Period-doubling Bifurcations Research Articles

A discrete two time scales model of a size-structured population of parasitized trees.

The work presented a general discrete-time model of a population of trees affected by a parasite. The tree population was considered size-structured, and the parasite was represented by a single scalar variable. Parasite dynamics were assumed to act on a faster timescale than tree dynamics. The model was studied based on an associated nonlinear matrix model, in which the presence of the parasites was only reflected in the value of its parameters. For the model in all its generality, an explicit condition of viability/extinction of the parasite/tree community was found. In a simplified model with two size-classes of trees and particular forms of the vital rates, it was shown that the model undergoes a transcritical bifurcation and, likewise, a period-doubling bifurcation. It was found that, for any tree fertility rate that makes them viable without a parasite, if the parasite sufficiently reduces the survival of young trees, it can lead to the extinction of the entire community. The same cannot be assured if the parasite acts on adult trees. In situations where a high fertility rate coupled with a low survival rate of adult trees causes a non-parasitized population of trees to fluctuate, a parasite sufficiently damaging only young trees can stabilize the population. If, instead, the parasite acts on adult trees, we can find a destabilization condition on the tree population that brings them from a stable to an oscillating regime.

Neimark-Sacker bifurcation, chaos, and local stability of a discrete Hepatitis C virus model

<p>In this paper, we explore the bifurcation, chaos, and local stability of a discrete Hepatitis C virus infection model. More precisely, we studied the local stability at fixed points of a discrete Hepatitis C virus model. We proved that at a partial infection fixed point, the discrete HCV model undergoes Neimark-Sacker bifurcation, but no other local bifurcation exists at this fixed point. Moreover, it was also proved that period-doubling bifurcation does not occur at liver-free, disease-free, and total infection fixed points. Furthermore, we also examined chaos control in the understudied discrete HCV model. Finally, obtained theoretical results were confirmed numerically.</p>

Controllable multistability of fractional-order memristive Henon map and its application in video encryption

In recent years, the use of discrete memristors to enhance chaotic maps has received increasing attention. The introduction of memristors increases the complexity of chaotic maps, making them suitable for engineering applications based on chaotic systems. In this work, a fractional-order discrete memristor exhibiting local activity and controllable asymptotic stability points is constructed by using multiband nonlinear functions. The locally active property of this memristor is demonstrated by using the power-off plot and DC <i> <b>v</b> </i> -<i> <b>i</b> </i> plot. It is then introduced into the Henon map to construct a fractional-order memristive Henon map that can generate any number of coexisting attractors. Simulation results show that the number of fixed points in the system is controlled by the memristor parameters and related to the number of coexisting attractors, thus achieving controllable homogeneous multistability. The complex dynamical behaviors of this map are analyzed by using phase portraits, bifurcation diagrams, maximum Lyapunov exponent (MLE), and attractor basins. Numerical simulations show that the fractional-order map can generate various periodic orbits, chaotic attractors, and period-doubling bifurcations. The system is then implemented on an ARM digital platform. The experimental results are consistent with the simulation results, confirming the accuracy of the theoretical analysis and its physical feasibility. Finally, a parallel video encryption algorithm is designed by using the chaotic sequence iteratively generated by fraction-order memory Henon mapping, which mainly includes frame pixel scrambling and diffusion. Comprehensive security analyses are conducted, proving the robustness and reliability of the proposed encryption scheme. The results show that the encryption algorithm can effectively protect video information. In the future, we will explore other methods of constructing chaotic or hyperchaotic systems with controllable multistability and study their circuit implementation, synchronization control, and chaos-based engineering applications.

Bifurcation and hybrid control of a discrete eco-epidemiological model with Holling type-III.

In this paper, a three dimensional discrete eco-epidemiological model with Holling type-III functional response is proposed. Boundedness of the solutions of the system is analyzed. Existence condition and stability of all fixed points are discussed for the proposed model. Furthermore, we obtained the transcritical bifurcation surfaces of the system by bifurcation theory. Based on the explicit criteria for the Neimark Sacker bifurcation and flip bifurcation, we obtained that the system undergoes these two types of bifurcations at the positive fixed point. Then we apply a hybrid control strategy that based on both parameter perturbation and a state feedback strategy to control the Neimark-Sacker bifurcation. Finally, some numerical simulations are carried out to support the analytical results.

