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This paper investigates bifurcations analysis and resonances in a discrete-time prey-predator model analytically and numerically as well. The local stability conditions of all the fixed points in the system are determined. Here, codim-1 and codim-2 bifurcation including multiple and generic bifurcations in the discrete model are explored. The model undergoes fold bifurcation, flip bifurcation, Neimark–Sacker bifurcation and resonances 1:2, 1:3, 1:4 at different fixed points. Using the critical normal form theorem and bifurcation theory, normal form coefficients are calculated for each bifurcation. The different bifurcation curves of fixed points are drawn which validate the analytical findings. The numerical simulation gives a wide range of periodic cycles including codim-1 bifurcation and resonance curves in the system. The results in this manuscript reveal that the dynamics of the discrete-time model in both single-parameter and two-parameter spaces are inherently rich and complex. The resonance bifurcation in the discrete-time map indicates that both species coincide till order 4 in stable periodic cycles near some critical parametric values.

We use direct numerical simulations to study convection in rotating Rayleigh-B & eacute;nard convection in horizontally confined geometries of a given aspect ratio, with the walls held at fixed temperatures. We show that this arrangement is unconditionally unstable to flow that takes the form of wall-adjacent convection rolls. For wall temperatures close to the temperatures of the upper or lower boundaries, we show that the base state undergoes a Hopf bifurcation to a state comprised of spatiotemporal oscillations - 'wall modes' - precessing in a retrograde direction. We study the saturated nonlinear state of these modes, and show that the velocity boundary conditions at the upper and lower boundaries are crucial to the formation and propagation of the wall modes: asymmetric velocity boundary conditions at the upper and lower boundaries can lead to prograde wall modes, while stress-free boundary conditions at both walls can lead to wall modes that have no preferred direction of propagation.

In this paper, we propose a reaction-diffusion genetic regulatory network with Neumann boundary conditions. We incorporate the diffusion factors into the delayed mathematical model of genetic networks, and with the time delays, we demonstrate the local stability and Hopf bifurcation in accordance with the Hopf bifurcation theory. When the sum of the delays exceeds the critical value, a Hopf bifurcation occurs. Finally, we select the appropriate system parameters and present a numerical simulation to illustrate the theoretical analysis.

A dynamic bifurcation analysis on a three-species cooperating model was presented and it was proved that the problem bifurcated an attractor as the parameter λ crossed the critical value λ 0 . The analysis was based on the attractor bifurcation theory together with the central manifold reduction.

In this paper, we have considered a deterministic epidemic model with logistic growth rate of the susceptible population, non-monotone incidence rate, nonlinear treatment function with impact of limited hospital beds and performed control strategies. The existence and stability of equilibria as well as persistence and extinction of the infection have been studied here. We have investigated different types of bifurcations, namely Transcritical bifurcation, Backward bifurcation, Saddle-node bifurcation and Hopf bifurcation, at different equilibrium points under some parametric restrictions. Numerical simulation for each of the above-defined bifurcations shows the complex dynamical phenomenon of the infectious disease. Furthermore, optimal control strategies are performed using Pontryagin’s maximum principle and strategies of controls are studied for two infectious diseases. Lastly using efficiency analysis we have found the effective control strategies for both cases.

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation x ¨ + a x ˙ + b x ˙ | x ˙ | + c x + d x 3 = 0 for the case d > 0 , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case d < 0 , i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

A series of bifurcations from period-1 bursting to period-1 spiking in a complex (or simple) process were observed with increasing extra-cellular potassium concentration during biological experiments on different neural pacemakers. This complex process is composed of three parts: period-adding sequences of burstings, chaotic bursting to chaotic spiking, and an inverse period-doubling bifurcation of spiking patterns. Six cases of bifurcations with complex processes distinguished by period-adding sequences with stochastic or chaotic burstings that can reach different bursting patterns, and three cases of bifurcations with simple processes, without the transition from chaotic bursting to chaotic spiking, were identified. It reveals the structures closely matching those simulated in a two-dimensional parameter space of the Hindmarsh–Rose model, by increasing one parameter I and fixing another parameter r at different values. The experimental bifurcations also resembled those simulated in a physiologically based model, the Chay model. The experimental observations not only reveal the nonlinear dynamics of the firing patterns of neural pacemakers but also provide experimental evidence of the existence of bifurcations from bursting to spiking simulated in the theoretical models.

Mixed-mode oscillations (abbreviated as MMOs) belong to a typical kind of fast/slow dynamical behavior, and how to investigate the mechanism is an important problem in nonlinear dynamics. In this paper, we explore the MMOs induced by the bifurcation delay phenomenon and twist of the trajectories in space based on a coupled system consisting of a Van der Pol system and a Duffing oscillator with two potential wells. Regarding the low-frequency external excitation as a generalized state variable, we obtain the traditional fast and slow subsystems. Appling the equilibrium analysis and bifurcation theory, the stability critical conditions of the equilibrium and the generation conditions of fold and Hopf bifurcation are also presented. To analyze the critical conditions clearly, the two-parameter bifurcation and one-parameter bifurcation diagrams are performed by using numerical simulation method. The bifurcation characteristics are studied, especially the effects of parameter δ on the bifurcation structures. We find that the fast subsystem performs different dynamical behaviors such as fold bifurcation of limit cycles, period-doubling bifurcations, inverse-period-doubling bifurcations and chaos, when parameter δ is taken at different values. By using phase diagrams, time series, maximum Lyapunov exponent diagrams, three-dimensional phase diagrams and superimposed diagrams, the mechanisms of the MMOs are investigated numerically in detail. The Hopf bifurcation delay can lead the trajectories to arrive at the vector fields of the equilibrium point and limit cycles. In addition, the chaotic behaviors can be found on the route of period doubling, which lead to the chaotic spiking-state-oscillations types. Our findings are helpful to understand the generation of the MMOs and intensify the understanding of some special dynamical behaviors on the MMOs.

Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, we study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, we use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.