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Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators

Faranak Rajabi, Frédéric Gibou, Jeff Moehlis

arXiv:2609.16449v1math.OCeess.SYmath.NA

TL;DR

The paper asks whether phase-only control misses lower-cost mechanisms for strongly coupled oscillators. It solves the four-dimensional HJB equation for full-state feedback and finds a sharp switch from near-cycle steering to transient amplitude collapse, with broader off-cycle evidence beyond the Stuart-Landau-like model.

  • Problem

    Phase reduction constrains oscillator control to the limit cycle, leaving uncertain whether strong coupling makes qualitatively different full-state strategies energetically preferable.

  • Method

    The authors solve a four-dimensional HJB equation for two diffusively coupled Stuart-Landau-like oscillators and compare its feedback with reduced and direct-optimization baselines.

  • Results

    The optimal strategy switches sharply near c∗ = 0.227 from near-cycle steering to collapsing one oscillator’s amplitude before rebuilding it, while phase-reduced and direct optimizers miss the lower-cost trajectory.

  • Takeaways & Limitations

    Temporary off-cycle excursions can reduce the cost of overcoming strong synchronizing coupling, with nonisochronicity providing an additional phase-velocity benefit.

  • Takeaways & Limitations

    Extending grid-based HJB solutions to three or more oscillators is limited by the curse of dimensionality, motivating approximate solvers.

Abstract

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We solve the four-dimensional Hamilton-Jacobi-Bellman (HJB) equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. A sweep over coupling strength reveals a sharp change in the numerically optimal strategy: below a threshold coupling value, the controller steers the phase difference toward anti-phase while keeping both oscillators near the limit cycle, as reduced-order models would suggest. Above this threshold, the HJB solution changes qualitatively; the controller transiently collapses one oscillator's amplitude to near zero, thereby enabling large phase repositioning near the origin before rebuilding its amplitude. Direct gradient-based and stochastic optimization do not recover this lower-cost collapse trajectory from the initializations considered, suggesting that it occupies a region of the control landscape that is difficult to access by direct search. A joint sweep over nonisochronicity and coupling shows that collapse can occur even for an isochronous oscillator: phase repositioning near the origin can favor an off-cycle strategy. Nonisochronicity provides an additional energetic benefit through a phase-velocity surplus at small amplitude, quantitatively accounting for the observed reduction in control cost. Comparisons with uncoupled and coupled phase-reduced baselines show that phase models become increasingly inaccurate and cost significantly more energy for strong coupling. Results for coupled Van der Pol oscillators further demonstrate that exploitation of off-cycle dynamics is not specific to the Stuart-Landau-like oscillators.

I. INTRODUCTION

The paper asks whether full-state optimal control reveals mechanisms missed by phase-only models when coupling is strong. Solving the full HJB problem identifies near-cycle and amplitude-collapse strategies, while direct optimizers may miss the lower-cost basin.

  • I. INTRODUCTION: The control problem has clinical and infrastructure relevance because pathological synchrony and oscillator-based grid control create energy and stability concerns.The introduction connects these applications to battery-intensive stimulation and cascading desynchronization failures.
  • I. INTRODUCTION: Phase-amplitude reductions extend beyond standard phase reduction, but the paper investigates whether full-state optimization exposes additional off-cycle behavior.The related work includes adaptive phase-amplitude and dynamic-programming approaches for stronger inputs.
  • I. INTRODUCTION: Full-state HJB control reveals qualitatively different strategies that phase reduction cannot represent near strong perturbations or coupling.Phase reduction constrains dynamics to the limit cycle, whereas full-state methods can expose off-cycle mechanisms.
  • I. INTRODUCTION: At weak coupling, optimal trajectories remain near the limit cycle; at stronger coupling, one oscillator collapses toward the origin before returning to the cycle.Near the origin, phase becomes highly sensitive to state perturbations, enabling large phase repositioning.
  • I. INTRODUCTION: HJB methods provide global-optimum guarantees under their admissible-control formulation, unlike direct methods that return initialization-dependent local optima.The paper uses this distinction to compare full-state HJB solutions with gradient-based and evolutionary trajectory optimization.

