Qualitative Theory of Dynamical Systems最新文献

We first investigate a class of generalized non-homogeneous two-dimensional hybrid systems and derive formulas for their solutions. We then obtain the transfer matrix and characteristic equation specific to this system type. Examples are provided to illustrate our theory.

We study the bifurcation problem of limit cycles in near-Hamiltonian systems near a double homoclinic loop on the cylinder. We obtain a sufficient condition to find a lower bound of the maximal number of limit cycles near the loop by the coefficients of the expansions of the three Melnikov functions corresponding to the three families of periodic orbits near the double homoclinic loop. We also provide an application of our main results to a class of cylindrical systems.

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center.

Recalling that at any regular point we always have a unique particular solution curve passing through it. In this work it is constructed such particular solution curve not passing through the nilpotent singularity but as close as we want to the singularity. By product the existence of such particular curve allows to use it to determine necessary conditions to have a center for nilpotent singularities in the plane. Several involve methods to solve the center problem are known all based in the existence of a change of variables and a scaling transformation of time bringing any differential system with a nilpotent center into a time-reversible system. Here we present a new algebraic method based on the existence of such particular solution curve not passing through the singular point and the involution associated to the nilpotent system with a center. The algebraic method needs the computation of this particular curve up to certain order, which can be done with the help of an algebraic manipulator. Finally a new algebraic method is derived computing the vanishing of a unique function which really gives a scalar method for computing the necessary conditions.

This paper investigates the existence and upper semi-continuity of the pullback measure attractors of the non-autonomous stochastic 3D globally modified Navier–Stokes equations driven by nonlinear noise. Firstly, we introduce the abstract theory of pullback measure attractors and asymptotic compactness of such equations. Then, the existence of the pullback measure attractors is shown for such equations. Furthermore, the upper semi-continuity of these attractors is also obtained as the noise intensity tends to zero.

In this paper, we consider nearly integrable reversible systems, whose unperturbed part has a degenerate equilibrium point and a degenerate frequency mapping. Based on the topological degree theory and some KAM techniques, we prove that the non-twist lower dimensional invariant torus with prescribed frequencies persists under small perturbations.

We consider the state-dependent delay differential equation (SDDE) obtained by adding delayed perturbation to a one-dimensional ODE with a degenerate equilibrium. We prove the existence of the response solution of the equation, i.e., the quasi-periodic solution with the same frequency as the forcing. The novelty of our paper is to provide a concrete example to discuss the smoothness issues of SDDE, especially showing the analyticity of quasi-periodic solutions in some probability sense.

In this paper, we consider a class of asymptotically linear damped vibration problems with resonance at infinity. Compared with the existing results, under this resonance condition the functional corresponding to the problem may not satisfy the compactness condition. By combining the penalized functional technique, Morse theory and two critical point theorems, we obtain the existence and multiplicity of nontrivial periodic solutions.

In this paper, we investigate a variable-coefficient extended Korteweg-de Vries equation with an external-force term in fluid mechanics and plasma dynamics. Under certain variable-coefficient constraints, we get the Painlevé integrable property of that equation. With the truncated Painlevé expansion and Hirota method, we work out some bilinear forms, bilinear Bäcklund transformations under certain variable-coefficient constraints. With the bilinear forms, multi-soliton solutions are constructed. Based on those solutions, multi-complex-soliton solutions are derived through the complex forms of the Hirota method. Influences of the variable coefficients on the multi-soliton solutions are discussed graphically. We find that (i) different types of the one-soliton profiles and soliton interactions can be seen with the changes of variable coefficients; (ii) the amplitudes of those solitons are influenced under the dissipative and cubic-nonlinear coefficients; (iii) the characteristic lines and velocities of those solitons are influenced under the dissipative, dispersive coefficients and external-force term; (iv) the backgrounds of those solitons are influenced under the external-force term. Additionally, the influences of the variable coefficients on the complex solitons are similar to the influences on the real solitons.

The utilization of quaternions in nonlinear systems with input delays is presented in this paper, aiming to investigate the controllability of nonlinear quaternion-valued systems (QVSs). In view of the non-commutativity of quaternion multiplication, the system is decomposed into four real-valued subsystems. The existence and uniqueness of solutions for QVSs with input delays are demonstrated using the contraction mapping principle and Laplace transform. To achieve system controllability amidst distinct input delays, two categories of Gram matrices along with their respective control functions are established, and their feasibility are demonstrated by Arzela-Ascoli theorem and the Schaefer fixed point theorem. Finally, numerical simulations are conducted to validate the theoretical findings.