Period-doubling Bifurcations Research Articles

Discontinuity-Induced Dynamics in the Conductance-Based Adaptive Exponential Integrate-and-Fire Model

In this article, we present a computational study of the Conductance-Based Adaptive Exponential (CAdEx) integrate-and-fire neuronal model, focusing on its multiple timescale nature, and on how it shapes its main dynamical regimes. In particular, we show that the spiking and so-called delayed bursting regimes of the model are triggered by discontinuity-induced bifurcations that are directly related to the multiple-timescale aspect of the model, and are mediated by canard solutions. By means of a numerical bifurcation analysis of the model, using the software package coco, we can precisely describe the mechanisms behind these dynamical scenarios. Spike-increment transitions are revealed. These transitions are accompanied by a fold and a period-doubling bifurcation, and are organised in parameter space along an isola periodic solutions with resets. Finally, we also unveil the presence of a homoclinic bifurcation terminating a canard explosion which, together with the presence of resets, organises the delayed bursting regime of the model.

Complex dynamics in an atmospheric pressure glow discharge in helium: transition from stationary to periodic oscillatory, and fully chaotic states

Abstract This work deals with the numerical study of spontaneous temporal oscillations in an atmospheric pressure glow discharge (APGD) in helium. The transition of helium APGD from stationary to periodic oscillatory state through the Hopf bifurcation, and further from periodic to chaotic oscillations through period-doubling bifurcations is explored. The choice of the discharge and external electric circuits parameters is guided by the relevant experiments. The ballast resistance and supply voltage of the external circuit play the role of control parameters. The method is based on the stability analysis of stationary states of the discharge. 

The stability diagram predicting parameter regimes at which stable and oscillatory states of the APGD can be expected is obtained. The effects of the discharge parameters (such as the gas gap, secondary electron emission coefficients, and capacitance in the external electric circuit) on the bifurcation curves are identified. 

The Lorenz map and corresponding period-doubling bifurcation diagram characterizing transition to chaotic oscillations in helium APGD with an increase in the control parameter are derived. The value of the capacitance in the external circuit plays a critical role in the dynamical behavior of the discharge. Decreasing its value contributes to the dissipation/damping of the system, whereas increasing it enhances the irregular behavior of the system.

Period-doubling bifurcation and polarization dynamics of vector solitons in a fiber laser

Period-doubling bifurcation and polarization dynamics of vector solitons in a fiber laser

Mixed-mode oscillations and chaos in a complex chemical reaction network involving heterogeneous catalysis.

The nonlinear dynamical behavior in a complex isothermal reaction network involving heterogeneous catalysis is studied. The method first determines the multiple steady states in the reaction network. This is followed by an analysis of bifurcation continuations to identify several kinds of bifurcations, including limit point, Bogdanov-Takens, generalized Hopf, period doubling, and generalized period doubling. Numerical simulations are performed around the period doubling and generalized period doubling bifurcations. Rich nonlinear behaviors are observed, including simple sustained oscillations, mixed-mode oscillations, non-mixed-mode chaotic oscillations, and mixed-mode chaotic oscillations. Concentration-time plots, 2D phase portraits, Poincaré maps, maximum Lyapunov exponents, frequency spectra, and cascade of bifurcations are reported. Period-doubling and period-adding routes leading to chaos are observed. Maximum Lyapunov exponents are positive for all the chaotic cases, but they are also positive for some non-chaotic orbits. This result diminishes the reliability of using maximum Lyapunov exponents as a tool for determining chaos in the network under study.

A novel discrete memristive hyperchaotic map with multi-layer differentiation, multi-amplitude modulation, and multi-offset boosting.

In recent years, the introduction of memristors in discrete chaotic map has attracted much attention due to its enhancement of the complexity and controllability of chaotic maps, especially in the fields of secure communication and random number generation, which have shown promising applications. In this work, a three-dimensional discrete memristive hyperchaotic map (3D-DMCHM) based on cosine memristor is constructed. First, we analyze the fixed points of the map and their stability, showing that the map can either have a linear fixed point or none at all, and the stability depends on the parameters and initial state of the map. Then, phase diagrams, bifurcation diagrams, Lyapunov exponents, timing diagrams, and attractor basins are used to analyze the complex dynamical behaviors of the 3D-DMCHM, revealing that the 3D-DMCHM enters into a chaotic state through a period-doubling bifurcation path, and some special dynamical phenomena such as multi-layer differentiation, multi-amplitude control, and offset boosting behaviors are also observed. In particular, with the change of memristor initial conditions, there exists an offset that only homogeneous hidden chaotic attractors or a mixed state offset with coexistence of point attractors and chaotic attractors. Finally, we confirmed the high complexity of 3D-DMCHM through complexity tests and successfully implemented it using a digital signal processing circuit, demonstrating its hardware feasibility.