Global Dynamics of a Fisheries Economic Model with Gradient Adjustment

Taking into account the nonlinear demand function, we have developed a multi-agent fishery economic model, where a multitude of agents are bounded by rationality. The fishing decisions of these agents are driven by a profit gradient mechanism. To assess the local stability of the system, stability analysis is performed with the Jury criterion. The investigation has revealed the presence of two conventional paths to chaos, namely, the flip bifurcation and the Neimark–Sacker bifurcation. This was achieved by mapping the stability regions and stability curves of the Nash equilibrium. The multistability of the system is further explored on two-dimensional planes on which the influence of joint parameters on the system’s stability is demonstrated. The existence of Arnold’s tongue has demonstrated unparalleled complexity and intricate interactions across different scales of the system. Both critical curves and basins of attraction are illustrated to gain insight into global bifurcations. The chaotic attractor is found to be confined within specific boundaries. The findings clearly show higher maximum instantaneous demand, relatively slower adjustment speed, and lower price sensitivity. Arguably, a controlled cost would lead to sustainable fishing resources. Moreover, the results also suggest that the agents would benefit more from confined conditions.

Dynamical analysis of a discrete two-patch model with the Allee effect and nonlinear dispersal.

The dynamic behavior of a discrete-time two-patch model with the Allee effect and nonlinear dispersal is studied in this paper. The model consists of two patches connected by the dispersal of individuals. Each patch has its own carrying capacity and intraspecific competition, and the growth rate of one patch exhibits the Allee effect. The existence and stability of the fixed points for the model are explored. Then, utilizing the central manifold theorem and bifurcation theory, fold and flip bifurcations are investigated. Finally, numerical simulations are conducted to explore how the Allee effect and nonlinear dispersal affect the dynamics of the system.

New Class of Discrete-Time Memristor Circuits: First Integrals, Coexisting Attractors and Bifurcations Without Parameters

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua’s circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds.

On bifurcations of chaotic attractors in a pulse width modulated control system

This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.

Stability and bifurcation analysis of a discrete predator-prey system of Ricker type with refuge effect.

The refuge effect is critical in ecosystems for stabilizing predator-prey interactions. The purpose of this research was to investigate the complexities of a discrete-time predator-prey system with a refuge effect. The analysis investigated the presence and stability of fixed points, as well as period-doubling and Neimark-Sacker (NS) bifurcations. The bifurcating and fluctuating behavior of the system was controlled via feedback and hybrid control methods. In addition, numerical simulations were performed as evidence to back up our theoretical findings. According to our findings, maintaining an optimal level of refuge availability was critical for predator and prey population cohabitation and stability.

Stability and bifurcation analysis of a discrete-time plant-herbivore model with harvesting effect

<abstract><p>The dynamics of plant-herbivore interactions are essential for understanding ecosystem stability and resilience. This article investigated the effects of incorporating a harvesting effect on the dynamics of a discrete-time plant-herbivore system. An analysis was performed to determine the existence and stability of fixed points. In addition, studies have shown that the system experienced transcritical, period-doubling, and Neimark-Sacker bifurcations. Moreover, we provided numerical simulations to substantiate our theoretical results. Our research indicated that harvesting in excessive amounts may have negative effects on the populations of both plants and herbivores. However, when harvesting was done at moderate levels, it promoted the coexistence and stability of both populations. The findings of our analysis provided a deep understanding of the intricate dynamics of ecological systems and underscored the need to use sustainable harvesting methods for the management and preservation of ecosystems.</p></abstract>

Complex firing activities and bifurcations in memristor-coupled Hindmarsh–Rose neuron

Due to the unique synaptic plasticity and memory effect, a memristor can not only mimic biological synapses but also characterize the influence of external electromagnetic radiation. In this paper, a ReLU-type non-ideal memristor with a simple structure is first coupled to a classical three-dimensional Hindmarsh–Rose neuron to describe the electromagnetic induction effect, which can show period-doubling, period-adding, and saddle-node bifurcations by varying the coupling strength of the memristor. Furthermore, complex discharge behaviors of the system, including bursting discharge and spiking discharge, are exhibited, and some coexisting discharge modes associated with initial values are also presented. Finally, an analog circuit scheme consuming fewer circuit components is designed, and it was found that the experimental results are consistent with the numerical results.

