Qualitative Theory of Dynamical Systems最新文献

In this article, we introduce a new fractional laser chaotic system derived from the Lorenz–Haken equations. We investigate the complex dynamics of the proposed system consisting chaos, stability, control and synchronization of chaos. Moreover, we numerically reveal the nonlinear dynamics of the fractional laser chaotic system through the phase portraits, time histories and bifurcation diagrams. Also, we indicate the chaotic behaviors of the system by means of Lyapunov exponents, bifurcation diagrams versus all parameters along the state variables, phase portraits and time histories with different trajectories and initial conditions. The necessary conditions to eliminate the chaotic vibration of the system are obtained via the feedback control procedure. Meanwhile, a synchronization mechanism based on the feedback control technique is achieved for coupled fractional laser chaotic systems. Furthermore, we show that the fractional derivative order is very effective on reducing the irregular and chaotic behaviors of the system.

In the article Secelean and Wardowski (Qual Theory Dyn Syst 21:162, 2022), there were introduced iterated function systems involving certain condition of orbital type. The attractors of such IFSs substantially extend the family of fractal objects investigated so far. In the present work we provide some further properties and give the answer regarding the continuity dependence of this type of attractors with respect to the continuous change of the parameters that describe the maps of the given orbital IFS and the initially chosen compact subset.

The main aim of this paper is to study the existence and uniqueness solutions for the nonlinear Hilfer pantograph fractional differential equations. This paper initiates with the persistence of the nonlinear Hilfer pantograph fractional differential equation. Also, it extended to the fractional integrodifferential equation. The premises are attained by using the fixed-point theorem. Ultimately, numerical examples are furnished to demonstrate our outcomes.

In this research, an examination is conducted on a model derived from the inquiry into the efficacy of HIV Pre-Exposure Prophylaxis (PrEP) within high-risk populations. To achieve this objective, we employ an SI model coupled with an age-structured equation to delineate the dynamics of individuals under PrEP protection, incorporating an infected-dependent rate of new users recruited from the susceptible population. This nonlinear term is contingent upon a factor dedicated to the political or economic context of a government. Local asymptotic stability for both disease-free and endemic equilibria is established, and global asymptotic stability for the disease-free steady-state is demonstrated. To address the system’s behavior, a reduction of the partial differential equation is undertaken, presenting it as a coupled system of differential equations and a delayed difference equation. Lastly, persistence is substantiated when the endemic equilibrium is realized.

In this paper, we propose a novel Caputo-type fractional-order cascade tri-neuron Hopfield neural network (HNN) taking no connection between the first and third neuron. We analyse the symmetry and dissipativity of the system using divergence and transformations. The stability of the equilibrium points is checked by fixing the synaptic weights. To further analyse the dynamics of the HNN system, we derive a numerical solution by using the Adams–Bashforth–Moulton method along with its stability analysis. We performed several graphical simulations, considering two synaptic weights as adjustable variables, and explored the fact that the HNN system shows various periodic and chaotic attractors. The reason for proposing a fractional-order HNN is that such a system has limitless memory, which can improve the system’s controllability for a wide range of real-world phenomena with important applications. Also, the proposed fractional-order HNN shows better convergence compared to the integer-order case.

In this investigation, the (3+1)-D potential Yu–Toda–Sasa–Fukuyama (YTSF) equation that arises in physical dynamics is studied for passing the painlevé test and obtaining many various exact solutions. The governing equation has many applications in fluid mechanics. Firstly, we applied the painlevé property for the governing equation and proved that the equation passes the painlevé test. After that, we utilized symmetry analysis to convert the governing equation to various ordinary differential equations. Subsequently, we obtained a new type of exact solutions for YTSF equation by using an Algorithm–Riccati method. The obtained solutions contained several arbitrary constants and functions that enhance the dynamic behaviors of these solutions. The obtained solutions include hyperbolic and trigonometric functions and represent kink wave, singularity wave and solitary wave solutions.

In this work, we study the approximate controllability problem for a system of nonlocal integro-differential equations with impulsive effects. We start by investigating the existence and uniqueness of solutions for this system. The results are derived using the theory of resolvent operators combined with fixed point theory in a generalized Banach space. Next, we examine approximate controllability without necessarily requiring the nonlinear terms to be uniformly bounded. In particular, we do not impose here the compactness condition for either the resolvent operator or the state-dependent function in the nonlocal condition, as is commonly found in the literature. Finally, we provide an example to demonstrate the abstract results of this work.

In this paper, we aim to find a mild solution for a delay control system described by nonlinear fractional evolution differential equations in Banach spaces while being subjected to nonlocal conditions. Further, we explore the sufficient conditions for the approximate controllability of the nonlinear fractional control system, assuming that the associated linear system is approximately controllable. We provide some applications at the end to demonstrate our proposed results.

In this paper, we investigate the dynamics of a class of discrete predator–prey model with alternative prey. We prove the boundedness of the solution, the existence and local/global stability of equilibrium points of the model, and verify the existence of flip bifurcation and Neimark-Sacker bifurcation. In addition, we use the maximum Lyapunov exponent and isoperimetric diagrams to verify the existence of periodic structures namely Arnold tongue and the shrimp-shaped structures in bi-parameter spaces of a class of predator–prey model.

We introduce three types of topological pressures and measure-theoretic pressures, present three variational principles between these measure-theoretic pressures and the corresponding topological pressures, and show that the upper capacity topological pressure of the whole space is determined by the Pesin–Pitskel topological pressure of dynamical balls under some suitable assumptions. Moreover, we show that these measure-theoretic pressures are equivalent for nonsingular measures.