Analytic Prediction of Isolated Forced Response Curves from Spectral Submanifolds

Sten Ponsioen; Tiemo Pedergnana; George Haller

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Analytic Prediction of Isolated Forced Response Curves from Spectral Submanifolds

Sten Ponsioen, Tiemo Pedergnana, George Haller

arXiv:1812.06664v1Dec 17, 2018math.DSnlin.CD

TL;DR

Isolated forced responses in nonlinear multi-degree-of-freedom systems are difficult to detect because standard continuation and frequency sweeps generally miss detached branches. The paper uses time-periodic spectral submanifolds to derive analytic forced-response and isola predictions, with higher-order refinements that remain accurate for high-dimensional systems where numerical continuation is infeasible.

Problem
Isolas are difficult to identify, and a general analytical criterion for predicting them in multi-degree-of-freedom systems without costly numerical simulations has been unavailable.
Method
The method uses two-dimensional time-periodic SSM reduced dynamics to extract forced-response curves and analytically predict isolas, with refinements generated by SSMtool.
Results
The SSM-based predictions remain accurate for a 100-dimensional system, including the predicted unstable-FRC frequency region confirmed by a full numerical frequency sweep.
Takeaways & Limitations
SSM reductions provide analytic forced-response and isola predictions for arbitrary multi-degree-of-freedom mechanical systems without costly full-system numerical simulations.
Takeaways & Limitations
For high-dimensional systems, numerical continuation becomes unfeasible, so verification relies on discrete periodic responses from long-term integration rather than a continuous full-system FRC.

Abstract

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We show how spectral submanifold theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of isolated forced responses that will generally be missed by numerical continuation techniques. Our analytic predictions can be refined to arbitrary precision via an algorithm that does not require the numerical solutions of the mechanical system. We illustrate all these results on low- and high-dimensional nonlinear vibration problems. We find that our SSM-based forced-response predictions remain accurate in high-dimensional systems, in which numerical continuation of the periodic response is no longer feasible.

1 Introduction

The paper addresses the difficulty of detecting isolated forced responses in nonlinear multi-degree-of-freedom systems, where continuation and frequency sweeps generally miss detached branches. It develops an SSM-based reduced-order methodology for analytically predicting these isolas and refining the predictions without solving the full mechanical system.

Isolas can merge with the main FRC after small forcing-amplitude changes, causing unexpected increases in response amplitude.
Numerical continuation and frequency sweeps generally miss isolas because they start from or remain on non-isolated response branches.
Existing analytical studies focused mainly on specific low-dimensional systems or restricted nonlinearities, leaving no general criterion for multi-degree-of-freedom systems.
The proposed approach uses mathematically rigorous spectral submanifolds whose two-dimensional reduced dynamics provide exact single-degree-of-freedom models for individual vibration modes.
Cubic-order SSM dynamics yields hand-computable first-order isola predictions, while higher-order refinements are recursively constructible in SSMtool.
System set-up: The systems considered are periodically forced nonlinear mechanical models with analytic nonlinearities and forcing independent of positions and velocities.

3 Non-autonomous spectral submanifolds and their reduced dynamics

The paper constructs time-periodic SSMs and reduces the forced dynamics to a two-dimensional polar system whose fixed points define the forced-response curve. The resulting zero-level-set formulation also predicts isolated branches that numerical continuation would require prior knowledge to detect.

The SSM is the smoothest invariant manifold perturbing from a selected spectral subspace under periodic forcing, with its construction based on non-resonant modal spectra.
Under non-resonance conditions, a unique two-dimensional, time-periodic, analytic SSM exists and supports polynomial reduced dynamics satisfying an invariance equation.
A time-periodic SSM can contain multiple limit cycles at one forcing frequency, including cycles belonging to an isola.
In polar coordinates, the reduced radial dynamics decouples from the phase in the unforced limit, so nontrivial periodic orbits arise only after forcing is introduced.
The forced-response curve is obtained from the zero-level set of the reduced amplitude equation, with two K± segments meeting where the square-root discriminant vanishes.
The zero-level set can predict isolas, whereas numerical continuation must start on an isola and therefore assumes prior knowledge of the isolated branch.

