A construction of stable subharmonic orbits in monotone time-periodic dynamical systems

We consider a family of abstract semilinear elliptic-like equationsB(t,u _o (t))=0 for an unknown functionu ₀ (t) parametrized by the time-variablet0 and valued in a Banach spaceX. Suppose that bothB(.,u) andu ₀ areT-periodic in timet, and each Fréchet derivative generates an exponentially decaying, analyticC ₀-semigroup inX. We show that, for every small >0, the abstract parabolic-like evolution equationdu /dt=B(t,u _(t) ),t0, has a linearly stableT-periodic solutionu nearu ₀. Given any integern2, we construct examples ofB andu ₀ such that the minimum periods ofB(.,u) andu ₀, respectively, are=T/n andT. Thenu (t), t0, is alinearly stable subharmonic orbit of minimum periodT for our -periodic evolution equation. The corresponding dynamical systems are strongly monotone.This work was supported in part by, Vanderbilt University Research Council.     

Quasisolutions and dynamics of time-periodic nonquasimonotone reaction-diffusion systems

In this paper, using the upper-lower solution method, we investigate the properties of periodic quasisolutions for time-periodic nonquasimonotone reaction-diffusion systems, and present some existence, asymptotic behavior results for two ecological models.     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations). Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 115–126, January, 1998.     

On stabilization of discrete monotone dynamical systems

A stabilization theorem for discrete strongly monotone and nonexpansive dynamical systems on a Banach lattice is proved. This result is applied to a periodic-parabolic semilinear initial-boundary value problem to show the convergence of solutions towards periodic solutions.     

On the linearization of infinite-dimensional random dynamical systems

We present a new version of the Grobman–Hartman's linearization theorem for random dynamics. Our result holds for infinite-dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the nonlinear perturbations, we do not require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we do not prescribe any behavior for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are Hölder continuous when restricted to bounded subsets of the space.     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations).     

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed.     

Stability and convergence in discrete convex monotone dynamical systems

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on n letters, where n is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.     

From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.     

Poisson stable motions of monotone nonautonomous dynamical systems

In this paper, we study the Poisson stability(in particular, stationarity, periodicity, quasiperiodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations(ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.     

LDG method for reaction-diffusion dynamical systems with time delay

In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction-diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.     

Massera's theorem for monotone dynamical systems in three dimensions

In this paper we give a variant of the classical second Massera's theorem for three-dimensional periodic systems, that satisfy different types of monotonicity assumptions.     

Attractors of nonlinear dynamical systems with a weakly monotone measure

This paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure. It is assumed that the measure of a set of points in the phase space is increasing along the flow. We prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions. For a system of autonomous ordinary differential equations, we present attractivity conditions in terms of the divergence with a density function. Unlike previous results in the literature, our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure. As an example, the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions.     

Some remarks on a boundedness assumption for monotone dynamical systems

We study the consequences of the assumption that all forward orbits are bounded for monotone dynamical systems. In particular, it turns out that this assumption has more implications than is immediately apparent.

     

Linear time-periodic dynamical systems: an H2 analysis and a model reduction framework

Linear time-periodic (LTP) dynamical systems frequently appear in the modelling of phenomena related to fluid dynamics, electronic circuits and structural mechanics via linearization centred around known periodic orbits of nonlinear models. Such LTP systems can reach orders that make repeated simulation or other necessary analysis prohibitive, motivating the need for model reduction. We develop here an algorithmic framework for constructing reduced models that retains the LTP structure of the original LTP system. Our approach generalizes optimal approaches that have been established previously for linear time-invariant (LTI) model reduction problems. We employ an extension of the usual H₂ Hardy space defined for the LTI setting to time-periodic systems and within this broader framework develop an a posteriori error bound expressible in terms of related LTI systems. Optimization of this bound motivates our algorithm. We illustrate the success of our method on three numerical examples.     

Saddle-point behavior for monotone semiflows and reaction-diffusion models

The saddle-point behavior is established for monotone semiflows with weak bistability structure and then these results are applied to three reaction-diffusion systems modeling man-environment-man epidemics, single-loop positive feedback and two species competition, respectively.