Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

• Torus bifurcations, also called Neimark–Sacker bifurcations in a Poincaré map, create two-dimensional invariant tori carrying quasi-periodic responses.Source: §1 · p. 1
• Quasi-periodic responses have been reported in beams, plates, shells, and cyclic structures with internal resonances such as 3:1, 1:1, and 1:1:2.Source: §1 · p. 1
• Existing approaches include direct integration, shooting, harmonic balance, and boundary-value discretizations, but several require stability, convergence, or substantial computational effort.Source: §1 · pp. 1–2
• These numerical methods are effective for low-dimensional systems but impractical for high-dimensional finite-element models, motivating reduced-order modeling for efficient quasi-periodic response computation.Source: §1 · p. 2
• The study extends Part I, where periodic forced-response curves were computed by locating equilibria of reduced-order models on resonant SSMs.Source: §1 · pp. 2–3
• The paper uses SSM reduction to obtain two-dimensional tori from reduced-model limit cycles, infer their stability, analyze their bifurcations, and compute bifurcating three-dimensional tori.Source: §1 · pp. 2–3

System setup

• The system is modeled as a weakly forced, damped first-order dynamical system, with harmonically forced mechanical equations as a special second-order case.Source: §2.1 · p. 3
• Internal resonance is checked using approximate relationships among eigenvalues of the linearized system, such as a 1:3 resonance between two weakly damped modes.Source: §2.1 · p. 3
• The analysis excludes internal resonances between nonlinear frequencies in conservative systems because the paper assumes small forcing amplitudes and focuses on damped forced responses.Source: §2.1 · pp. 3–4

SSM definition and reduction

• A nonautonomous SSM is a time-periodic invariant manifold tangent to a selected master spectral subspace and smoothly perturbed by nonlinear terms and external forcing.Source: §2.2 · p. 4
• The SSM embedding maps reduced coordinates and forcing phase into the full phase space, while its reduced vector field governs the lower-dimensional dynamics.Source: §2.2 · p. 4
• The unique smoothest SSM is selected by the spectral quotient, whose value depends on the real parts of the linear eigenvalues.Source: §2.2 · p. 4

Computation of the resonant SSM

• The reduced dynamics are expanded in modal parameterization coordinates, with leading-order nonautonomous terms and an embedding map computed from the SSM invariance equations.Source: §2.3 · pp. 4–5
• The forcing amplitude enters through the external-force vector, and the embedding separates autonomous and nonautonomous contributions whose coefficients are computed with the reduced-model expansion.Source: §2.3 · p. 5

Assumptions

• The theorem applies for sufficiently small forcing amplitude under the stated internal- and external-resonance conditions.The resonant SSM is constructed under the internal resonance condition and forcing resonance with the master eigenvalues.Source: §3 · p. 5

Periodic-orbit reduction

• Fixed points of the leading-order reduced dynamics correspond to periodic solutions on the resonant SSM, with matching stability when hyperbolic.The coordinate transformations simplify the reduced dynamics and convert periodic-orbit computation into fixed-point location.Source: §3 · p. 6

Bifurcation correspondence

• A saddle-node bifurcation of a reduced fixed point corresponds to a saddle-node bifurcation of the associated periodic orbit on the SSM.Source: §3 · p. 6
• A Hopf bifurcation of a reduced fixed point corresponds to a torus bifurcation of the associated periodic orbit on the SSM.Source: §3 · p. 6

Computational significance

• SSM reduction avoids expensive monodromy calculations and numerical event-function issues by solving periodic orbits as fixed points.Source: §3 · pp. 6–7

Two-dimensional tori

• Each reduced-model limit cycle produces a two-dimensional invariant torus in the full system through the SSM mapping.A Hopf bifurcation of a reduced fixed point creates the limit cycle that initiates this construction.Source: §4.1 · p. 7

Two-dimensional torus bifurcations

• Reduced periodic-orbit saddle-node, period-doubling, and torus bifurcations correspond to quasi-periodic saddle-node, period-doubling, and Hopf bifurcations of two-dimensional tori.Source: §4.1 · p. 7