4 Analytic criterion for isolas

The paper gives an analytic criterion linking non-spurious transverse zeros of the reduced backbone function a(ρ) to nearby isolas. Higher-order approximations test whether zeros are genuine, while cubic-order formulas additionally predict disconnection and merger with the main FRC.

A non-spurious transverse zero ρ0 of a(ρ) implies an isola near the damped backbone curve for sufficiently small forcing amplitude.
Increasing Taylor-series order distinguishes genuine zeros from spurious ones: genuine zeros remain bounded away from the convergence circle, whereas spurious zeros approach it.
The cubic-order approximation provides an analytically computable isola criterion after truncating the SSM parameterization and reduced dynamics.
Under the stated non-spuriousness and Re(γ1) > 0 conditions, the cubic criterion predicts an isola near the damped backbone curve.
The same criterion gives forcing-amplitude conditions for the isola to remain disconnected from, and approximately merge with, the main FRC.

5 Numerical Examples

The examples demonstrate that SSM-based reduced dynamics analytically predicts isolated forced responses and their mergers in low- and high-dimensional nonlinear systems, with accuracy verified against full-system computations where feasible.

Example 5.1.1: A non-spurious transverse zero of a(ρ) establishes an isola near the corresponding autonomous backbone amplitude and predicts its approximate merger forcing amplitude.The zero is verified by persistence under increasing Taylor orders and by its location within the convergence domain.
Example 5.1.1: The two-dimensional SSM-reduced forced response curve perfectly predicts the full two-degree-of-freedom system’s isola-merger behavior near the analytically predicted forcing amplitudes.For ε = 0.0027 the isola remains separate, whereas for ε = 0.0029 it has merged with the main branch.
Discretized Bernoulli beam: For the discretized Bernoulli beam, the SSM reduction predicts an isola and its merger with the main FRC in a 100-dimensional system where numerical continuation is infeasible.A discrete full-system frequency sweep confirms the predicted unstable frequency region.

6 Conclusion

The paper uses exact reduced dynamics on two-dimensional time-periodic SSMs to extract FRCs and predict isolas in multi-degree-of-freedom mechanical systems without costly numerical simulations. Symbolic dependence on forcing amplitude exposes isolas that continuation can miss, while high-dimensional beam results remain accurate against full-system frequency sweeps.

Conclusion: Exact reduced dynamics on two-dimensional time-periodic SSMs extracts FRCs and predicts isolas for arbitrary multi-degree-of-freedom mechanical systems without costly numerical simulations.A cubic-order approximation gives analytic predictions, and higher-order refinements are available through SSMtool.
Conclusion: A discrete frequency sweep verifies the SSM prediction for the full 100-dimensional system, including the frequency region where the FRC becomes unstable.Continuous numerical continuation is infeasible at this dimensionality.
Conclusion: Because the reduced equations depend symbolically on forcing amplitude, they approximate FRCs across forcing amplitudes and reveal isolas through transverse intersections of reduced zero-level sets.Numerical continuation generally misses these branches because it must be initialized on an isolated solution branch.

A Coefficient equations for the the non-autonomous SSMs

The appendix derives coefficient equations for the non-autonomous SSM by expanding the embedding and reduced dynamics in forcing amplitude and parameterization coordinates. Resonant terms are identified and transferred to the reduced dynamics to avoid small denominators near resonance.

Coefficient expansion: The non-autonomous SSM construction expands W(s, φ) and R(s, φ) in ε, then Taylor expands their autonomous and non-autonomous parts in the parameterization coordinates s.The autonomous SSM and reduced dynamics are solved first before computing the non-autonomous correction.
Coefficient equations: For each multi-index power and coordinate row, collecting equal-order terms in the non-autonomous invariance equation produces recursive coefficient equations.The appendix introduces multi-index notation and derives the coefficient equation for the kth-power term of the ith row.
Solution cases: The |k| = 0 and |k| > 0 cases are solved separately, with the latter using the general solution of the non-autonomous invariance equation.The forcing terms are O(ε), so the leading-order terms are independent of φ.
Resonant terms: Near resonance, resonant coefficients that grow for small damping and Ω ≈ Imλ1 are removed from the SSM embedding and included in the non-autonomous reduced dynamics.This choice avoids small denominators in W1(s, φ).

