Qualitative Theory of Dynamical Systems最新文献

The current study examines the (2 + 1)-dimensional fractional Chaffee–Infante (FCI) model, which is a nonlinear evolution equation that characterizes the processes of pattern generation, reaction-diffusion, and nonlinear wave propagation. The construction of analytical solutions involves the use of analytical methods, namely the Khater III and improved Kudryashov schemes. The He’s Variational Iteration method is employed as a numerical approach to validate the accuracy of the obtained solutions. The main objective of this study is to get novel analytical and numerical solutions for the FCI model, with the intention of gaining a deeper understanding of the system’s dynamics and its possible implications in the fields of fluid mechanics, plasma physics, and optical fiber communications. The study makes a valuable contribution to the area of nonlinear science via the use of innovative analytical and numerical methodologies in the FCI model. This research enhances our comprehension of pattern creation, reaction–diffusion phenomena, and the propagation of nonlinear waves in diverse physical scenarios.

The current study introduces the generalised New Extended Direct Algebraic Method (gNEDAM) for producing and examining propagation of kink soliton solutions within the framework of the Conformable Kolmogorov–Petrovskii–Piskunov Equation (CKPPE), which entails conformable fractional derivatives into account. The primary justification around employing conformable derivatives in this study is their special ability to comply with the chain rule, allowing for in the solution of aimed nonlinear model. The CKPPE is a crucial model for a number of disciplines, such as mathematical biology, reaction-diffusion mechanisms, and population increase. CKPPE is transformed into a Nonlinear Ordinary Differential Equation by the proposed gNEDAM, and many kink soliton solutions are found by applying the series form solution. These kink soliton solutions shed light on propagation mechanisms within the framework of the CKPPE model. Furthermore, our research offers multiple graphical depictions that facilitate the examination and analysis of the propagation patterns of the identified kink soliton solutions. Through the integration of mathematical biology and reaction-diffusion principles, our research broadens our comprehension of intricate occurrences in various academic domains.

Shallow water waves are seen in magnetohydrodynamics, atmospheric science, oceanography and so on. In this article, we study a ((3+1))-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff equation with the time-dependent coefficients for the shallow water waves. N-soliton solutions are obtained via the simplified Hirota method. Via the N-soliton solutions, we present the elastic interactions between the two solitons and among the three solitons. Some other analytic solutions are constructed through the tanh method and ((frac{G'}{G^{2}}))-expansion method.

In this article, the integrability and linearizability of a class of cubic (Z_2)-equivariant systems (dot{x}=-frac{1}{2}x-a_{21}y+frac{1}{2}x^3+a_{21}x^2y+a_{12}xy^2+a_{03} y^3,, dot{y}=(-q-b_{21})y+b_{21}x^2y+b_{12}xy^2+b_{03}y^3, ) are studied. For any positive integer q, we obtain the first three saddle quantities of the above systems by theoretical analysis. Moreover, for any positive integer q, we derive the necessary and sufficient conditions for the linearizability of the above systems under some assumptions.

To study the impact of stochastic environmental variations on the transmission dynamics of HTLV-I infection, a stochastic HTLV-I infection model with a nonlinear CTL immune response is developed. By selecting an appropriate stochastic Lyapunov functional, we discussed the qualitative behavior of the stochastic HTLV-I infection model, such as existence and uniqueness, stochastically ultimate bounded, and uniformly continuous. We find adequate criteria for the presence of a distinct ergodic stationary distribution of the HTLV-I system when the stochastic basic reproduction number is bigger than one by a careful mathematical examination of the HTLV-I infection model. Furthermore, when the stochastic fundamental reproduction number ((R_0^{E})) is smaller than one, we provide sufficient circumstances for the extinction of the diseases. To illustrate our analytical conclusions, we ran numerical simulations. We also plotted the time series evolution of the CTL immune response, healthy CD4+T cells, latently infected CD4+T cells, and actively infected CD4+T cells in relation to the white noise. In the numerical simulation, we investigate that small intensities of a single white noise can sustain a very slight fluctuation in each population. The high intensities of only one white noise can maintain the irregular recurrence of each population. Both the deterministic and stochastic models have the same solution if the random noises are too small.

This research paper study the existence, uniqueness and Ulam–Hyers stability of the solutions of a certain system of thegeneralized Caputo fractional differential equations in the context of the generalized metric spaces. The existence and uniqueness theorems are proved by using the Krasnoselskii’s and Perov’s fixed point theorems under the Bielecki norm with a Lipschitzian matrix in the generalized metric spaces. Moreover, the Ulam–Hyers stability analysis is conducted based on the Urs’s criterion. An example, lastly, is proposed to check the efficiency of the above-mentioned theorems. The results are novel and provide extensions to some of the findings known in the literature.

We consider k-dimensional discrete-time systems of the form (x_{n+1}=F(x_n,ldots ,x_{n-k+1})) in which the map F is continuous and monotonic in each one of its arguments. We define a partial order on ({mathbb {R}}^{2k}_+), compatible with the monotonicity of F, and then use it to embed the k-dimensional system into a 2k-dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest.

This study aims to explore the complexity of a discrete-time predator–prey system with a weak Allee effect. The existence and stability of fixed points, as well as period-doubling and Neimark–Sacker bifurcations, are all investigated. The system’s bifurcating and fluctuating behavior is controlled using feedback and hybrid control techniques. Additionally, numerical simulations are performed as evidence to support theoretical results. From an ecological perspective, these findings suggest that the Allee effect plays a pivotal role in shaping predator–prey dynamics. The moderate Allee effect fosters stability in both predator and prey populations, promoting coexistence and persistence within ecosystems. However, the disproportionate impact on predator populations underscores predators’ vulnerability to changes in prey behavior and availability, highlighting the importance of considering indirect effects in ecological modeling and conservation efforts.

In this paper we consider a forced oscillator with a discontinuous restoring force. By the Aubry–Mather theory we prove that there exist infinitely many periodic and quasiperiodic solutions. The proof relies on analysing the generating function of the system. The approach is applicable to studying the dynamics of more general forced nonsmooth oscillators of Hamiltonian type.

We are concerned with the existence of homoclinic solutions for the nonlinear problems

where (omega in mathbb {R},~k>0) are real constants, and (f: mathbb {R}^{3}rightarrow mathbb {R}) is an (L^{1}-)Carathéodory function. Under some suitable conditions, the existence of homoclinic solutions for problem (P) and the corresponding coupled systems are provided. The proofs of the main results are based on the method of upper and lower solutions.