International Journal of Bifurcation and Chaos

This article is an informal auto-biographical memoir by Mike Thompson, reflecting in retirement on his scientific researches in nonlinear phenomena, wandering pictorially from shell buckling, through bifurcations and chaos to climate tipping points. Some ideas and advice to young researchers are offered whenever it seems appropriate. Two research groups at University College London, and their two IUTAM Symposia are given some prominence, as are the ten years editing the Philosophical Transactions of the Royal Society.

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of known results on bifurcations of these contours. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software matcont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.

This article reviews recent progress in signal frequency up-conversion and down-conversion, both theory and experiments with network implementations. The fundamental idea is to exploit the inherent symmetry of networks to produce collective behavior in which certain oscillators tend to oscillate at different frequencies. This concept is significantly different from other techniques, e.g. master-slave systems, in the sense that the collective behavior arises naturally from the mutual interactions of the individual units, and without any external forcing. In this manuscript, we present a comprehensive review of the basic ideas, methods, and experiments of the symmetry-based phenomenon of frequency conversion. In addition, we present a review of a device implementation of a broad spectrum analyzer, which motivated the development of systematic methods to up- and down-convert frequencies of oscillations. This device is made up of large parallel arrays of analog nonlinear oscillators with the ability to receive complex signals containing multiple frequencies and instantaneously lock-on or respond to a received signal in a few oscillation cycles.

Drug combination has become an attractive strategy against complex diseases, despite the challenges in handling a large number of possible combinations among candidate drugs. How to detect effective drug combinations and determine the dosage of each drug in the combination is still a challenging task. When regarding a drug as a perturbation, we propose a bifurcation-based approach to detect synergistic combinatorial perturbations. In the approach, parameters of a dynamical system are divided into two groups according to their responses to perturbations. By combining two parameters chosen from two groups, three types of combinations can be obtained. Synergism for different perturbation combinations can be detected by relative positions of the bifurcation curve and the isobole. The bifurcation-based approach can be used not only to detect combinatorial perturbations but also to determine their perturbation quantities. To demonstrate the effectiveness of the approach, we apply it to the epithelial-to-mesenchymal transition (EMT) network. The approach has implications for the rational design of drug combinations and other combinatorial control, e.g. combinatorial regulation of gene expression.

In this paper, we obtain an upper bound for the number of small-amplitude limit cycles produced by Hopf bifurcation in one particular type of rational Liénard systems in the form of $ẋ=y-q_n(x)/p_m(x)$, $ẏ=-g(x)$, where $q_n(x)$ and $p_m(x)$ are polynomials in $x$ with degrees $n$ and $m$, respectively. Furthermore, we show that the upper bound presented here is sharp in the case of $m=n$.

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations.

In this paper, we investigate the dynamics of an electromechanical system consisting of a DC motor-driving arm within a circular periodic potential created by three permanent magnets. Two configurations of the circular potential appear when one varies the positions of the magnets and the length of the DC motor, respectively. Two different forms of input signal are used: DC and AC voltage sources. For each case, conditions under which the mechanical arm can perform a complete rotation are obtained. Under the DC voltage excitation, the arm oscillates and then is stabilized at an equilibrium position for a DC voltage lower than a critical value $E_C$. When the DC voltage is higher than the critical value $E_C$, the arm performs large amplitude motions (complete rotation). Submitted to an AC voltage with amplitude lower than a critical value, the mechanical arm exhibits sinusoidal oscillations around the equilibrium position $0$ with amplitudes less than one turn. Angular oscillations with amplitudes greater than one turn are observed when the voltage amplitude is higher than the critical value. Bifurcation diagrams show that the simple system can enter chaotic regime with the amplitudes of angular oscillations varying erratically from small to high values.

Electrical activity occurs in the cell membrane of cardiomyocytes. This electrical activity forms the action potential that generates pumping of the heart. An abnormality in the action potential turns into arrhythmia, which may cause sudden death. Studies of arrhythmias using mathematical models are important to reduce the risk of sudden death. In this study, we investigate bifurcations related to the generation of early afterdepolarizations (EADs) in a mathematical model. We clarify the transition process from a normal state to a persistent EAD through a transient EAD while changing only one parameter (multiple of conductance of L-type calcium channel current) value. The dependence of the transient EAD generation on parameters is shown through bifurcation analysis in a [Na]i-parameterized system.

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder $O(2)\times \mathrm{\Gamma }\times {\mathbb{Z}}_2$-equivariant degree theory. Here, $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system, $\mathrm{\Gamma }$ reflects extra spacial symmetries of the system, and ${\mathbb{Z}}_2$ is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with $\mathrm{\Gamma }=D_6$ — the dihedral group of order 12.

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

We report on a stripe-core mixed spiral chimera in a system of nonlocally coupled phase oscillators, located on the spherical surface, where the spiral wave consisting of phase-locked oscillators is separated by a stripe-type region of incoherent oscillators into two parts. We analyze the existence and stability of the stripe-core mixed spiral chimera state rigorously, on the basis of the Ott–Antonsen reduction theory. The stability diagram for the stationary states including the spiral chimeras as well as incoherent state is presented. Our stability analysis reveals that the stripe-core mixed spiral chimera state emerges as a unique attractor and loses its stability via the Hopf bifurcation. We verify our theoretical results using direct numerical simulations of the model system.

