iteration scheme. Finally, numerical simulations are done not only to validate the theoretical conclusions, but also to further clarify the impacts of parameters on the three species.

Characterizing the relationship between time series is an important issue in many fields, in particular, in many cases there is a nonlinear correlation between series. This paper provides a new method to study the relationship between time series using the perspective of complex networks. This method converts a time series into a distance matrix and constructs a sequence of nearest neighbor networks, so that the nonlinear relationship between time series is expressed as similarity between networks. In addition, based on the surrogate series, we applied $Z$-score to characterize the level of significance and analyzed some benchmark models. We not only use the artificial dataset and the real dataset to verify the effectiveness of the proposed method, but also analyze its robustness, which provides an alternative method for detecting nonlinear relationships.

Records for observing dynamics are usually complied by a form of time series. However, time series can be a challenging type of dataset for deep neural networks to learn. In deep neural networks, pairs of inputs and outputs are usually fed for constructive mapping. Such inputs are typically prepared as static images in successful applications. And so, here we propose two methods to prepare such inputs for learning the dynamical properties behind time series. In the first method, we simply array a time series in the shape of a rectangle as an image. In the second method, we convert a time series into a distance matrix using delay coordinates, or an unthresholded recurrence plot. We demonstrate that the second method performs well in inferring a slow driving force from observations of a forced system within which there are symmetry and almost invariant subsets.

For one-dimensional wave equations with the van der Pol boundary conditions, there have been several different ways in the literature to characterize the complexity of their solutions. However, if the right-end van der Pol boundary condition contains a source term, then a considerable technical difficulty arises as to how to describe the complexity of the system. In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li–Yorke chaos. For this end, we consider a system $(I,f_{\infty })$ induced by a sequence of continuous maps and its functional envelope $(L^1(I,I),H_{\infty })$, and show that, under some considerable condition, $H_{\infty }$ is transitive if and only if $f_{\infty }$ is weakly mixing of order $3$; $H_{\infty }$ is Li–Yorke chaotic and sensitive if $f_{\infty }$ is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.

The mechanisms of species coexistence make ecologists fascinated, although theoretical work shows that omnivory can promote coexistence of species and food web stability, it is still a lack of the general mechanisms for species coexistence in the real food webs, and is unknown how omnivory affects the interactions between competitor and predator. In this work, we first establish an omnivorous food web model with a competitor based on two natural ecosystems (the plankton community and fig–fig wasp system). We analyze the changes of both food web structure and stability under the different resource levels and predation preference of the generalist/top predator. The results of model analyses show that weak predation strength can promote stable coexistence of predators and prey. Moreover, the evolutionary trend of food web structure changes with the relative predation strength is more diverse than the relative competition strength, and an integration of both omnivory, increased competition, top-down control and bottom-up control can promote species diversity and food web stability. Our theoretical predictions are consistent with empirical data in the plankton community: the lower concentration of nutrient results in a more stable population dynamics. Our theoretical work could enrich the general omnivorous theory on species coexistence and system stability in the real food webs.

This paper is concerned with chaotification of first-order partial difference equations. Two criteria of chaos for the difference equations with general controllers are established, and all the controlled systems are proved to be chaotic in the sense of Li–Yorke or of both Li–Yorke and Devaney by applying the coupled-expanding theory of general discrete dynamical systems. The controllers used in this paper can be easily constructed, facilitating the chaotification of first-order partial difference equations. For illustration, two illustrative examples are provided.

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles $S^{(2)}$ or $S^{(3)}$ under the perturbations of piecewise smooth polynomials with degree $n$. Roughly speaking, for $n\in \mathbb{N}$, a polycycle $S^{(n)}$ is cyclically ordered collection of $n$ saddles together with orbits connecting them in specified order. The discontinuity is on the line $y=0$. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary $S^{(2)}$ and $S^{(3)}$ are respectively $25n+149$ and $25n+115$ (taking into account the multiplicity).

This paper aims to study the dynamic behavior of a logistic model with feedback control and Allee effect. We prove the origin of the system is always an attractor. Further, if the feedback control variable and Allee effect are big enough, the species goes extinct. According to the analysis of the Jacobian matrix of the corresponding linearized system, we obtain the threshold condition for the local asymptotic stability of the positive equilibrium point. Also, we study the occurrence of saddle-node bifurcation, supercritical and subcritical Hopf bifurcations with the change of parameter. By calculating a universal unfolding near the cusp and choosing two parameters of the system, we can draw a conclusion that the system undergoes Bogdanov–Takens bifurcation of codimension-2. Numerical simulations are carried out to confirm the feasibility of the theoretical results. Our research can be regarded as a supplement to the existing literature on the dynamics of feedback control system, since there are few results on the bifurcation in the system so far.

