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This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamic behaviors and chaos synchronization of the Chen system described by Caputo–Hadamard fractional derivative. First, the existence and uniqueness of a solution to the Chen system with Caputo–Hadamard derivative are proved by qualitative analysis. Further, the stability of equilibria of the considered system is analyzed with the aid of Routh–Hurwitz criteria. Meanwhile, the bifurcation condition of the Caputo–Hadamard Chen system is compared with the integer-order Chen system, where the differences between the two systems are demonstrated numerically. In the study of chaos synchronization of the drive–response Chen systems with Caputo–Hadamard derivative, two control schemes are developed: three nonlinear controllers and single linear controller. The feasibility of two control schemes is verified, and the synchronization performances of these two schemes are compared by numerical simulations. Based on this, the influence of the fractional-order on chaos synchronization performance is illustrated as well. [ABSTRACT FROM AUTHOR]

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly. [ABSTRACT FROM AUTHOR]

Based on the feature of piecewise linear (PWL) functions, a nonlinear feedback of a higher-order system can be transformed to an approximately equivalent PWL function so as to ease system implementation in engineering. As an example, a cubic feedback term can be approximately equivalently transformed to be a PWL function. Since the PWL function can be expressed by many simple functions such as signum function and absolute-valued function, the cubic term can be approximately equivalently replaced with these functions. Consequently, the method of approximate equivalence is employed in the JCS-08-13-2022 (JCS) chaotic system for simple circuit design and implementation. In this approach, the widely used multipliers are avoided and the circuits become more economical and also more robust. In this paper, the cubic Chua's resistor is equivalently approximately replaced by a PWL function. To show the effectiveness of the approximate equivalence, numerical simulations are demonstrated and verified by circuit implementation. [ABSTRACT FROM AUTHOR]