Non-autonomous Perturbations Research Articles

Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

SIR models with vital dynamics, reinfection, and randomness to investigate the spread of infectious diseases

We investigate SIR models with vital dynamics, reinfection, and randomness at the transmission coefficient and recruitment rate. Initially, we conduct an extensive analysis of the autonomous scenario, covering aspects such as local and global well-posedness, the existence and internal structure of attractors, and the presence of gradient dynamics. Subsequently, we explore the implications of small nonautonomous random perturbations, establishing the continuity of attractors and ensuring their topological structural stability. Additionally, we study scenarios in which both the transmission coefficient and the recruitment rate exhibit time-dependent or random behavior. For each scenario, we establish the existence of attractors and delineate conditions that determine whether the disease is eradicated or reaches an endemic state. Finally, we depict numerical simulations to illustrate the theoretical results.

Bifurcations of degenerate homoclinic solutions in discontinuous systems under non-autonomous perturbations.

The main aim of this paper is to study bifurcations of bounded solutions from a degenerate homoclinic solution for discontinuous systems under non-autonomous perturbations. We use Lyapunov-Schmidt reduction to give bifurcation equations and prove that a single parameter is enough to unfold two distinct homoclinic solutions bifurcated from the unperturbed degenerate homoclinic solution. Furthermore, we give an example of a periodically perturbed piecewise smooth differential equation in R4 to support our conclusions.

Subharmonic Melnikov functions and nonintegrability for autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems near periodic orbits

We study autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems and give sufficient conditions for their real-analytic non-integrability near periodic orbits of the unperturbed systems such that the first integrals and commutative vector fields depend analytically on the small parameter by using the subharmonic Melnikov functions. Moreover, we show that autonomous dissipative perturbations prevent real-analytic integrability of these systems. Our results reveal that the perturbed systems can be real-analytically non-integrable even if there is no homoclinic/heteroclinic orbit in the unperturbed systems. We illustrate our theory with a periodically forced duffing equation and a damped Morse oscillator.

Normalized solutions for critical growth Schrödinger equations with nonautonomous perturbation

This paper is dedicated to investigating the existence of solutions characterized by a predefined L2-norm in the context of the nonlinear Sobolev critical Schrödinger equation −Δu+λu=|u|2∗−2u+V(x)|u|p−2u,inRN,∫RNu2dx=a2,u∈H1(RN).Here, N≥3, a>0 and 2<p<2∗, where 2∗=2NN−2 represents the critical Sobolev exponent. It is worth noting that the parameter λ functions as a Lagrange multiplier within this framework.

Resilience of dynamical systems

AbstractStability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the ‘global stability’ of a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline ‘resilience’. Unfortunately, there are many different variants and explanations of resilience, and often, the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalised to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain nonautonomous perturbations to demonstrate how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.

Stability of non-instantaneous impulsive systems in Hilbert spaces

In this paper, we introduce the concepts of Lyapunov exponent and Lyapunov regularity in Hilbert spaces, which is similar to that in the classical theory. As an application, under sufficiently small perturbations, we establish the asymptotic stability of the zero solution of non-instantaneous impulses nonautonomous nonlinear perturbations systems, which linear system is Lyapunov regular.

Robustness of 3D Navier–Stokes System with Increasing Damping

The principal objective of the paper is the study of the three-dimensional Navier–Stokes system with non-autonomous perturbation force term and increasing damping term, which often appears in the fluid system within saturated porous media and other complex media. With some suitable assumptions on the system parameters and external force term, based on the known result on global well-posedness, the existence of pullback attractors is educed, and the system robustness is shown via the upper semicontinuity of system attractors as the perturbation parameter approaches a certain value.

A Note on Forced Oscillations in Systems on a Plane

A sufficient condition for the existence of forced oscillations in nonautonomous systems on a plane is presented under the assumption that the magnitude of the nonautonomous perturbation is small. An advantage of the results presented over analytic methods is that they can be applied in degenerate systems as well.

The KAM theorem with a large perturbation and application to the network of Duffing oscillators

We prove that there is an invariant torus with the given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic external forces possesses Lagrange stability for almost all initial data.