This paper deals with planar piecewise linear slow–fast Liénard differential systems with three zones separated by two vertical lines. We show the existence and uniqueness of canard limit cycles for systems with a unique singular point located in the middle zone.

Herein, an asymmetric image encryption algorithm based on RSA cryptosystem and a fractional-order chaotic system is proposed. Its security depends on RSA algorithm. First, a pair of public and private keys is generated by RSA algorithm. Subsequently, a random message shown as plaintext key information is encrypted by the public key and RSA to achieve ciphertext key information. Next, a new transformation map is established to generate the initial key according to the ciphertext key information. Subsequently, the initial key is substituted into a fractional hyperchaotic system equation to calculate the keystream. Finally, permutation and diffusion operations are employed to encrypt a plain image to obtain the final cipher image. In the proposed algorithm, different keys for encryption and decryption are designed under an asymmetric architecture. The RSA algorithm and fractional chaotic system are combined to encrypt images; in particular, a fast algorithm for computing power multiplication is employed, which significantly improves the encryption effect and enhances the security. Simulation results show that the proposed algorithm is effective and applicable to image protection.

In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite-equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.

In this paper, the influence of PM${}_{2.5}$ on children’s respiratory diseases is taken as the main research focus. Based on the real monitoring data of children’s respiratory diseases in Anhui province, the traditional model is modified substantially, leading to the establishment of two mathematical models. First of all, considering that the PM${}_{2.5}$ changes over time, a nonautonomous air pollution-related disease model is constructed to study its permanence and extinction. Furthermore, regarding lag days of PM${}_{2.5}$ exposure, an air pollution-related disease model with the lag effect is installed and its local and global stabilities and Hopf bifurcation are investigated. Meanwhile, the above two models are numerically simulated, respectively. Our study demonstrates that the threshold conditions of permanence and extinction are obtained by the nonautonomous air pollution-related disease model, and the optimal parameters are obtained through the annual revision of the data by integrating the mathematical model, such that the number of children with respiratory diseases in the future can be checked and predicted. Also our study finds that the lag days of PM${}_{2.5}$ exposure have little effect on children with respiratory diseases in the air pollution-related disease model with a lag effect, but the PM${}_{2.5}$ has a tremendous influence on the number of patients. Once the lag days are combined with the effect of the PM${}_{2.5}$, it can have a significant impact on the patients’ number, e.g. an emergence of periodic oscillations, with an approximate period of 11 days in Anhui Province, due to the Hopf bifurcation.

It has been known for almost 40 years that general planar quadratic polynomial systems can have four limit cycles. Recently, four limit cycles were also found in near-integrable quadratic polynomial systems. To help more people to understand limit cycles theory, the visualization of such four numerically simulated limit cycles in quadratic systems has attracted researchers’ attention. However, for near-integral systems, such visualization becomes much more difficult due to limitation on choosing parameter values. In this paper, we start from the simulation of the well-known quadratic systems constructed around the end of 1979, then reconsider the simulation of a recently published quadratic system which exhibits four big size limit cycles, and finally provide a concrete near-integral quadratic polynomial system to show four normal size limit cycles.

In this paper, we present a study on a mutual interference host-parasitoid model with Beverton–Holt growth. It is well known that, mutual interference of parasites has a stabilizing influence on the dynamics of the host-parasitoid model since the variance in searching efficiency, with parasite density, significantly depends on parasites’ mutual interference. Thus, we have incorporated a mutual interference functional response into a host-parasitoid model to characterize such a phenomenon. The qualitative behaviors of the present model is investigated in this paper. Firstly, the existence and local stability of the model fixed points are discussed. Then, using perturbation method and normal form theory, we derived the emergence conditions of Neimark–Sacker bifurcation of the model. Furthermore, chaotic behavior of the model in the sense of Marotto is proved. In order to control chaotic behavior of the present model, we apply OGY feedback control strategy. Finally, numerical simulations are provided to support our theoretical discussion.

In this paper, a new class of systems with nonclassical jump resonance behavior is presented. Although jump resonance has been widely studied in the literature, this contribution refers to systems presenting a multiple hysteresis jump resonance phenomenon, meaning that the frequency response of the system presents more hysteresis windows nested within the same range of frequency. The analytical conditions for observing this type of behavior are derived and a design strategy to obtain multiple hysteresis jump resonance in circuits and systems presented.