In dynamical systems, a spectral submanifold (SSM) is the unique smoothest

invariant manifold In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. Examples include the slow manifold, center manifold, stable manifold, stable manifold, unsta ...

serving as the nonlinear extension of a spectral subspace of a linear dynamical system under the addition of nonlinearities. SSM theory provides conditions for when invariant properties of eigenspaces of a linear dynamical system can be extended to a nonlinear system, and therefore motivates the use of SSMs in

nonlinear dimensionality reduction Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-d ...

Definition

Consider a nonlinear ordinary differential equation of the form :

\frac = Ax + f_0(x),\quad x\in \R^n,

with constant matrix

\ A\in \R^

and the nonlinearities contained in the smooth function

f_0 = \mathcal(, x, ^2)

. Assume that

\text \lambda_j < 0

for all eigenvalues

\lambda_j,\ j = 1,\ldots, n

A

, that is, the origin is an asymptotically stable fixed point. Now select a span

E = \text\, \

m

eigenvectors

v^E_

A

. Then, the eigenspace

E

is an invariant subspace of the linearized system :

\frac = Ax,\quad x\in \R^n.

Under addition of the nonlinearity

f_0

to the linear system,

E

generally perturbs into infinitely many invariant manifolds. Among these invariant manifolds, the unique smoothest one is referred to as the spectral submanifold. An equivalent result for unstable SSMs holds for

\text \lambda_j > 0

Existence

The spectral submanifold tangent to

E

at the origin is guaranteed to exist provided that certain non-resonance conditions are satisfied by the eigenvalues

\lambda^E_i

in the spectrum of

E

. In particular, there can be no linear combination of

\lambda^E_i

equal to one of the eigenvalues of

A

outside of the spectral subspace. If there is such an outer resonance, one can include the resonant mode into

E

and extend the analysis to a higher-dimensional SSM pertaining to the extended spectral subspace.

Non-autonomous extension

The theory on spectral submanifolds extends to nonlinear non-autonomous systems of the form :

\frac = Ax + f_0(x) + \epsilon f_1(x, \Omega t),\quad \Omega\in \mathbb^k,\ 0\le \epsilon \ll 1,

with

f_1 : \R^n \times \mathbb^k \to \R^n

a quasiperiodic forcing term.

Significance

Spectral submanifolds are useful for rigorous nonlinear dimensionality reduction in dynamical systems. The reduction of a high-dimensional phase space to a lower-dimensional manifold can lead to major simplifications by allowing for an accurate description of the system's main asymptotic behaviour. For a known dynamical system, SSMs can be computed analytically by solving the invariance equations, and reduced models on SSMs may be employed for prediction of the response to forcing. Furthermore these manifolds may also be extracted directly from trajectory data of a dynamical system with the use of machine learning algorithms.

References

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External links