Infinite-dimensional Dynamical Systems Research Articles

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1988, xvi +500pp.

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1988, xvi +500pp.

Existence of SRB measures for a class of partially
 hyperbolic attractors in banach spaces

In this paper, we study the existence of SRB measures for infinite dimensional dynamical systems in a Banach space. We show that if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has an SRB measure.

Collective phase reduction of globally coupled noisy dynamical elements.

We formulate a theory for the collective phase reduction of globally coupled noisy dynamical elements exhibiting macroscopic rhythms. We first transform the Langevin-type equation that represents a group of globally coupled noisy dynamical elements into the corresponding nonlinear Fokker-Planck equation and then develop the phase reduction method for limit-cycle solutions to the nonlinear Fokker-Planck equation. The theory enables us to describe the collective dynamics of a group of globally coupled noisy dynamical elements by a single degree of freedom called the collective phase. As long as the group collectively exhibits macroscopic rhythms, the theory is applicable even when the coupling and noise are strong; it is also independent of the assumption that each element of the group is a self-sustained oscillator. We also provide a simple and accurate numerical algorithm for the collective phase description method and numerically illustrate the theory using a group of globally coupled noisy FitzHugh-Nagumo elements.

Modeling and analysis of the secondary routine dose against measles in China

Measles remain to be an important global public health issue in China. In spite of large coverage rates of the first dose of Measles mumps rubella (MMR) combination vaccines (MMR1), large numbers of measles cases continue to be reported in China in recent years due to the high incidence and the low coverage of the second MMR vaccine dose (MMR2). This paper is devoted to modeling the combined effects of MMR1 and MMR2 coverage rates on the controlling of measles. To do that, we propose and study a robust time-delayed compartment measles infection model where MMR2 is followed after a fixed time interval of MMR1, and the combined elements of infection and mass immunization are also considered. By using the methods of Lyapunov functional and the uniform theory for infinite-dimensional dynamical systems, a threshold dynamics determined by the basic reproduction number Re_{0} is established: the measles can be eradicated if Re_{0}<1, whereas the disease persists if Re_{0}>1. Moreover, it is shown that the endemic equilibrium is locally asymptotically stable once Re_{0}>1. Numerical simulations are performed to support the theoretical results and to consider the effects of MMR2 on the controlling of measles. Our results show that to eliminate measles in China, we should have MMR1 coverage rates larger than 88.01% based on perfect MMR2 coverage, and have MMR2 coverage rates larger than 92.63% based on perfect MMR1 coverage; Moreover, our simulations suggest that there is a risk that paradox of vaccination against measles in China may occur: that is, the final size of infected individuals may even increase in spite of the increase of MMR2 coverage rates.

Necessary and sufficient conditions for the existence of time-dependent global attractor and application

In this paper, we are concerned with infinite dimensional dynamical systems in time-dependent space. First, we characterize some necessary and sufficient conditions for the existence of the time-dependent global attractor by using a measure of noncompactness. Then, we give a new method to verify the sufficient condition. As a simple application, we prove the existence of the time-dependent global attractor for the damped equation in strong topological space.

Almost-periodic solutions of an almost-periodically forced wave equation

This work focuses on a wave equation with almost-periodic forcing ψ(ωt,x) subject to Dirichlet boundary conditions. For the equation we obtain almost-periodic solutions corresponding to full-dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on the reducibility via an improved KAM theory.

Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"

Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"

Asymptotic Analysis for Randomly Forced MHD

We consider the three-dimensional magnetohydrodynamics (MHD) equations in the presence of a spatially degenerate stochastic forcing as a model for magnetostrophic turbulence in the Earth's fluid core. We examine the multiparameter singular limit of vanishing Rossby number $\varepsilon$ and magnetic Reynold's number $\delta$, and establish that (i) the formal limit (with $\varepsilon=\delta=0$), a stochastically driven active scalar equation, possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converges weakly, as $\varepsilon,\delta \rightarrow 0$, to the unique invariant measure of the limit equation. This latter convergence result does not require any conditions on the relative rates at which $\varepsilon, \delta$ decay. Our analysis of the limit equation relies on a recently developed theory of hypoellipticity for infinite-dimensional stochastic dynamical systems. We carry out a detailed study of the interactions between ...

Integrable Nonlocal Nonlinear Equations

A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced “potential” is symmetric thus the nonlocal NLS equation is also symmetric. In this paper, new reverse space‐time and reverse time nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space‐time, and in some cases reverse time, nonlocal NLS, modified Korteweg‐deVries (mKdV), sine‐Gordon, (1 + 1) and (2 + 1) dimensional three‐wave interaction, derivative NLS, “loop soliton,” Davey–Stewartson (DS), partially symmetric DS and partially reverse space‐time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space‐time and reverse time nonlocal discrete nonlinear Schrödinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlevé type equations are derived from the reverse space‐time and reverse time nonlocal NLS equations.