Co-evolutionary dynamics for two adaptively coupled Theta neurons.

Natural and technological networks exhibit dynamics that can lead to complex cooperative behaviors, such as synchronization in coupled oscillators and rhythmic activity in neuronal networks. Understanding these collective dynamics is crucial for deciphering a range of phenomena from brain activity to power grid stability. Recent interest in co-evolutionary networks has highlighted the intricate interplay between dynamics on and of the network with mixed time scales. Here, we explore the collective behavior of excitable oscillators in a simple network of two Theta neurons with adaptive coupling without self-interaction. Through a combination of bifurcation analysis and numerical simulations, we seek to understand how the level of adaptivity in the coupling strength, a, influences the dynamics. We first investigate the dynamics possible in the non-adaptive limit; our bifurcation analysis reveals stability regions of quiescence and spiking behaviors, where the spiking frequencies mode-lock in a variety of configurations. Second, as we increase the adaptivity a, we observe a widening of the associated Arnol'd tongues, which may overlap and give room for multi-stable configurations. For larger adaptivity, the mode-locked regions may further undergo a period-doubling cascade into chaos. Our findings contribute to the mathematical theory of adaptive networks and offer insights into the potential mechanisms underlying neuronal communication and synchronization.

Bifurcations on a discrete–time SIS–epidemic model with saturated infection rate

In this paper, we explore the complex dynamics of a discrete-time SIS (Susceptible-Infected-Susceptible)-epidemic model. The population is assumed to be divided into two compartments: susceptible and infected populations where the birth rate is constant, the infection rate is saturated, and each recovered population has a chance to become infected again. Two types of mathematical results are provided namely the analytical results which consist of the existence of fixed points and their dynamical behaviors, and the numerical results, which consist of the global sensitivity analysis, bifurcation diagrams, and the phase portraits. Two fixed points are obtained namely the disease-free and the endemic fixed points and their stability properties. Some numerical simulations are provided to present the global sensitivity analysis and the existence of some bifurcations. The occurrence of forward and period-doubling bifurcations has confirmed the complexity of the solutions.

Exploring complex dynamics in a Stackelberg Cournot duopoly game model

Abstract In this study, we investigate the dynamics of a Cournot duopoly game model with bounded rational players, incorporating a leader-follower mechanism where the first player acts as a leader and the second as a follower, aware of the leader’s production. We examine the existence and stability of all fixed points in the model and use center manifold and bifurcation theory to analyze the occurrence and direction of period-doubling and Neimark-Sacker bifurcations at the positive fixed point. To control bifurcation and chaos, feedback control and hybrid control methods are applied. Numerical examples are provided to confirm our theoretical results and reveal the model’s complex dynamics. Our results highlight the critical role of the leader’s strategic decisions, particularly the adjustment speed parameter v 1, in driving the system from stability to chaos, affecting both firms and leading to significant shifts in market dynamics.

Chaotic behavior and multistability of a tree trunk structure driven by self-and parametric hydrodynamic forces

Abstract This study investigates the chaotic behavior of a tree trunk under dynamic wind loads, focusing on control strategies and multistability. We consider time-varying wind speeds and analyze a specific case where hydrodynamic drag forces align with the flow velocity. The stability of the model’s equilibrium points is analyzed theoretically and numerically. Melnikov’s method is employed to identify conditions for homoclinic bifurcation. Numerical simulations employing basin of attraction confirm the analytical predictions. Our findings show a decrease in the threshold for chaos with increasing amplitudes of external excitation, damping coefficient, and parametric damping. The global dynamics are explored numerically using a fourth-order Runge-Kutta method. When solely subjected to external excitation, the system exhibits period doubling bifurcations, multiperiodic oscillations, mixed-mode oscillations, and chaos. Conversely, with self- and parametric drag forces, the system displays reverse periodic bifurcations, periodic bubbling oscillations, antimonotonicity, transient chaos, and chaos. Poincaré maps analyze the geometric structure of chaotic attractors, revealing a strong influence of dimensionless drag parameters. These parameters can be manipulated to control or eliminate chaos. Furthermore, the system exhibits multistability, with coexisting attractors. Beyond its application in protecting infrastructure from wind damage, this research can contribute to ecological balance by improving our understanding of tree wind resistance.

