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Spectral submanifold

Indynamical systems, a spectral submanifold (SSM) is the unique smoothest invariant manifold serving as the nonlinear extension of a spectral subspace of a linear dynamical system under the addition of nonlinearities.^[2] 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.

Definition[edit]

Consider a nonlinear ordinary differential equation of the form

${\displaystyle {\frac {dx}{dt}}=Ax+f_{0}($x),\quad x\in \mathbb {R} ^{n},}

with constant matrix ${\displaystyle \ A\in \mathbb {R} ^{n\times n}}$ and the nonlinearities contained in the smooth function ${\displaystyle f_{0}={\mathcal {O}}(|x|^{2})}$.

Assume that ${\displaystyle {\text{Re}}\lambda _{j}<0}$ for all eigenvalues ${\displaystyle \lambda _{j},\ j=1,\ldots ,n}$of${\displaystyle A}$, that is, the origin is an asymptotically stable fixed point. Now select a span ${\displaystyle E={\text{span}}\,\{v_{1}^{E},\ldots v_{m}^{E}\}}$of${\displaystyle m}$ eigenvectors ${\displaystyle v_{i}^{E}}$of${\displaystyle A}$. Then, the eigenspace ${\displaystyle E}$ is an invariant subspace of the linearized system

${\displaystyle {\frac {dx}{dt}}=Ax,\quad x\in \mathbb {R} ^{n}.}$

Under addition of the nonlinearity ${\displaystyle f_{0}}$ to the linear system, ${\displaystyle E}$ 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 ${\displaystyle {\text{Re}}\lambda _{j}>0}$.

Existence[edit]

The spectral submanifold tangent to ${\displaystyle E}$ at the origin is guaranteed to exist provided that certain non-resonance conditions are satisfied by the eigenvalues ${\displaystyle \lambda _{i}^{E}}$ in the spectrum of ${\displaystyle E}$.^[3] In particular, there can be no linear combination of ${\displaystyle \lambda _{i}^{E}}$ equal to one of the eigenvalues of ${\displaystyle A}$ outside of the spectral subspace. If there is such an outer resonance, one can include the resonant mode into ${\displaystyle E}$ and extend the analysis to a higher-dimensional SSM pertaining to the extended spectral subspace.

Non-autonomous extension[edit]

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

${\displaystyle {\frac {dx}{dt}}=Ax+f_{0}($x)+\epsilon f_{1}(x,\Omega t),\quad \Omega \in \mathbb {T} ^{k},\ 0\leq \epsilon \ll 1,}

with ${\displaystyle f_{1}:\mathbb {R} ^{n}\times \mathbb {T} ^{k}\to \mathbb {R} ^{n}}$aquasiperiodic forcing term.^[4]

Significance[edit]

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.^[5] 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.^[6]

Furthermore these manifolds may also be extracted directly from trajectory data of a dynamical system with the use of machine learning algorithms.^[7]

See also[edit]

• Invariant manifold
• Nonlinear dimensionality reduction
• Lagrangian coherent structure

References[edit]

External links[edit]

• Tool for automated SSM computation