In this paper, we investigate an epidemic model with quarantine and recovery-age effects. Reformulating the model as an abstract nondensely defined Cauchy problem, we discuss the existence and uniqueness of solutions to the model and study the stability of the steady state based on the basic reproduction number. After analyzing the distribution of roots to a fourth degree exponential polynomial characteristic equation, we also derive the conditions of Hopf bifurcation. Numerical simulations are performed to illustrate the results.

In this paper, we study the dynamics of phytoplankton–zooplankton system with delay, where delay means that releasing toxin for phytoplankton is not instantaneous. First, we prove the positivity and boundedness of solutions, discuss the Hopf bifurcation caused by delay. Furthermore, we study the property of Hopf bifurcation by center manifold and normal form. Then, we study the global existence of bifurcated periodic solution. Finally by simulation, we show the influence of delay, disease spread and recovery from infected to susceptible on the dynamics of phytoplankton–zooplankton system.

Layered structures are an object of interest for theoretical and experimental reasons. In this work, we study coupled map lattice on a ladder. The ladder consists of two one-dimensional chains coupled at every point. We study linearly and nonlinearly coupled logistic maps in this system and study transition to nonzero persistence, in particular. We coarse-grain the variable value by assigning spin $+1$ ($-1$) to sites that have value greater (less) than the fixed point and compute the number of sites that have not changed their spin values at all even times till the given time $t$. The fraction of such sites at a given time $t$ is known as persistence. In our system, we observe a power-law of persistence at the critical value of coupling. This transition is also accompanied by long-range antiferromagnetic ordering for nonlinear coupling and long-range ferromagnetic ordering for linear coupling. The number of domain walls decay as $1/\sqrt{t}$ at the critical point in both cases. The persistence exponent is 0.375 for a nonlinear case with two layers which is an exponent for the voter model on the ladder as well as for the Ising model at zero temperature or voter model in 1D. For linear coupling, we obtain a smaller persistence exponent.

This paper studies the dynamics and bifurcations of a vibration-assisted, regenerative, nonlinear turning-tool system using an implicit mapping method. Machine vibration has been studied for a century for the improvement of machine accuracy and metal removal rate. In fact, this problem is unsolved yet. This is because such dynamical systems are involved in nonlinearity, discontinuity and time-delay. Thus, a comprehensive understanding of nonlinear machining dynamics with time-delay is indispensable. In this paper, period-$m$ motions in the turning machine-tool system are studied through specific mapping structures, and the corresponding stability and bifurcations of the period-$m$ motion are determined through the eigenvalue analysis. The analytical bifurcation scenarios for two sets of sequential period-$m$ motions in a turning-tool system are presented. Numerical simulations of period-$m$ motions are carried out to verify the prediction of periodic motions. The complex dynamics of vibration-assisted machining with strong nonlinearity are presented, which can provide a good overview for nonlinear dynamics of machine-tool systems.

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in []. Similar to that method, for an $n$ degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a $2n-2$ dimensional geometrical object in the energy surface of a $2n-1$ dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.

Memristive synaptic weight is a changeable connection synaptic weight. It reflects the self-adaption physical processing in biological neurons. To study its dynamical effect, this paper presents a memristive synaptic weight-based tabu learning neuron model. It is constructed by replacing the resistive self-connection synaptic weight in the tabu learning neuron with a memristive self-connection synaptic weight. The equilibrium point of the memristive tabu learning model is time-varying and switches between no equilibrium state and line equilibrium state with the change of the external current. Particularly, the stability of the line equilibrium state closely relies on the initial state of the memristor, resulting in the emergence of coexisting infinitely many nonchaotic attractors. By employing the bifurcation plots, Lyapunov exponents, and phase plots, this paper numerically reveals the initial state-switched coexisting bifurcation behaviors and initial state-relied extreme multistability, and thereby discloses the coexisting infinitely many nonchaotic attractors composed of mono-periodic, multiperiodic, and quasi-periodic orbits. In addition, PSIM circuit simulations and printed-circuit board-based experiments are executed and the coexisting infinitely many nonchaotic attractors are realized physically. The results well verify the numerical simulations.

Detection of chaos in experimental data is a crucial issue in nonlinear science. Historically, one of the first evidences of a chaotic behavior in experimental recordings came from laser physics. In a recent work, a Minimal Universal Model of chaos was developed by revisiting the model of laser with feedback, and a first electronic implementation was discussed. Here, we propose an upgraded electronic implementation of the Minimal Universal Model, which allows for a precise and reproducible analysis of the model’s parameters space. As a marker of a possible chaotic behavior the variability of the spiking activity that characterizes one of the system’s coordinates was used. Relying on a numerical characterization of the relationship between spiking activity and maximum Lyapunov exponent at different parameter combinations, several potentially chaotic settings were selected. The analysis via divergence exponent method of experimental time series acquired by using those settings confirmed a robust chaotic behavior and provided values of the maximum Lyapunov exponent that are in very good agreement with the theoretical predictions. The results of this work further uphold the reliability of the Minimal Universal Model. In addition, the upgraded electronic implementation provides an easily controllable setup that allows for further developments aiming at coupling multiple chaotic systems and investigating synchronization processes.