Structure-preserving analysis on chaotic characteristics of transverse vibration for embedded double-walled carbon nanotube

Abstract Analysing the ultra-high frequency vibrational characteristics of carbon nanotubes, especially on the chaotic characteristics, is a key scientific problem in the dynamic design of the carbon nanotube devices. Considering the van der Waals force between the inner layer and the outer layer of the embedded double-walled carbon nanotube, and the effects of the elastic medium as well as the effects of the simple harmonic external excitation, the coupling dybamic model describing the transverse vibration of the embedded double-walled carbon nanotube is presented. The generalized multi-symplectic formulations with an explicit multi-symplectic structure residual are deduced by introducing the dual momenta. The Preissmann approach, which has been proved to be a structure-preserving method that can be used to reproduce the chaotic characteristics of carbon nanotubes, is employed to discrete the generalized multi-symplectic formulations. The numerical results imply that, the transverse vibration of the embedded double-walled carbon nanotube subjected to the external excitation larger than the critical external excitation will enter the chaotic state through a period-doubling bifurcation path. In addition, the critical external excitation for the chaos of the inner layer carbon nanotube’s transverse vibration is larger than that of the outer layer carbon nanotube’s transverse vibration. The above findings reported in this paper provide some guidance for the dynamic design of the carbon nanotube devices directly.

Dynamic complexity of a discretized predator-prey system with Allee effect and herd behaviour

This paper investigates the dynamics of a discrete-time predator-prey system in which the prey population is impacted by the Allee effect. Possible fixed points in the system are studied for their existence and topological categorization. Moreover, the presence and direction of period-doubling and Neimark-Sacker bifurcations at the interior fixed point are examined through the application of bifurcation theory and the centre manifold theorem. A hybrid approach is adopted to control chaotic behaviour and prevent bifurcations. Numerical examples are provided to substantiate our theoretical findings. The Allee effect has been shown to affect the dynamics of the system using numerical simulations. The moderate Allee effect stabilizes predator and prey populations, facilitating ecological cohabitation and persistence.

Dynamics of a Discrete Population Model

In order to explore the influences of the strong Allee effect on population evolution in two-patch environments, this paper constructs a population model with discrete diffusion and analyzes its dynamics. Using the center manifold theorem, we discuss the codimension-one bifurcation, such as flip bifurcation, of the model. The qualitative structures and stability of the model at the degenerate fixed point are obtained from the approximation of flow. Based on the theoretical analysis results, we investigate the influences of the strong Allee effect and diffusion on population evolution. The qualitative structures of degenerate fixed point and numerical simulation show that the strong Allee effect will make the population tend to a stable size when the two initial population patch sizes are equal and exceed a threshold. But the stable size of population does not exceed the carrying capacity, depending on the constant Allee effect and carrying capacity, which differ from the conclusion of logistic model that population size tends to the carrying capacity. However, if the two initial population patches are equal in size and below this threshold, or if the sizes of the initial population patches are unequal, the population size will decrease. The fixed point becomes degenerate with respect to the strong Allee effect causing the population tend to a stable size below carrying capacity, which exacerbates the complexity of population evolution.

Nonlinear dynamics of a Darwinian Ricker system with strong Allee effect and immigration

In this paper, we investigate the complex dynamics of a Darwinian Ricker system through a comprehensive qualitative and dynamical analysis. Our research shows that the system exhibits Neimark–Sacker bifurcation, period-doubling bifurcation, and codimension-two bifurcations associated with 1:2, 1:3, and 1:4 resonances. These findings are derived using bifurcation and center manifold theories. We numerically illustrate all bifurcation results and chaotic features, providing a thorough understanding of the system’s behavior. This detailed examination of the Darwinian Ricker system, with a focus on the interplay between immigration and the strong Allee effect, enhances our understanding of the intricate mechanisms driving population dynamics. Furthermore, it highlights the significant implications for ecological modeling, particularly in predicting ecosystem responses to external perturbations such as climate change and species invasions.

