Inertial manifold

In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems. Inertial manifolds are finite-dimensional, smooth, invariant manifolds that contain the global attractor and attract all solutions exponentially quickly. Since an inertial manifold is finite-dimensional even if the original system is infinite-dimensional, and because most of the dynamics for the system takes place on the inertial manifold, studying the dynamics on an inertial manifold produces a considerable simplification in the study of the dynamics of the original system. In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems. Inertial manifolds are finite-dimensional, smooth, invariant manifolds that contain the global attractor and attract all solutions exponentially quickly. Since an inertial manifold is finite-dimensional even if the original system is infinite-dimensional, and because most of the dynamics for the system takes place on the inertial manifold, studying the dynamics on an inertial manifold produces a considerable simplification in the study of the dynamics of the original system. In many physical applications, inertial manifolds express an interaction law between the small and large wavelength structures. Some say that the small wavelengths are enslaved by the large (e.g. synergetics). Inertial manifolds may also appear as slow manifolds common in meteorology, or as the center manifold in any bifurcation. Computationally, numerical schemes for partial differential equations seek to capture the long term dynamics and so such numerical schemes form an approximate inertial manifold. Consider the dynamical system in just two variables  p ( t ) {displaystyle p(t)} and  q ( t ) {displaystyle q(t)} and with parameter  a {displaystyle a} : Hence the long term behavior of the original two dimensional dynamical system is given by the 'simpler' one dimensional dynamics on the inertial manifold  M {displaystyle {mathcal {M}}} , namely  d p d t = a p − 1 1 + 2 a p 3 {displaystyle {frac {dp}{dt}}=ap-{frac {1}{1+2a}}p^{3}} . Let u ( t ) {displaystyle u(t)} denote a solution of a dynamical system. The solution  u ( t ) {displaystyle u(t)} may be an evolving vector in H = R n {displaystyle H=mathbb {R} ^{n}} or may be an evolving function in an infinite-dimensional Banach space  H {displaystyle H} . In many cases of interest the evolution of  u ( t ) {displaystyle u(t)} is determined as the solution of a differential equation in  H {displaystyle H} , say d u / d t = F ( u ( t ) ) {displaystyle {du}/{dt}=F(u(t))} with initial value u ( 0 ) = u 0 {displaystyle u(0)=u_{0}} .In any case, we assume the solution of the dynamical system can be written in terms of a semigroup operator, or state transition matrix, S : H → H {displaystyle S:H o H} such that u ( t ) = S ( t ) u 0 {displaystyle u(t)=S(t)u_{0}} for all times t ≥ 0 {displaystyle tgeq 0} and all initial values  u 0 {displaystyle u_{0}} .In some situations we might consider only discrete values of time as in the dynamics of a map. An inertial manifold for a dynamical semigroup  S ( t ) {displaystyle S(t)} is a smooth manifold  M {displaystyle {mathcal {M}}} such that The restriction of the differential equation  d u / d t = F ( u ) {displaystyle du/dt=F(u)} to the inertial manifold  M {displaystyle {mathcal {M}}} is therefore a well defined finite-dimensional system called the inertial system.Subtly, there is a difference between a manifold being attractive, and solutions on the manifold being attractive.Nonetheless, under appropriate conditions the inertial system possesses so-called asymptotic completeness: that is, every solution of the differential equation has a companion solution lying in  M {displaystyle {mathcal {M}}} and producing the same behavior for large time; in mathematics, for all  u 0 {displaystyle u_{0}} there exists  v 0 ∈ M {displaystyle v_{0}in {mathcal {M}}} and possibly a time shift  τ ≥ 0 {displaystyle au geq 0} such that dist ( S ( t ) u 0 , S ( t + τ ) v 0 ) → 0 {displaystyle { ext{dist}}(S(t)u_{0},S(t+ au )v_{0}) o 0} as  t → ∞ {displaystyle t o infty } . Researchers in the 2000s generalized such inertial manifolds to time dependent (nonautonomous) and/or stochastic dynamical systems (e.g.)