Standing Waves in Near-Parallel Vortex Filaments

A model derived in (Klein et al., J Fluid Mech 288:201–248, 1995) for n near-parallel vortex filaments in a three dimensional fluid region takes into consideration the pairwise interaction between the filaments along with an approximation for motion by self-induction. The same system of equations appears in descriptions of the fine structure of vortex filaments in the Gross–Pitaevski model of Bose–Einstein condensates. In this paper we construct families of standing waves for this model, in the form of n co-rotating near-parallel vortex filaments that are situated in a central configuration. This result applies to any pair of vortex filaments with the same circulation, corresponding to the case n = 2. The model equations can be formulated as a system of Hamiltonian PDEs, and the construction of standing waves is a small divisor problem. The methods are a combination of the analysis of infinite dimensional Hamiltonian dynamical systems and linear theory related to Anderson localization. The main technique of the construction is the Nash–Moser method applied to a Lyapunov–Schmidt reduction, giving rise to a bifurcation equation over a Cantor set of parameters.

On the computation of attractors for delay differential equations

In this work we present a novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems. It extends a classical subdivision technique [Dellnitz/Hohmann 1997] for the computation of such objects of finite dimensional systems to the infinite dimensional case by utilizing results on embedding techniques for infinite dimensional systems. We show how to implement this approach for the analysis of delay differential equations and illustrate the feasibility of our implementation by computing invariant sets for three different delay differential equations.

Global Attractors in Impulsive Infinite-Dimensional Systems

We study the existence of global attractors in discontinuous infinite-dimensional dynamical systems, which may have trajectories with infinitely many impulsive perturbations. We also select a class of impulsive systems for which the existence of global attractors is proved for weakly nonlinear parabolic equations.

Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions

For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.

Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$

We consider a damped forced nonlinear Schrodinger-Poisson system in $R^3$. This provides us with an infinite-dimensional dynamical system in the energy space $H^1(R^3)$. We prove the existence of a finite dimensional global attractor for this dynamical system.

SRB measures for a class of partially hyperbolic attractors in Hilbert spaces

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is Hölder continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and the splitting is Hölder continuous, then there exist at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set.

Numerical algorithms for stationary statistical properties of dissipative dynamical systems

It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. The main concern of this manuscript is numerical methods for dissipative chaotic infinite dimensional dynamical systems that are able to capture the stationary statistical properties of the underlying dynamical systems. We first survey results on temporal and spatial approximations that enjoy the desired properties. We then present a new result on fully discretized approximations of infinite dimensional dissipative chaotic dynamical systems that are able to capture asymptotically the stationary statistical properties. The main ingredients in ensuring the convergence of the long time statistical properties of the numerical schemes are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors of the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval $[0,1]$ modulo an initial layer, uniformly with respect to initial data from the union of the global attractors. The two conditions are reminiscent of the Lax equivalence theorem where stability and consistency are needed for the convergence of a numerical scheme. Applications to the complex Ginzburg-Landau equation and the two-dimensional Navier-Stokes equations in a periodic box are discussed.

Long-time convergence of numerical approximations for 2D GBBM equation

We study the long-time behavior of the finite difference solution to the generalized BBM equation in two space dimensions with dirichlet boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems. Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.

On the equations of the hydrodynamic type

The conditions of stability of ‘pure’ equilibrium states of finite-dimensional dynamical systems of the hydrodynamic type are indicated. Integrable chains allowing full sets of quadratic first integrals are found. The conditions of existence of invariant Gaussian measures in the infinite-dimensional case are discussed. The existence of such measures makes it possible to establish the properties of recurrence of solutions of infinite-dimensional dynamical systems of the hydrodynamic type.

From phenomena of synchronization to exact synchronization and approximate synchronization for hyperbolic systems

In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations, and will be connected with the control theory via boundary controls in a finite time interval. More precisely, various kinds of exact boundary synchronization and approximate boundary synchronization will be introduced and realized by means of fewer boundary controls for a coupled system of wave equations with Dirichlet boundary controls. Moreover, as necessary conditions for various kinds of approximate boundary synchronization, criteria of Kalman's type are obtained. Finally, some prospects will be given.

High–low frequency slaving and regularity issues in the 3D Navier–Stokes equations

The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the $3D$ Navier-Stokes equations. A set of dimensionless frequencies $\{\tilde{\Omega}_{m}(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the $\tilde{\Omega}_{m}$ ($m>1$) are slaved to $\tilde{\Omega}_{1}$ (the global enstrophy) in the form $\tilde{\Omega}_{m} = \tilde{\Omega}_{1}\mathcal{F}_{m}(\tilde{\Omega}_{1})$. This is shaped by the constraint of two Holder inequalities and a time average from which emerges a form for $\mathcal{F}_{m}$ which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions $1 \leq \lambda_{m}(\tau) \leq 4$, where curves of constant $\lambda_{m}$ form the boundaries between tongue-shaped regions. In regions where $2.5 \leq \lambda_{m} \leq 4$ and $1 \leq \lambda_{m} \leq 2$ the Navier-Stokes equations are shown to be regular\,: numerical simulations appear to lie in the latter region. Only in the central region $2 < \lambda_{m} < 2.5$ has no proof of regularity been found.