Qualitative Theory of Dynamical Systems最新文献

In this paper, we study the impulsive evolution equation with state-dependent delay by the theory of operator semigroup in Banach spaces. Under conditions that both nonlinearity and impulsive functions are Lipschitz continuous, we obtain the existence and uniqueness results of mild solution. Furthermore, we prove the differentiability of a semi-flow defined by a continuously differentiable solution operator under the appropriate condition.

This article explores the existence and approximate controllability of integral solutions for Hilfer fractional evolution equations in a non-dense domain. Leveraging the well-known generalized Banach contraction theorem, we establish both the existence and uniqueness of the integral solution. Furthermore, we adopt a sequential approach to derive results related to approximate controllability, without relying on the compactness of semigroups or the uniform boundedness of nonlinear functions. To validate our findings, we present and discuss an illustrative example.

In this paper, we show the approximate controllability for a class of semilinear fractional stochastic systems in abstract space with the Riemann–Liouville fractional derivative. The key of the proof is the existence of the mild solution for the proposed problem. These results are based on new properties of the operator obtained by the subordination principle, compact semigroup and Schauder fixed point theorem. Here we obtain the compactness of the solution operator by using Arzelà–Ascoli theorem. As an application, we establish the approximate controllability of the stochastic Rayleigh–Stokes problem for a generalized second grade fluid.

In this paper, we study the number of limit cycles H(n) bifurcating from the piecewise smooth system formed by the quadratic reversible system (r22) for (yge 0) and the cubic system ({dot{x}} =ybigl (1+{{bar{x}}}^2+y^2bigr )), ({dot{y}} =-{bar{x}}bigl (1+{{bar{x}}}^2+y^2bigr )) for (y<0) under the perturbations of polynomials with degree n, where ({{bar{x}}}=x-1). By using the first-order Melnikov function, it is proved that (2n+3le H(n)le 2n+ 7) for (nge 3) and the results are sharp for (n=0,1,2).

Numerous studies have investigated the stability of highly nonlinear stochastic systems (HNSSs). However, previous works have primarily focused on either deterministic or random switchings. In this paper, we examine HNSSs with delays and two switching modes. First, we introduce a hybrid switching rule and construct a stopping time in segments, dividing the switching interval of the entire system into a deterministic switching interval and a stochastic switching interval. Second, we establish the existence and boundedness of the global solution of the system by using the Lyapunov function and the average dwell time method. Additionally, we prove the asymptotic stability and exponential stability of the system without relying on the linear growth condition (LGC). Finally, we provide an illustrative example to demonstrate the validity of the obtained results.

This paper aims to demonstrate the Chebyshev property of the linear space (V={sum _{i=0}^{2}alpha _ioint _{Gamma _h}x^{2i}ytextrm{d}x:alpha _0,alpha _1,alpha _2in mathbb {R},,hin Sigma }) (which is equivalent to that every function of V has at most 2 zeros, counted with multiplicity), with three hyperelliptic Abelian integrals (oint _{Gamma _h}x^{2i}ytextrm{d}x ,(i=0,1,2)) as generators, where (Gamma _h) is an oval determined by (H(x,y)=frac{y^2}{2}+Psi (x)=h), and (Psi (x)) is an even polynomial of indefinite degree with real non-Morse critical points. As an application, we can obtain the exact upper bound for the number of zeros of a class of hyperelliptic Abelian integrals related to some planar polynomial Hamiltonian systems with two cusps and a nilpotent center.

In this paper, we consider a delayed patch-constructed Nicholson’s blowflies system in almost periodic environment. By combining the innovative inequality technique with the basic properties of almost periodic functions and the fluctuation lemma, some testable criteria are achieved to verify the global exponential stability of the addressed almost periodic system under more general conditions, which improve and complement the existing literature. In particular, the assumptions employed in the established exponential stability criteria are sharp when the addressed system degenerates into the scalar Nicholson’s blowflies equation. Moreover, a numerical example is presented to illustrate the effectiveness of the theoretical results.

The primary aim of this study is to analyze the chaotic dynamics of a conformable maturity structured cell partial differential equation of order (zin (0,1)), which extends the classical Lasota equation. To examine the chaotic behavior of our model’s solution, we initially extend certain criteria of linear chaos to conformable calculus. This extension is crucial because the solution of our model does not generate a classical semigroup but rather a (c_0)-z-semigroup. For the velocity term of our model, (B(mathfrak {w})=mu mathfrak {w}), where (mu in mathbb {C}), and the term source (g(mathfrak {w}, vartheta (textsf{r}, mathfrak {w}))), we utilize spectral properties of the z-infinitesimal generator to demonstrate chaotic behavior in the space (C(textrm{J}_0, mathbb {C})), (textrm{J}_0:=[0,+infty )). Furthermore, by employing conformable admissible weight functions and setting (B(mathfrak {w})=1), we establish chaos in the solution z-semigroup, this time within the space (C_{0}(textrm{J}_0, mathbb {C})).

In this paper, the authors present new oscillation criteria for the noncanonical second-order delay differential equation with mixed nonlinearities

using an arithmetic–geometric mean inequality. We establish our results first by transforming the studied equation into canonical form and then applying a comparison technique and integral averaging method to get new oscillation criteria. Examples are provided to illustrate the importance and novelty of their main results.

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.