A. Two Coupled Stuart-Landau-Like Oscillators

The model consists of two diffusively coupled Stuart-Landau-like oscillators with controlled x-channel actuation and a unit-circle limit cycle. Its anti-phase orbit is a natural target but becomes unstable beyond weak coupling.

  • A. Two Coupled Stuart-Landau-Like Oscillators: The system comprises two diffusively coupled oscillators with x-channel control and coupling, parameterized by a, ω, b, c, and bounded inputs.The admissible control satisfies ui ∈ [−umax, umax].
  • A. Two Coupled Stuart-Landau-Like Oscillators: The uncoupled polar dynamics are ṙ = ar(1 − r2) and ϕ̇ = ω + br2 sin 2ϕ, producing a stable unit-circle limit cycle.The extra sin 2ϕ term distinguishes this system from the standard Stuart-Landau normal form.
  • A. Two Coupled Stuart-Landau-Like Oscillators: The anti-phase solution is a natural target, but for c ≳ 0.015 it is unstable and trajectories near it return to the in-phase solution.This instability motivates explicitly anchoring the radii in the terminal cost.
  • A. Two Coupled Stuart-Landau-Like Oscillators: The nonisochronicity parameter b satisfies b < ω, with the reported default setting a = ω = 1 and b = 0.8.The bound keeps Ω = √(ω2 − b2) real, while the period diverges as b approaches 1.
  • A. Two Coupled Stuart-Landau-Like Oscillators: The oscillator-swap symmetry is broken by the initial condition, allowing the controller to select which oscillator collapses.The system is Z2-equivariant, but the chosen initial state is asymmetric.

B. Optimal Control Objective

The objective targets the entire anti-phase orbit rather than one absolute phase while penalizing terminal radial error. This prevents the zero state from becoming a trivial zero-cost solution.

  • B. Optimal Control Objective: The selected objective uses R = 10 and γ = 1000 to balance control effort with terminal targeting.These parameters appear in the running and terminal-cost specification.
  • B. Optimal Control Objective: The terminal objective targets the entire anti-phase orbit, so any unit-radius anti-phase pair is acceptable regardless of absolute phase.The distance term uses oscillator sums and radii rather than absolute phase.
  • B. Optimal Control Objective: The terminal cost explicitly penalizes deviations of both radii from one, preventing the trivial origin state from achieving zero terminal cost.The radial terms (ri − 1)2 enforce return toward the limit cycle.

IV. HJB SOLUTION METHOD

The four-dimensional HJB method computes a full-state value function and feedback control for the coupled oscillator system. A GPU-accelerated WENO solver makes these grid-based calculations practical and numerically stable over the reported tests.

  • IV. HJB SOLUTION METHOD: The HJB formulation evolves a value function V(z, τ) over the full state z ∈ R4 in backward time τ = Tf − t.The value function satisfies the Hamilton-Jacobi-Bellman equation with terminal condition determined by the target cost.
  • IV. HJB SOLUTION METHOD: The Hamiltonian combines the uncontrolled dynamics with convex-dual running costs for the two bounded control channels.The gradient is p = ∇zV, and the control penalty is Ru2 under |u| ≤ umax.
  • IV. HJB SOLUTION METHOD: The minimizing feedback control is obtained by saturating the value-gradient-based command at the input bound umax.The saturation map clips commands between −umax and umax.
  • IV. HJB SOLUTION METHOD: The PDE is solved on [−1.5, 1.5]4 with a fifth-order WENO scheme, local Lax-Friedrichs flux splitting, SSP-RK3 stepping, and GPU acceleration.A single NVIDIA A100 solves a full-state instance in approximately five minutes.
  • IV. HJB SOLUTION METHOD: Numerical checks show domain enlargement changes optimal cost by less than 0.3%, while grid refinement changes reported costs by less than 1%.The tested trajectories remain within ||z(t)||∞ < 1.2.