Dynamical and Phase Analyses on Chaos Control of Discrete Predator–Prey Systems of Mixed Holling Types with Density-Dependent Birth Rates

This paper discusses the effect of density-dependent birth rate in a discrete predator–prey system with mixed functional responses of Holling types I and III. We use the Beverton–Holt function to modify the birth rate parameter of the prey species. The steady states and their stability are established. The criteria for flip bifurcation (FB) and Neimark–Sacker bifurcation (NSB) are proposed analytically using the center manifold theorem and normal bifurcation theory. Using the state feedback control method, it is shown that chaotic orbit can be stabilized at an unstable steady-state. This study deduces that the density of predator species is negatively affected by the density-dependent birth rate of prey species. Moreover, density-dependent birth rate can increase the possibility of extinction in prey species, which means that this birth rate may cause Allee effect. Some numerical simulations are given to verify these analytical results.

Complexity and response of bio-inspired energy harvesters based on wing-beat pattern

This paper aims to investigate the dynamical mechanism of bio-inspired energy harvesters based on wing-beat pattern under harmonic excitation. Due to the existence of the gravity force in the established model, the harmonic balance method is utilized to calculate the theoretical results, which has the advantage to keep the influence of gravity force. Multiple solutions are found in the high frequency region, and they are very close in the amplitude of displacement and voltage due to the special structure of the bio-inspired energy harvester. Direct time-domain analysis verifies the effectiveness of theoretical results. The influence mechanism of the equivalent stiffness is also explored, which leads to the appearance of different states. Then, the root mean square (RMS) voltage and average power are analyzed. It is observed that a smaller damping coefficient and equivalent capacitance enhance the average electrical output and achieve greater output power. Subsequently, the bifurcation and complexity properties of the harvester are discussed. Complex phenomena are observed under different external excitations and equivalent damping, including double periodic bifurcation, multiple periodic bifurcation, and chaos phenomena. The complexity analyses confirm the effectiveness of the bifurcation results. The distribution of complexity also exhibits significant fluctuations, closely correlated with the trend of the bifurcation diagram.

Operational Modal Analysis of Self-excited Vibrations in Milling Considering Periodic Dynamics

Abstract A new method is presented to identify the dynamics of regenerative chatter from measured process vibrations in milling. This method combines the synchronous once-per-revolution sampling of process vibrations with Operational Modal Analysis to estimate the Floquet multipliers of the delayed linear time-periodic dynamics in milling, all from the natural process vibrations without external excitation. The identified multipliers quantify vibration stability, enabling chatter prediction before it occurs. In addition to this, they can be used to calibrate physics-based chatter models based on vibration measurements solely within the stable region. The method's accuracy in identifying Floquet multipliers is validated through extensive numerical simulations and two experimental case studies. The results show that chatter due to both Hopf and period-doubling bifurcations can be predicted from the process vibrations during stable cuts. Moreover, the experimental case studies demonstrate a vibration measurement system for implementing the presented method in standard milling operations and confirm its effectiveness in practice.

Asymmetric Duffing oscillator: the birth and build-up of a period-doubling cascade

Asymmetric Duffing oscillator: the birth and build-up of a period-doubling cascade

The topological characteristics of the bifurcation and chaos in the motion of combustion fronts in solids.

We investigate the geometric features in the bifurcation and chaos of a partial differential equation describing the unsteady combustion of solid propellants. Driven by the interaction of the unsteady combustion at the surface and the diffusion inside solids, the motion of the combustion fronts can be steady, harmonically oscillatory, and become more complicated to chaos through a series of bifurcations. We examined the dynamics in both free and forced oscillations. In the free oscillation, by varying a parameter related to the solid property, the intrinsic instability of the combustion is discovered. We find the typical period-doubling to chaos route and verify it via both qualitative and quantitative universalities. In the forced oscillation case, the system is perturbed by an external pressure excitation, leading to a more complicated bifurcation diagram with richer dynamics. Concentrating on the topological characteristics of the periodic orbits, we discover two new types of bifurcation other than the period-doubling bifurcation. In present work, we extract a series subtle topological structures from an infinite-dimensional dynamical systems governed by a partial differential equation with free boundary. We find the results provide an explanation for the period-3 orbits in the experimental data of a full-scale motor.

Spatiotemporal Dynamics of an Intraguild Predation Model with Intraspecies Competition

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions.

A three-dimensional discrete fractional-order HIV-1 model related to cancer cells, dynamical analysis and chaos control

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using $C_{0}$ complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper.