Small Nonlinear Perturbations Research Articles

Hyers–Ulam stability of integral equations with infinite delay

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay.

On Convergence Rate Bounds for a Class of Nonlinear Markov Chains

A new approach is developed for evaluating the convergence rate for nonlinear Markov chains (MC) based on the recently developed spectral radius technique of Markovian coupling for linear MC and the idea of small nonlinear perturbations of linear MC. The method further enhances recent advances in the problem of convergence for such models.

Controlled bio-inspired self-organised criticality.

Complex biological systems are considered to be controlled using feedback mechanisms. Reduced systems modelling has been effective to describe these mechanisms, but this approach does not sufficiently encompass the required complexity that is needed to understand how localised control in a biological system can provide global stable states. Self-Organised Criticality (SOC) is a characteristic property of locally interacting physical systems, which readily emerges from changes to its dynamic state due to small nonlinear perturbations. These small changes in the local states, or in local interactions, can greatly affect the total system state of critical systems. It has long been conjectured that SOC is cardinal to biological systems, that show similar critical dynamics, and also may exhibit near power-law relations. Rate Control of Chaos (RCC) provides a suitable robust mechanism to generate SOC systems, which operates at the edge of chaos. The bio-inspired RCC method requires only local instantaneous knowledge of some of the variables of the system, and is capable of adapting to local perturbations. Importantly, connected RCC controlled oscillators can maintain global multi-stable states, and domains where power-law relations may emerge. The network of oscillators deterministically stabilises into different orbits for different perturbations, and the relation between the perturbation and amplitude can show exponential and power-law correlations. This can be considered to be representative of a basic mechanism of protein production and control, that underlies complex processes such as homeostasis. Providing feedback from the global state, the total system dynamic behaviour can be boosted or reduced. Controlled SOC can provide much greater understanding of biological control mechanisms, that are based on distributed local producers, with remote consumers of biological resources, and globally defined control.

Asymptotic behaviour of solutions of quasilinear differential-algebraic equations

This paper is concerned with the asymptotic behavior of solutions of linear differential-algebraic equations (DAEs) under small nonlinear perturbations. Some results on the asymptotic behavior of solutions which are well known for ordinary differential equations are extended to DAEs. The main tools are the projector-based decoupling and the contractive mapping principle. Under certain assumptions on the linear part and the nonlinear term, asymptotic behavior of solutions are characterized. As the main result, a Perron type theorem that establishes the exponential growth rate of solutions is formulated.

Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations

<p style='text-indent:20px;'>This paper addresses the following question: "Suppose that a state-feedback controller stabilizes an infinite-dimensional linear continuous-time system. If we choose the parameters of an event/self-triggering mechanism appropriately, is the event/self-triggered control system stable under all sufficiently small nonlinear Lipschitz perturbations?" We assume that the stabilizing feedback operator is compact. This assumption is used to guarantee the strict positiveness of inter-event times and the existence of the mild solution of evolution equations with unbounded control operators. First, for the case where the control operator is bounded, we show that the answer to the above question is positive, giving a sufficient condition for exponential stability, which can be employed for the design of event/self-triggering mechanisms. Next, we investigate the case where the control operator is unbounded and prove that the answer is still positive for periodic event-triggering mechanisms.</p>

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t^{-frac{1}{3}} for position and momentum correlations and as t^{-frac{2}{3}} for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t^{-frac{1}{4}} for position and momentum correlators and with rate t^{-frac{1}{2}} for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.

Hyers–Ulam stability for hyperbolic random dynamics

We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations.

Quasi-shadowing for partially hyperbolic dynamics on Banach spaces

A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points (xn)n∈Z such that xn+1 is obtained from the image of xn by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also obtain a continuous time version of this result. As an application of our main result, we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction.

Perturbations of delay equations

We study the behavior of an exponential dichotomy for a nonautonomous linear delay equation under sufficiently small linear and nonlinear perturbations. In particular, we establish a partial version of the Grobman–Hartman theorem and we use the characterization of an exponential dichotomy in terms of an admissibility property to establish the robustness property.

On the Reducibility of a Class of Nonlinear Periodic Hamiltonian Systems with Degenerate Equilibrium

In this paper, we prove the existence of periodic solutions of a class of Hamiltonian systems with degenerate equilibriums under small nonlinear periodic perturbations. Actually we prove that the periodic Hamiltonian systems with small perturbations can be reducible to a periodic Hamiltonian system with an equilibrium by a periodic symplectic mapping. This result is a reformulation of the result in Lu and Xu (Nonlinear Differ Equ Appl 21:361–370, 2014) in the case of Hamiltonian systems.

StabilIty of Scalar Nonlinear Fractional Differential Equations with Linearly Dominated Delay

Abstract In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.