V. MAIN RESULTS

The HJB solution reveals two qualitatively distinct strategies: near-cycle phase steering at weak coupling and transient amplitude collapse at stronger coupling. Collapse drives one oscillator near the origin, enables phase repositioning, and is followed by natural amplitude recovery.

  • A. Two Qualitatively Distinct Strategies: At c = 0.10, both oscillators remain near the limit cycle while small perturbations steer their phase difference toward 180°.The minimum amplitude remains above r1,min > 0.85, consistent with phase-reduction intuition.
  • A. Two Qualitatively Distinct Strategies: At c = 0.30, HJB control collapses oscillator 1 to r1,min = 0.11 while oscillator 2 remains near the limit cycle, then rebuilds r1 to 0.98 by Tf.The trajectory spirals inward, accumulates a phase advantage near the origin, and spirals back toward approximate anti-phase.
  • A. Two Qualitatively Distinct Strategies: The transition is sharp: c = 0.20 stays near-cycle with r1,min = 0.77, whereas c = 0.30 produces deep collapse, with no observed intermediate excursion.The dense sweep resolves the crossover to Δc < 0.006 and finds no trajectory with r1,min between 0.13 and 0.72.

B. Sharp Strategy Transition

The computed optimal strategy switches abruptly near c* ≈ 0.227 rather than gradually increasing its off-cycle excursion. After the switch, optimal cost decreases despite stronger coupling, consistent with collapse becoming energetically advantageous.

  • B. Sharp Strategy Transition: c* = 0.227 ± 0.003 separates near-cycle steering from collapse, with c = 0.2247 yielding r1,min = 0.728 and c = 0.2299 yielding r1,min = 0.097.The switch occurs over Δc = 0.0052, narrower than 0.006 at the reported resolution.
  • B. Sharp Strategy Transition: The optimal cost decreases between c = 0.20 and c = 0.30 despite stronger coupling opposition, indicating the efficiency of the collapse strategy.The cost curve is therefore non-monotonic across the transition.
  • C. Physical Mechanism: Phase Repositioning and Phase-Velocity Surplus: Near the origin, phase sensitivity enables large repositioning without continuously opposing synchronizing dynamics on the limit cycle.At b = 0.8, the phase-velocity surplus is δω = 0.4, while holding near the origin costs Ehold ≈ Rc²τ/2 and yields δω·τ radians of free phase.
  • C. Physical Mechanism: Phase Repositioning and Phase-Velocity Surplus: At b = 0.8 and c = 0.30, holding near the origin costs Ehold ≈ 2.7 while producing 2.4 radians of free phase repositioning.Collapse is favorable when on-cycle repositioning of the same 2.4 radians costs more than 2.7.

D. Role of Nonisochronicity

Amplitude collapse occurs across the nonisochronicity sweep when coupling is sufficiently strong, including the isochronous case. Nonisochronicity changes the energetic benefit and threshold through the phase-velocity surplus, while long periods can delay collapse.

  • D. Role of Nonisochronicity: Collapse occurs for all b values at sufficiently strong coupling, including b = 0, where the estimated threshold lies in c* ∈ (0.20, 0.30).Thus phase repositioning near the origin can favor an off-cycle strategy without nonisochronicity.
  • D. Role of Nonisochronicity: The strategy-change threshold is non-monotonic in b: c* ∈ (0.10, 0.20) for b = 0.2 to 0.6, while c* > 0.50 at b = 0.9.At b = 0.9 and c = 0.40, the trajectory only grazes low amplitude and does not engage the holding mechanism.
  • D. Role of Nonisochronicity: At c = 0.30, optimal cost falls from J* = 19.93 at b = 0 to J* = 11.53 at b = 0.8, a 42% reduction.The reduction corresponds to δω increasing from 0 at b = 0 to 0.4 at b = 0.8 and 138° of free phase repositioning.
  • D. Role of Nonisochronicity: At b = 0.9, the long period T = 14.4 provides enough time for on-cycle steering, delaying collapse to higher coupling.

E. Quantitative Verification: Free Phase Budget

At c = 0.30, collapse gains phase near the origin and lowers cost, while direct optimization does not readily recover the HJB trajectory.