Stability and bifurcation analysis of a discrete Leslie predator-prey system via piecewise constant argument method

<abstract><p>The objective of this study was to analyze the complex dynamics of a discrete-time predator-prey system by using the piecewise constant argument technique. The existence and stability of fixed points were examined. It was shown that the system experienced period-doubling (PD) and Neimark-Sacker (NS) bifurcations at the positive fixed point by using the center manifold and bifurcation theory. The management of the system's bifurcating and fluctuating behavior may be controlled via the use of feedback and hybrid control approaches. Both methods were effective in controlling bifurcation and chaos. Furthermore, we used numerical simulations to empirically validate our theoretical findings. The chaotic behaviors of the system were recognized through bifurcation diagrams and maximum Lyapunov exponent graphs. The stability of the positive fixed point within the optimal prey growth rate range $ A_1 < a < A_2 $ was highlighted by our observations. When the value of $ a $ falls below a certain threshold $ A_1 $, it becomes challenging to effectively sustain prey populations in the face of predation, thereby affecting the survival of predators. When the growth rate surpasses a specific threshold denoted as $ A_2 $, it initiates a phase of rapid expansion. Predators initially benefit from this phase because it supplies them with sufficient food. Subsequently, resource depletion could occur, potentially resulting in long-term consequences for populations of both the predator and prey. Therefore, a moderate amount of prey's growth rate was beneficial for both predator and prey populations.</p></abstract>

Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

<p>Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.</p>

Local Stability, Period-Doubling and 1:2 Resonance Bifurcation for a Discretized Chemical Reaction System

We investigate a discrete counterpart of planar dynamical system of nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction. Chemical reactions exhibit a wide range of dynamical behavior. We show how the theoretical analysis provides insight into the potential behavior of chemical reaction systems, determining the areas of parametric space which indicate scenarios for local stability, then for one type of bifurcation co-dimension one and one type of bifurcation co-dimension two. Precisely, we prove the existence of period-doubling bifurcation and 1:2 resonance bifurcation also, by using the center manifold theorem and the technique of normal forms. All mathematical investigations are illustrated with numerical examples, bifurcation diagrams, Lyapunov exponents and phase portraits.

Bifurcation analysis and chaos control of a discrete fractional-order Leslie-Gower model with fear factor

<p>This study focused on the dynamical behavior analysis of a discrete fractional Leslie-Gower model incorporating antipredator behavior and a Holling type Ⅱ functional response. Initially, we analyzed the existence and stability of the model's positive equilibrium points. For the interior positive equilibrium points, we investigated the parameter conditions leading to period-doubling bifurcation and Neimark-Sacker bifurcation using the center manifold theorem and bifurcation theory. To effectively control the chaos resulting from these bifurcations, we proposed two chaos control strategies. Numerical simulations were conducted to validate the theoretical results. These findings may contribute to the improved management and preservation of ecological systems.</p>

Stability analysis and chaos control in a discrete predator-prey system with Allee effect, fear effect, and refuge

<abstract><p>This paper investigates the complex dynamical behavior of a discrete prey-predator system with a fear factor, a strong Allee effect, and prey refuge. The existence and stability of fixed points in the system are discussed. By applying the central manifold theorem and bifurcation theory, we have established the occurrence of various types of bifurcations, including flip bifurcation and Neimark-Sacker bifurcation. Furthermore, to address the observed chaotic behavior in the system, three controllers were designed by employing state feedback control, OGY feedback control, and hybrid control methods. These controllers serve to control chaos in the proposed system and identify specific conditions under which chaos or bifurcations can be stabilized. Finally, the theoretical analyses have been validated through numerical simulations.</p></abstract>

Bifurcation Analysis of the Dynamics in COVID‐19 Transmission through Living and Nonliving Media