Nonuniform spectrum on Banach spaces

For a sequence of bounded linear operators acting on a Banach space, we consider the notion of nonuniform spectrum. This is defined in terms of the existence of nonuniform exponential dichotomies with an arbitrarily small nonuniform part and can be seen as a nonuniform version of the spectrum introduced by Sacker and Sell in the case of a single trajectory, although now in the infinite-dimensional setting. We give a complete characterization of all possible forms of the nonuniform spectrum for sequences of compact linear operators and, more generally, for sequences of bounded linear operators satisfying a certain asymptotic compactness. Moreover, we provide explicit examples of sequences of compact linear operators acting on the l2 space of sequences of real numbers for all the possible forms of the nonuniform spectrum. As nontrivial applications, we show that the nonuniform spectrum of a Lyapunov regular sequence is the set of Lyapunov exponents and that the asymptotic behavior persists under sufficiently small nonlinear perturbations, in the sense that the lower and upper Lyapunov exponents of the perturbed dynamics belong to a connected component of the nonuniform spectrum. Finally, we obtain appropriate versions of the results for nonuniformly hyperbolic cocycles.

Lyapunov type equation for discrete exponential trichotomies

For a nonautonomous dynamics obtained by a sequence of linear operators acting on an arbitrary Hilbert space, we give a complete characterization of the notion of a uniform exponential trichotomy in terms of what can be considered to be a discrete version of the Lyapunov equation. We then use this characterization to study the stability of exponential trichotomies under small linear and nonlinear perturbations.

Stability by linear approximation for time scale dynamical systems

We study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of matrices we formulate conditions, sufficient for stability by linear approximation. For non-periodic time scales we use techniques of central upper Lyapunov exponents (a common tool of the theory of linear ODEs) to study stability of solutions. Also, time scale versions of the famous Chetaev's theorem on conditional instability are proved. In a nutshell, we have developed a completely new technique in order to demonstrate that methods of non-autonomous linear ODE theory may work for time-scale dynamics.

Nonuniform Spectrum on the Half Line and Perturbations

For a one-sided nonautonomous dynamics defined by a sequence of invertible matrices, we develop a spectral theory (in the sense of Sacker and Sell) for the notion of a nonuniform exponential dichotomy with an arbitrarily small nonuniform part. We emphasize that this notion is ubiquitous in the context of ergodic theory, unlike the notion of a uniform exponential dichotomy. In particular, we show that each Lyapunov exponent belongs to one interval of the spectrum. We also consider a class of sufficiently small nonlinear perturbations of a linear dynamics satisfying a nonuniform bounded growth condition and we show that each solution is either eventually zero or the Lyapunov exponents belong to one interval of the spectrum.

Lyapunov theorems for exponential dichotomies in Hilbert spaces

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

Tempered exponential dichotomies: admissibility and stability under perturbations

ABSTRACTWe give a characterization of the notion of a tempered exponential dichotomy on a Banach space in terms of an admissibility property. We note that for a linear cocycle over a measure-preserving transformation satisfying a certain integrability assumption, it follows from the multiplicative ergodic theorem that the dynamics admits a tempered exponential dichotomy if and only if all Lyapunov exponents are non-zero almost everywhere. As a consequence of our approach, we give a new proof of the robustness property of the notion of a tempered exponential dichotomy under sufficiently small linear perturbations and we establish a version of the Grobman–Hartman theorem yielding the existence of topological conjugacies between a linear dynamics with a tempered exponential dichotomy and any sufficiently small nonlinear perturbation. In addition, we show that the conjugacy maps vary continuously with the perturbation.

Stable invariant manifolds with impulses and growth rates

This paper is devoted to the study of stable and unstable behaviors with growth rates $e^{c\rho(t)}$ for impulsive differential equations, where $\rho(t)$ is some increasing continuous function. By the techniques of impulsive analysis, we obtain the existence of stable invariant manifolds for the impulsive perturbed equation provided that the linear equation admits a ρ-nonuniform exponential dichotomy and f, g are sufficiently small Lipschitz perturbation. We also consider the case of exponential contraction and show that the asymptotic stability persists under sufficiently small nonlinear perturbations. In addition, we study how the manifolds vary with the perturbations.

Existence results for a nonlinear transmission problem

Let Ωo and Ωi be open bounded regular subsets of Rn such that the closure of Ωi is contained in Ωo. Let fo be a regular function on ∂Ωo and let F and G be continuous functions from ∂Ωi×R to R. By exploiting an argument based on potential theory and on the Leray–Schauder principle we show that under suitable and completely explicit conditions on F and G there exists at least one pair of continuous functions (uo,ui) such that{Δuo=0in Ωo∖clΩi,Δui=0in Ωi,uo(x)=fo(x)for all x∈∂Ωo,uo(x)=F(x,ui(x))for all x∈∂Ωi,νΩi⋅∇uo(x)−νΩi⋅∇ui(x)=G(x,ui(x))for all x∈∂Ωi, where the last equality is attained in certain weak sense. A simple example shows that such a pair of functions (uo,ui) is in general neither unique nor locally unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we prove the existence of at least one classical solution which is in addition locally unique.

Stability of L1 contractions

AbstractThe notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.