Three-dimensional tori

• A two-dimensional invariant torus of the reduced dynamics generically maps to a three-dimensional quasi-periodic invariant torus of the full system.The full-system torus has the same stability type as the corresponding reduced two-dimensional torus.Source: §4.2 · pp. 7–8
• Rational independence of the forcing and reduced-torus frequencies makes the three-torus foliated by a one-parameter family of two-tori; rational dependence yields dense quasi-periodic filling.Source: §4.2 · p. 7

Scope and limitation

• The analysis uses truncated reduced dynamics, which numerical experiments indicate is accurate, while persistence under higher-order terms remains beyond the paper’s scope.The tori are only weakly normally hyperbolic, although the numerical experiments generally indicate persistence.Source: §4.2 · p. 8

SSM-po toolbox

• SSM-po continues reduced-model periodic orbits and maps them to two-dimensional invariant tori of the full system, including SN, PD, and TR bifurcations.Continuation uses the po toolbox of coco.Source: §5.1 · p. 8

SSM-tor toolbox

• SSM-tor continues two-dimensional reduced tori and maps them to three-dimensional invariant tori of the full system.The parameter continuation uses the Tor toolbox.Source: §5.1 · pp. 8–9

Computational decomposition

• SSM analysis separates autonomous SSM construction, reduced-dynamics continuation, nonautonomous SSM computation, and evaluation of the mapping to physical coordinates.Source: §5.2 · p. 9

Efficiency

• The reduced dynamics simplify computation to fixed points, periodic orbits, and two-tori in low-dimensional leading-order models, while mapping evaluations can be parallelized.Source: §5.2 · p. 9

Continuation workflow

• The workflow computes lower-dimensional invariant sets first, then switches from periodic responses to two- and three-dimensional tori at the relevant bifurcations.Source: §5.2 · p. 9

Two coupled nonlinear oscillators in resonance

• SSM-predicted Hopf and saddle-node bifurcations agree with full-system continuation, including matching torus-bifurcation curves for the coupled oscillators.The reduced model detects four saddle-node and two Hopf fixed points, and the Hopf continuation agrees with torus-bifurcation continuation of the original system.Source: §6.1 · p. 10
• Four-minute SSM continuation matches the Tor-toolbox results obtained in about one and a half hours, while also determining invariant-torus stability.The SSM results match Tor computations for quasi-periodic responses and additional frequency and Poincaré-section validations; stability types are validated by numerical integration.Source: §6.1 · p. 11

A Bernoulli beam with nonlinear support spring

• The beam reduction produces stable and unstable limit cycles and two-dimensional invariant tori, with HB1 yielding stable tori and HB2 yielding unstable tori.The resulting FRCs show coexistence of stable and unstable periodic or quasi-periodic responses over a finite frequency interval.Source: §6.2 · pp. 13–14
• Invariant-torus frequencies and amplitudes converge well by SSM expansion order 11, while order 7 already gives results close to convergence.The same convergence threshold applies to the two bifurcated tori and their two frequency components.Source: §6.2 · p. 15
• Three-dimensional invariant tori are obtained by continuing two-dimensional tori in the reduced dynamics and mapping them to physical coordinates.Their Poincaré-section intersections form two-dimensional invariant tori, consistent with the three-frequency structure of the full response.Source: §6.2 · p. 15