Transmission of COVID‐19 occurs either through living media, such as interaction with a sufferer, or nonliving objects contaminated with the virus. Recovering sufferers and disinfectant spraying prevent interaction between people and virus become the treatment to overcome it. In this research, we formulate a new mathematical model as a three‐dimensional ordinary differential equation system representing an interaction between viruses attached in nonliving media, susceptible, and infected subpopulations, including the treatment to investigate its effect. Disease‐free, sterile‐media endemic, and two nonsterile media endemic equilibriums exist in the model. The nonexistence of sterile‐media equilibria interpreting the nonendemic condition is achieved by crossing the branch point bifurcation of the equilibria point as the infected subpopulation recovery rate increases. Continuation of the limit cycle generated at a Hopf bifurcation point as susceptible‐coronavirus interaction prevention rate and period increase trigger two saddle‐node bifurcations and a branch point bifurcation of cycle. Stable symmetric cycles with decreasing amplitude that make the dynamic of subpopulation easier to control start to be gained at the branch point bifurcation of cycle between the two saddle‐node bifurcation points as the prevention rate increases. Some chaotic attractors which describe a complex and unpredictable pattern of the dynamic in the population are also found at inclination flip bifurcation by a continuation of a homoclinic orbit generated near the Bogdanov‐Takens bifurcation point as the prevention rate increases while the recovery rate decreases. Increasing the recovery and prevention rate along with avoiding an increase of the prevention rate while the recovery rate decreases becomes the treatment to optimize the effort in overcoming COVID‐19 transmission.

Energy function and complex dynamics from a jerk system

Jerk, as a mathematical concept, is used in mechanics to describe the rate of change of acceleration and plays a crucial role in the design of control systems for machines and vehicles. Therefore, it is important to master the various states and the energy released during the change of acceleration. This is why a new simple jerk function introduced afterward, energy released, is derived from a Hamilton function using the Helmholtz theorem. The condition of having a stable or unstable rate of change of acceleration is established using Hopf bifurcation theory. Some two-parameter stability charts are then computed for a suitable selection region of the study. Using some nonlinear analysis metrics, in the unstable region of the study, the occurrence of phenomena is found, such as reverse period doubling bifurcation, antimonotonicity, and hysteresis involving the coexistence of the states in the considered jerk system. An electronic circuit is built and used to implement the mathematical expression of the jerk equation and validate the result of the theoretical investigation.

Repertoire of dynamical states in dissimilarly coupled Van der Pol oscillators

We have considered dissimilarly coupled Van der Pol oscillators with an offset parameter which determines the degree of heterogeneity of the dissimilar coupling strength. Increasing degree of heterogeneity for decreasing values of the offset parameter results in a rich repertoire of bifurcation transitions and dynamical states including epochs of period doubling bifurcation. Two distinct multi-stable states are also observed along with several symmetry breaking dynamical states. We have deduced analytical stability conditions for Hopf and pitch-fork bifurcations through a linear stability analysis of symmetry preserving states, namely, trivial steady state and oscillation death state. The analytical conditions are found to match exactly with the simulation results in the two-parameter phase diagram. In addition to torus bifurcation, crisis and crisis induced intermittency routes to chaos are also observed for an appropriate heterogeneity of the dissimilar coupling strength. The period doubling bifurcation is characterized using the largest Lyapunov exponents of the dissimilarly coupled Van der Pol oscillators.

An Energy Transfer-Based Bifurcation Detection Method for Nonlinear Rotating Systems: Enables Accurate Capture of Period-Doubling Bifurcation and Instability

Rotor systems are widely used in industrial power generation and propulsion. Once the nonlinear contact stiffness and oil film force are taken into account, the dynamics and stability of rotor systems become quite complex, often accompanied by super-harmonic and chaotic motions. Furthermore, conventional methods face limitations in real-time detection of the bifurcations and complex nonlinear motions. This research investigates the bifurcations and stability induced by nonlinear factors in a rotor-bearing system from an energy perspective. Dynamic equations of a rotor-bearing system considering cubic term stiffness are established, the steady responses are obtained by the fourth-order Runge–Kutta method. The relationship between the bifurcations and energy transfers is analyzed numerically, the proposed stability criterion is validated by comparing the Lyapunov exponents. The bistable phenomenon of period-3[Formula: see text] ([Formula: see text]) motion is discussed in terms of numerical results and experiments which corresponds to the asymmetric jumps of the generalized energy. It is found that bifurcations and unstable motions of the nonlinear system can be captured accurately by detecting the energy transfers, the proposed generalized energy curve exhibits more detailed information than the conventional speed-up curve. These findings provide a new perspective on studying bifurcations and stability of rotating systems, which can be further applied in the condition monitoring, stability prediction as well as the design of nonlinear energy sink, representing significant progress in converting theory into engineering applications for rotor systems.