B Proof of Theorem 3.2

The proof derives the O(ε) reduced dynamics on a two-dimensional time-periodic SSM and converts its complex coordinates to polar coordinates relative to the forcing phase. Splitting the resulting equation into real and imaginary parts yields Theorem 3.2.

Complex reduced dynamics: The O(ε) reduced dynamics is first written in complex coordinates, whose two rows are complex conjugates by construction.This conjugate structure reflects the two-dimensional real SSM associated with a complex-conjugate modal pair.
Polar transformation: Introducing s1 = ρe^iθ and the phase difference ψ = θ − φ transforms the reduced dynamics into polar amplitude and phase equations.The equation is divided by e^iθ before the phase-coordinate substitution.
Theorem derivation: Separating the transformed equation into real and imaginary parts produces the result stated in Theorem 3.2.The proof closes after this decomposition.

C Extracting the forced response curve

The forced response curve is obtained from the common zeros of two reduced equations, whose transverse intersection forms a one-dimensional manifold that projects into amplitude–frequency space.

C Extracting the forced response curve: The zero problem defines the forced response curve as a one-dimensional submanifold of R3 near each regular solution.This follows when F(p)=0 and the Jacobian of F is surjective.
C Extracting the forced response curve: The two zero-level sets of F1 and F2 are locally two-dimensional submanifolds whose transverse intersection yields the forced response curve.The curve is then represented in the (Ω, ρ)-plane by projection.

D Proof of Theorem 3.3

The proof converts the phase dependence into an algebraic problem using the tangent half-angle substitution, then derives a quadratic equation whose roots determine the reduced forced-response amplitudes.

D Proof of Theorem 3.3: The tangent half-angle substitution rewrites sin(ψ) and cos(ψ) as rational functions of K.These identities eliminate the trigonometric phase terms from the reduced equations.
D Proof of Theorem 3.3: Substitution into the reduced amplitude equation produces a quadratic equation in K with coefficients determined by a(ρ), ε, f1(ρ, Ω), and f2(ρ, Ω).The resulting polynomial is then used to solve the phase condition algebraically.
D Proof of Theorem 3.3: The quadratic relation and trigonometric identities yield the amplitude characterization stated in Theorem 3.3.

E Proof of Theorem 4.1

Theorem 4.1 continues a non-spurious transverse zero on the autonomous backbone into a two-dimensional solution manifold, showing that an isola emerges for positive forcing.

E Proof of Theorem 4.1: A non-spurious transverse zero ρ0 of a(ρ), with a nonzero derivative, is considered on the autonomous backbone curve at ε=0.The backbone point satisfies Ω0=b(ρ0) and has constant phase.
E Proof of Theorem 4.1: Invertibility of the Jacobian with respect to ρ and Ω allows the solution to continue locally as a two-dimensional submanifold parameterized by ψ and ε.
E Proof of Theorem 4.1: For ε>0, an isola is born from the non-trivial transverse zero on the autonomous backbone curve.At fixed forcing amplitude, the isola is parameterized by ψ.

F Proof of Theorem 4.2

Theorem 4.2 identifies how zeros of the autonomous function generate folding points and isolas, including their separation from or merger with the main forced-response branch.

F Proof of Theorem 4.2: Zeros making the square-root term vanish lie on both the forced-response and autonomous backbone curves, where two FRC segments meet in a fold over Ω.
F Proof of Theorem 4.2: A zero of a(ρ) with nonzero derivative can generate an isola from a non-trivial transverse zero on the autonomous backbone.The result assumes the cubic-order zero is non-spurious.
F Proof of Theorem 4.2: For sufficiently small positive ε, three intersections with ±ε∥c1,0∥ produce three forced-response folding points over the Ω direction.
F Proof of Theorem 4.2: The folding points identify the main-branch maximum, isola minimum, and isola maximum amplitudes.ρa0 is the main FRC maximum, ρa1 the isola minimum, and ρb1 the isola maximum.
F Proof of Theorem 4.2: When the maximum amplitude of the main FRC branch meets the isola minimum, the isola becomes disconnected from the main FRC.