7 Conclusion

• SSM reduction predicts bifurcations of periodic and quasi-periodic responses in very high-dimensional forced-damped mechanical systems with internal resonance.The approach uses reduced-order models on spectral submanifolds to replace direct full-system bifurcation computations.Source: §7 · p. 23
• Leading-order resonant-SSM dynamics identify full-system invariant 2-tori and 3-tori through limit cycles and invariant 2-tori of the truncated reduced model.This enables detection of limit-cycle and torus bifurcations at dimensions and speeds previously described as unreachable or unattainable.Source: §7 · p. 23
• The open-source SSMTool-2.2 toolboxes automate detection and continuation of equilibria, periodic orbits, and invariant tori using established continuation frameworks.SSM-ep and SSM-po build on COCO toolboxes, while SSM-tor builds on Tor.Source: §7 · p. 23
• Across four examples, SSM results closely match semi-discretization for coupled oscillators, converge with expansion order for a beam, and are validated by numerical integration in finite-element examples.The examples include two- and three-dimensional invariant tori in a discrete Bernoulli beam and finite-element beam and plate models.Source: §7 · p. 24
• The study assumes harmonic external forcing, while extending SSM reduction to internal-resonance systems with quasi-periodic forcing is identified as future work.The paper notes that SSM existence for quasi-periodically forced systems has been discussed previously.Source: §7 · p. 24

8 Appendix

• The proof compares the truncated flow, averaged system, and full SSM dynamics through µ-close Poincaré maps and their fixed-point equations.The argument follows the chain P2/H2 → P0/H0 → P1/H1 to establish persistence.Source: §8.1 · p. 24
• The Cartesian reduced model is written as a linear part plus nonlinear, forcing, and periodic terms before a scaling transformation and averaging procedure are applied.The appendix identifies A as invertible and block diagonal, F as the nonlinear contribution, F_ext as a constant vector, and G(φ) as periodic.Source: §8.1 · p. 24
• The proof constructs auxiliary flows, period-T maps, and scaled zero functions to connect reduced-model fixed points with bifurcations of the full dynamics.These objects provide the formal framework for transferring local bifurcation properties.Source: §8.1 · p. 24
• For a saddle-node bifurcation, nearby fixed-point branches persist across the averaged, truncated, and full systems with the same local square-root structure.Closeness of the Jacobians also preserves the linearized stability of the corresponding branches.Source: §8.1 · p. 24
• For a Hopf bifurcation, nearby fixed-point curves persist and a complex-conjugate eigenvalue pair crosses the unit circle near the original parameter value.The proof also checks persistence of the non-strong-resonance condition required for the Neimark-Sacker bifurcation.Source: §8.1 · p. 25

8.2 Proof of Theorem 3

• A periodic reduced-model orbit generates a cross-section for constructing the corresponding Poincaré-map representation.The section is defined by sampling at times congruent modulo the reduced orbit period.Source: §8.2 · p. 25
• When the frequency ratio is rational, the Poincaré map has finitely many fixed points and the resulting response is periodic with a commensurate period.If the ratio equals m1/m2 in lowest terms, the period multiplier can be chosen as m2.Source: §8.2 · p. 25
• When the frequency ratio is irrational, the reduced periodic orbit corresponds to an invariant closed curve of the Poincaré map and a quasi-periodic orbit with frequencies r_dΩ and ω_s.The associated invariant object is a two-dimensional torus.Source: §8.2 · p. 25
• The periodic or quasi-periodic response inherits the stability type of the corresponding reduced periodic orbit or invariant Poincaré-map intersection.The proof states that the fixed-point algorithm gives the same stability results.Source: §8.2 · p. 25
• Periodic-orbit bifurcation types are defined by Floquet multipliers crossing the unit circle through 1, −1, or a nonresonant complex-conjugate pair.These correspond respectively to saddle-node, period-doubling, and torus bifurcations.Source: §8.3 · p. 25
• In high-dimensional systems, selecting Floquet multipliers closest to the unit circle avoids event-function degeneracy and enables torus-bifurcation detection.The paper applies this selection strategy in the von Kármán beam example.Source: §8.3 · pp. 25–26

