Non-resonant invariant foliations of quasi-periodically forced systems

Abstract

We show the existence and uniqueness of invariant foliations about invariant tori in analytic discrete-time dynamical systems. The parametrisation method is used prove the result. Our theory is a foundational block of data-driven model order reduction, that can only be carried out using invariant foliations. The theory is illustrated by two mechanical examples, where instantaneous frequencies and damping ratios are calculated about the invariant tori.

Figures (8)

• Figure 1: Invariant foliation about an invariant closed curve (torus), which is shown in black. The invariance equation maps each red curve into another red curve, which all lie on a green surface as parametrised by the $\boldsymbol{\theta}$ variable.
• Figure 2: A planar single mass oscillator in its equilibrium configuration. Gravity is ignored. The parameters are dimensionless $m=1$, $d=1$, $k_{1}=1$, $k_{2}$, $c_{1}=0.06$,$c_{2}=0.12$ and the forcing is $\boldsymbol{f}=A\left(\cos\left(\omega t+\pi/3\right),\cos\omega t\right)$, where $A$ is the forcing amplitude and $\omega$ is the forcing frequency.
• Figure 3: a) The invariant torus and the invariant manifold at one point along the torus. b) The spectrum of the linear dynamics about the torus. The stars denote the representative eigenvalues as selected by (\ref{['eq:NUM-evsel']}), the triangle denotes the eigenvalue use for model reduction. c) An incomplete representation of the invariant vector bundle: three out of four coordinates of the first vector that spans the two-dimensional invariant vector bundle.
• Figure 4: Instantaneous frequency b,e) and damping ratios c,f) about the invariant torus. The relative errors are shown in panels a,d). The green lines are calculated for the unforced system, the purple lines for the forced system with $A=0.03$. The calculations are carried out directly for the vector field (\ref{['eq:onemass-ode']}) (ODE), the generated Poincare map with sampling period $\Delta t=0.8$ (MAP) or through the invariant foliations of the Poincare map (FOIL). All polynomials were of order $7$ and the Fourier collocation used $\ell=7$ harmonics.
• Figure 5: A two-mass oscillator. The parameters are dimensionless $m=1$, $d=1$, $k_{1}=1$, $k_{2}=\sqrt{3}-A/2\sin^{2}\omega t-A/4\sin\omega t$, $k_{3}=k_{4}=0.02$, $c_{1}=0.03$,$c_{2}=0.04$ and $\omega=0.76$ is the forcing frequency.

Theorems & Definitions (8)

• Definition 1
• Remark 2
• Corollary 3
• Corollary 4
• Definition 5
• Theorem 6
• Theorem 7
• Theorem 8