Continuity and topological structural stability for nonautonomous random attractors

In this work, we study the continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study the existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish the lower semicontinuity of nonautonomous random attractors and to show that the gradient structure persists under nonautonomous random perturbations. Finally, we apply the abstract results in a stochastic differential equation and in a damped wave equation with a perturbation on the damping.

Evaluation on spectral submanifold based reduced models of a rotor/stator rubbing system with cross-coupling stiffness

The methodology for reduced order models of nonlinear dynamical systems is always of great interest in dynamics community. Recently, spectral submanifolds (SSMs) related to hyperbolic fixed points have been constructed and successfully used to build reduced order models for nonlinear systems with a large number of degrees of freedom. There are, however, few works on the spectral submanifolds associated with periodic orbits in literature. In this paper, SSM associated to the periodic orbit of a nonlinear normal mode (NNM) of a rotor/stator rubbing system with cross-coupling stiffness is first constructed following the parameterization method. The SSM with a two-dimensional slow sub-stable invariant manifold attached to the stable NNM with forward whirl also serves as a reduced order model of the system, whose dynamics on the parameter space conjugates the dynamics of the full system. Unlike the previous works, where reduced order models are mainly used to obtain backbones of conservative autonomous systems or FRCs of light damped forcing systems as the non-autonomous perturbation, the reduced order model derived from autonomous system are used to predict the self-excited dominant forcing responses of the rotor/stator rubbing system. As can be seen, the reduced order model is valid with good accuracy in some range of forcing amplitudes. Moreover, the modification of the reduced order model by considering forcing perturbation is also built and works well even in a larger range of forcing amplitudes.

On the admissibility of observation for perturbed non-autonomous linear systems

In this paper, we are concerned with unbounded observation operators for perturbed non-autonomous evolution equations. First, we briefly survey the necessary background about the concept of maximal regularity. Second, we establish the admissibility invariance of observation operators under non-autonomous Miyadera-Voigt perturbations. Finally, we illustrate our abstract result with an example.

Nonautonomous Perturbations of Morse–Smale Semigroups: Stability of the Phase Diagram

Nonautonomous Perturbations of Morse–Smale Semigroups: Stability of the Phase Diagram

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

Perturbation theory for Bose–Einstein condensates on bounded space domains

Bose–Einstein condensates (BECs), first predicted theoretically by Bose and Einstein and finally discovered experimentally in the 1990s, continue to motivate theoretical and experimental physics work. Although experiments on BECs are carried out in bounded space domains, theoretical work in the modelling of BECs often involves solving the Gross–Pitaevskii equation on unbounded domains, as the combination of bounded domains and spatial heterogeneity render most existing analytical approaches ineffective. Motivated by a lack of theory for BECs on bounded domains, we first derive a perturbation theory for both ground and excited stationary states on a given bounded space domain, allowing us to explore the role various forms of the self-interaction, external potential and space domain have on BECs. We are able to show that the shape and curvature of a space domain strongly influence BEC structure, and may be used as control mechanisms in experiments. We next derive a non-autonomous perturbation theory to predict BEC response to temporal changes in an external potential. In certain cases, our approach can be extended to unbounded domains, and we conclude by constructing a perturbation theory for bright solitons within external potentials on unbounded domains.

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

<p style='text-indent:20px;'>We study a class of non-autonomous linear boundary control and observation systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

Study of Periodic Orbits in Periodic Perturbations of Planar Reversible Filippov Systems Having a Twofold Cycle

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.

Dynamics of 2D Incompressible Non-autonomous Navier–Stokes Equations on Lipschitz-like Domains

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier–Stokes equations with a non-homogeneous boundary condition on Lipschitz-like domains. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimates on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is perturbed from a stationary force by a time-dependent perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we investigate the regularity of these attractors when smoother initial data are given. Our results are new even for smooth domains.

Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping

This paper is devoted to study the asymptotic behavior of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. We prove the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the sources terms. Moreover, we prove the upper-semicontinuity of pullback attractors with respect to non-autonomous perturbations.