On the theory of invariant manifolds
Analysing nonlinear dynamical systems with the use of invariant manifolds is an idea which has been intensively pursued since results towards the existence of such objects were provided in the middle of the 20th century. In contrast to single trajectory analysis, these objects do give conclusive information about the phase space structure. They are computationally robust also for systems with high sensitivity on initial conditions and form physically observable structures.
An early result on invariant manifolds was provided by Lyapunov in 1907, who proved the existence of continuous Lyapunov Subcenter Manifolds (LSM) for pairwise non-resonant spectra of Hamiltonian systems on center subspaces [1]. His results were later generalised by Kelley in 1967 who managed to establish differentiability of these LSMs under some slightly more restrictive assumptions [2].
For hyperbolic fixed points, the theorem of Hartman and Grobman guarantees the local topological equivalence of the full nonlinear dynamical system to its linear part [3]. This put the intuition that under the addition of nonlinearities the local system behaviour and stability is strongly related to the linear dynamics on solid ground. As the linear eigenvalues on the center subspace do not give any dynamics, they are determined entirely by the nonlinearity which is why the theorem only holds for fixed points of hyperbolic type.
An invariant extension to the flat center subspace is given by the center manifold. Its existence and stable as well as unstable counterparts along with their degree of smoothness were provided in the center manifold theorem by Kelley in 1967 [2]. In particular for center - stable systems this center manifold received attention as a candidate to analyse low dimensional models of a system to capture the long term behaviour of autonomous dynamical systems around their trivial fixed point.
In 1971, Fenichel put forward the concept of Normally Hyperbolic Invariant Manifolds. The name refers to the local rate of normal attraction to these manifolds. They form a generalisation of hyperbolic fixed points, as at any point on the manifold a clear distinction of dynamical regimes holds. Trajectories which start close to such manifolds decay exponentially fast to the manifold, in fact also exponentially faster than timescales of the dynamics on the manifold itself. Once close enough, the orbits synchronise to those on the manifold [4]. In latter work Fenichel provided persistence results of such manifolds under small perturbations, guaranteeing the existence of invariant hyperbolic structures in perturbed settings [5] [6]. An extensive review can be found in the book of Wiggins [7]. These manifolds are of high interest for model reduction, as obtaining the ROM on such a manifold promises to be an excellent candidate for the full system dynamics around a fixed point.
A proposal of similar spirit was made by Rosenberg in 1962 [8]. He put forward the notion of a nonlinear analogon of normal modes, the Nonlinear Normal Mode (NNM). A slightly relaxed version of his definition is given by the general closure of a periodic orbit. This includes invariant sets such as fixed points and invariant tori. The concept was generalised in 1993 by Shaw and Pierre, who defined NNMs as invariant manifolds which are tangent to spectral subspaces of the linear part of a dynamical system [9]. In a conservative setting these definitions smoothly transition into each other, as locally Shaw-Pierre type invariant manifolds are filled by Rosenberg Type modes. Such a connection does not exist in the dissipative setting, as Rosenberg Type NNMs appear isolated in phase space. Shaw Pierre Type manifolds are not bounded, so the two notions refer to clearly distinct objects.
An existence of Shaw Pierre Type NNMs is furthermore not trivial, and in particular they are in general not unique. This issue was addressed by Haller and Ponsioen recently in 2016 by their introduction of the Spectral Submanifold (SSM) [10]. It is the unique smoothest Shaw-Pierre Type NNM and exists under some non-resonance conditions on the linear spectrum, its existence is established via the parametrization method for invariant manifolds [11] [12]. By invariance of the SSM the reduced dynamics on this resonant manifold provide an exact reduced order model. The slowest SSM of a stable fixedpoint attracts neighbouring trajectories and is thus ideal for model reduction in autonomous systems. In a non-autonomous setting with small enough amplitude the tivial fixed point of a dynamical system perturbs into an invariant torus [13]. The SSM is then attached to this torus. Resonant steady state responses due to external and parametric excitation may then be obtained from the ROM of the resonant SSM. In order to capture internal, super and subharmonic resonances an a-priori spectral analysis of a truncated version of the spectrum of the dynamical system allows to construct the SSM over the appropriate spectral subspaces.
A first automated toolbox to compute these SSMs was published by Ponsioen and Haller in 2018 [14]. It allows for the computation of forced response curves via the ROM on two-dimensional SSMs and the extraction of backbone curves from the autonomous manifold. To do so the system has to be transferred to modal coordinates. The computational restrictions due to this are stressed by Jain and Haller [15]. As an alternative they put forward a routine for computing SSMs of arbitrary dimensions directly from the physical coordinates. They thus circumvent a full modal analysis which is illusive for FE-models with large systems sizes. In parallel the notion of Direct Normal Forms appeared in literature [16]. It features a direct computation of the normal form reduced dynamics from second order mechanical systems and boils down to an SSM computation directly from physical coordinates as proposed by Jain and Haller.