Affine-periodic Solutions Research Articles

Affine-periodic solutions for generalized ODEs and other equations

It is known that the concept of affine-periodicity encompasses classic notions of symmetries as the classic periodicity, anti-periodicity and rotating symmetries (in particular, quasi-periodicity). The aim of this paper is to establish the basis of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T> 0$ and an invertible $n\times n$ matrix $Q$, with entries in $\mathbb C$, we establish conditions for the existence of a $(Q,T)$-affine-periodic solution within the framework of nonautonomous generalized ODEs, whose integral form displays the nonabsolute Kurzweil integral, which encompasses many types of integrals, such as the Riemann, the Lebesgue integral, among others. The main tools employed here are the fixed point theorems of Banach and of Krasnosel'skiĭ. We apply our main results to measure differential equations with Henstock-Kurzweil-Stiejtes righthand sides as well as to impulsive differential equations and dynamic equations on time scales which are particular cases of the former.

AFFINE-PERIODIC SOLUTIONS FOR PERTURBED SYSTEMS

In this paper, we try to study the existence and uniqueness of affine-periodic solutions for the perturbed affine-periodic system. We prove that, under certain conditions, if the coefficient of the forced term is sufficiently small, then the system admits affine-periodic solutions which have the form of $z(t+T,\mu)=Qz(t,\mu)$ with some nonsingular matrix Q. Depending on the structure of Q, they may be periodic, anti-periodic, quasi-periodic or even unbounded spiral motions. The main tools we used are the theory of exponential dichotomy and Banach contraction mapping principle.

A generalized Massera theorem based on affine periodicity

In this study, we establish a Massera type theorem for first order linear dynamical systems by using a generalized periodicity notion, namely affine periodicity. We introduce an alternative boundedness concept for vector-valued functions, mQ,T-boundedness, and we obtain the relationship between the existence of mQ,T-bounded solutions and (Q,T) affine-periodic solutions of linear systems due to fixed point theory.

Affine-Periodic Solutions by Asymptotic Method

We consider the existence of affine-periodic solutions to the nonlinear ordinary differential equation: 0.1 $\label {ae} x^{\prime }=f(t,x) $ in ℝn, where f is continuous and ensures the existence and uniqueness of solutions with respect to initial conditions, and there exist T > 0 and Q ∈ GL(n) such that: 0.2 $\label {me} f(t+T,x)=Qf(t,Q^{-1}x) \quad \forall (t,x)\in \mathbb R\times \mathbb R^{n}. $ We utilize the asymptotic method to study the existence of Q-affine T-periodic solutions of Eqs. 0.1–0.2. Affine-periodic solutions exhibit a rich variety of complex spatiotemporal pattern, might be periodic, anti-periodic, quasi-periodic, and even unbounded spiral motions.

Affine-periodic solutions by asymptotic and homotopy equivalence

This paper studies the existence of affine-periodic solutions which have the form of x(t+T)=Qx(t) with some nonsingular matrix Q. Depending on the structure of Q, they can be periodic, anti-periodic, quasi-periodic or even unbounded. Krasnosel’skii–Perov type existence theorem, asymptotic and homotopy equivalence approaches are given.

Affine-periodic solutions for higher order differential equations

This paper is a continuous work of Liu et al. (2017, first order) and Xu et al. (2019, second order) for affine-periodic solutions to ordinary differential equations. It is a hard problem to obtain satisfied extremum principles. In this paper, we give several extremum principles (Theorem 2.1 and Lemma 2.2) for affine-periodic problems, especially for the case of higher order systems. By these extremum principles, we partly establish the existence of affine-periodic solutions for higher order ordinary differential equations.

Affine-Periodic Solutions for Impulsive Differential Systems

The affine-periodicity is a new periodic concept which has been founded in recent years. In this paper, we will discuss the existence of affine-periodic solutions for impulsive differential systems. Several existence theorems are proved for dissipative impulsive (functional) differential systems. Some applications are also given by combining Lyapunov’s methods.

LaSalle stationary oscillation theorem for affine periodic dynamic systems on time scales

In this paper, for affine periodic systems on time scales, we establish LaSalle stationary oscillation theorem to obtain the existence and asymptotic stability of affine periodic solutions on time scales. As applications, we present the existence and asymptotic stability of affine periodic solutions on time scales via Lyapunov’s method.