8.4 From periodic orbits to two-dimensional tori

• Phase shifts of a periodic reduced-model orbit represent the same closed curve and provide distinct initial states for torus reconstruction.Because the reduced vector field is autonomous, every shifted trajectory remains on the same periodic orbit.Source: §8.4 · p. 26
• The reduced dynamics associated with each phase shift are expressed as trajectories p_σ(t) in normal coordinates.The coordinate vector collects modal variables and their conjugates according to the chosen representation.Source: §8.4 · p. 26
• Sampling phase shifts over one reduced-orbit period yields a collection of trajectories whose period-T intersections approximate the invariant torus.The shifted initial points approximate the closed invariant intersection of the Poincaré section.Source: §8.4 · p. 26
• The normal-coordinate trajectories are mapped through the SSM parametrization into physical coordinates, where their collection approximates the invariant torus.The resulting trajectories z_σ(t) are assembled over t ∈ [0,T].Source: §8.4 · p. 26

• Tor supports switching from periodic-orbit continuation to invariant-torus continuation at TR bifurcations, including fixed or free rotation-number continuation and secondary branches.Source: §8.5 · p. 26
• The toolbox does not currently provide stability or bifurcation analysis for invariant tori.Source: §8.5 · p. 26
• An invariant torus is discretized through a PDE, Fourier expansion, ODE system, and multisegment boundary-value problem with all-to-all coupling conditions.Source: §8.5 · p. 26
• Tor approximates a torus with 2n_h+1 trajectories whose initial points lie on a closed invariant intersection curve and whose phases undergo rigid rotation over one torus period.Source: §8.5 · p. 27

From two-dimensional tori to three-dimensional tori

• A nondegenerate two-dimensional torus in the reduced model is represented by trajectories sampled over the slow period, while a degenerate limit cycle reduces to a single trajectory.Source: §8.6 · p. 27
• At a TR bifurcation, the bifurcating torus shares the periodic orbit’s phase-space geometry, allowing a single reduced-model trajectory to approximate it.Source: §8.6 · p. 27
• The slow-period trajectory covers the two-dimensional torus effectively when its sampled intersection phases return approximately to their initial values.Source: §8.6 · p. 27
• A trajectory set generated from the two-dimensional torus forms a tube whose center line effectively covers a two-dimensional torus, yielding an approximation to a three-dimensional torus.Source: §8.6 · p. 27
• Mapping the interpolated trajectories through the reduced-to-physical-coordinate map produces the corresponding three-dimensional torus in physical coordinates.Source: §8.6 · p. 27

8.7.1 Continuation of limit cycles

• The continuation path contains several saddle-node and period-doubling bifurcation limit cycles as Ω varies.Source: §8.7.1 · p. 28
• The limit-cycle size grows from zero after HB1 at Ω=0.98785, evolves nontrivially, and returns to zero at HB2 when Ω=1.0096.Source: §8.7.1 · p. 28

8.7.2 Two additional ways of validation via Tor toolbox

• The SSM-derived limit-cycle periods match well with those computed by the Tor toolbox.Source: §8.7.2 · p. 28
• SSM and Tor-toolbox invariant-intersection curves match well for sampled tori at Ω=0.98, 0.99, 1.00, and 1.01.Source: §8.7.2 · p. 28

8.7.3 Stability verification

• Forward integration of the original system from points on SSM-computed tori is used to verify whether representative stable and unstable torus types are preserved.Source: §8.7.3 · p. 28

8.8 Algorithm for selecting final time of direct numerical simulation

• The final simulation time is chosen long enough for dense torus coverage but short enough to keep computational cost reasonable.Source: §8.8 · p. 30
• SSM reduction supplies the excitation and internal frequencies, which determine the phase rotation and an approximate final time for forward simulation.Source: §8.8 · p. 30
• For stable reduced-model limit cycles, the number of integration cycles is selected from the largest nontrivial Floquet multiplier and a prescribed tolerance.Source: §8.8 · p. 30
• The simulations use Newmark-beta integration with 1000 steps per excitation cycle and measure amplitudes over a terminal time window.Source: §8.8 · p. 30
• In a high-dimensional plate example, each numerical integration took more than 12 hours, motivating a fixed choice of M=1 to reduce computational load.Source: §8.8 · p. 31