On the affine-periodic solutions of discrete dynamical systems

On the affine-periodic solutions of discrete dynamical systems

The existence of affine-periodic solutions for nonlinear impulsive differential equations

In this paper, we study the existence of affine-periodic solutions of nonlinear impulsive differential equations. The affine-periodic solutions have the form x(t+T)=Qx(t) with some nonsingular matrix Q. We give a theorem on the existence of the affine-periodic solutions, respectively, depending on wether operatorname{det}(I-Q) (I= identity matrix) is equal to 0 or not.

Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems

This paper concerns the existence of affine-periodic solutions for differential systems (including functional differential equations) and Newtonian systems with friction. This is a kind of pattern solutions in time-space, which may be periodic, anti-periodic, subharmonic or quasi periodic corresponding to rotation motions. Fink type conjecture is verified and Lyapunov's methods are given. These results are applied to study gradient systems and Newtonian (including Rayleigh or Lienard) systems. Levinson's conjecture to Newtonian systems is proved.

Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions

This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.

Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions

This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.

Affine-periodic solutions by averaging methods

This paper concerns the existence of affine-periodic solutions for perturbed affine-periodic systems.This kind of affine-periodic solutions has the form of $x(t+T)\equiv~Qx(t)$ with some nonsingular matrix $Q$, which may be quasi-periodic when $Q$ is an orthogonal matrix.It can be even unbounded but $\frac{x(t)}{|x(t)|}$ is quasi-periodic, like a helical line, for example $x(t)={\rm~e}^{at}(\cos~\omega~t,\sin~\omega~t)$, when $Q$ is not an orthogonal matrix.The averaging method of higher order for finding affine-periodic solutions is given by topological degree.

LaSalle type stationary oscillation theorems for Affine-Periodic Systems

The paper concerns the existence of affine-periodic solutions for affine-periodic (functional) differential systems, which is a new type of quasi-periodic solutions if they are bounded. Some more general criteria than LaSalle's one on the existence of periodic solutions are established. Some applications are also given.

Affine-periodic solutions for nonlinear differential equations

The existence of affine-periodic solutions is studied. These types of solutions may be periodic, harmonic or even quasi-periodic. Mainly, via the topological degree theory, a general existence theorem is proved, which asserts the existence of affine-periodic solutions, extending some classical results. The theorem is applied to establish the Lyapunov function type theorem and the invariant region principle relative to affine-periodic solutions.

AFFINE-PERIODIC SOLUTIONS AND PSEUDO AFFINE-PERIODIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH EXPONENTIAL DICHOTOMY AND EXPONENTIAL TRICHOTOMY

It is proved that every (<i>Q, T</i>)-affine-periodic differential equation has a (<i>Q, T</i>)-affine-periodic solution if the corresponding homogeneous linear equation admits exponential dichotomy or exponential trichotomy. This kind of "periodic" solutions might be usual periodic or quasi-periodic ones if <i>Q</i> is an identity matrix or orthogonal matrix. Hence solutions also possess certain symmetry in geometry. The result is also extended to the case of pseudo affine-periodic solutions.

Affine-periodic solutions for nonlinear dynamic equations on time scales

The existence of affine-periodic solutions for dynamic equations on time scales is studied. Mainly, via the topological degree theory, a general existence theorem is proved, which provides an effective method in the qualitative theory for nonlinear dynamic equations on time scales.

AFFINE-PERIODIC SOLUTIONS FOR DISCRETE DYNAMICAL SYSTEMS

The paper concerns the existence of affine-periodic solutions for discrete dynamicalsystems. This kind of solutions might be periodic, harmonic, quasi-periodic, even non-periodic.We prove the existence of affine-periodic solutions for discrete dynamical systems by using thetheory of Brouwer degree. As applications, another existence theorem is given via Lyapnovfunction.

Affine-Periodic Solutions for Dissipative Systems

As generalizations of Yoshizawa’s theorem, it is proved that a dissipative affine-periodic system admits affine-periodic solutions. This result reveals some oscillation mechanism in nonlinear systems.