  • Free phase budget: The free phase δω · τ matches the analytical prediction (ω −Ω)τ within 1° across nonisochronicity values at c = 0.30.Table II defines the collapse interval as the time oscillator 1 spends below r1 = 0.5.
  • Free phase budget: 42% lower cost at b = 0.8 reflects 138° of free phase repositioning and reduced residual steering after amplitude recovery.The optimizer's hold time decreases from 0.74 T at b = 0 to 0.57 T at b = 0.8 as phase return per unit time increases.
  • Direct optimization: Neither L-BFGS-B nor CMA-ES discovers collapse in the considered runs, with direct-search solutions costing at least 36% more or 2.1× the HJB cost.L-BFGS-B remains at r1,min > 0.69, while CMA-ES reaches r1,min = 0.99.

VII. COMPARISON WITH PHASE-REDUCED CONTROL

Full-state HJB control outperforms phase-reduced baselines at stronger coupling by exploiting amplitude collapse, a mechanism constrained out of phase-only models.

  • Phase-reduced comparison: At c ≥0.20, coupled PRC saturates near 171° and costs 1.3 to 2.3 times more energy than HJB, failing to reach the target.At c = 0.10, coupled PRC reaches 178.6° versus 140.3° for the uncoupled baseline.
  • Phase-reduced comparison: At c = 0.01, uncoupled PRC, coupled PRC, and HJB perform comparably, reaching 175 to 180° at similar cost.The HJB trajectory remains near the limit cycle in this weak-coupling regime.
  • Off-cycle mechanism: At c = 0.30, HJB reaches r1,min = 0.11 by collapsing amplitude, whereas phase-only models remain on the limit-cycle manifold.Fig. 5 shows similar near-limit-cycle trajectories at weak coupling but qualitatively different trajectories at c = 0.30.
  • Scope: The demonstrated collapse excursion, with |r −1| = 0.89, is considerably deeper off-cycle than regimes previously demonstrated by cited phase-amplitude control methods.Phase-amplitude reductions can include transverse dynamics, but the present excursion approaches the origin more deeply.

VIII. VALIDATION ON VAN DER POL OSCILLATORS

Coupled Van der Pol validation shows that strong coupling can make full-state HJB control exploit substantial departures from a non-circular limit cycle. This supports off-cycle control as a qualitative phenomenon beyond the Stuart-Landau-like model.

  • VIII. VALIDATION ON VAN DER POL OSCILLATORS: The Van der Pol test uses µ = 1 and umax = 0.5, with an egg-shaped limit cycle of amplitude ≈2 in x and period T = 6.66.Its geometry differs qualitatively from the unit-circle Stuart-Landau normal form.
  • VIII. VALIDATION ON VAN DER POL OSCILLATORS: The validation primarily tests whether strong coupling induces qualitatively different off-cycle control in an oscillator lacking the Stuart-Landau model’s simple phase-velocity decomposition.Because Van der Pol lacks the same analytically defined off-cycle phase-velocity surplus, the example tests mechanism generality rather than that decomposition.
  • VIII. VALIDATION ON VAN DER POL OSCILLATORS: At c = 0.25 and c = 0.30, optimal off-cycle deviations reach 1.09 and 1.20, respectively, versus below 0.80 for c ≤0.20.These deviations correspond to 55% of the Van der Pol limit-cycle amplitude at c = 0.25, while weak-to-moderate coupling stays below 40%.
  • VIII. VALIDATION ON VAN DER POL OSCILLATORS: The large excursions show that full-state HJB can exploit off-cycle dynamics despite qualitatively different limit-cycle geometry.The result is not specific to the Stuart-Landau-like algebraic structure.

X. CONCLUSION

The paper finds a sharp coupling-dependent transition in HJB-optimal control: near-cycle phase steering at weak coupling gives way to transient amplitude collapse at stronger coupling. This off-cycle strategy can reduce synchronizing-coupling costs, gains an additional phase-velocity benefit under nonisochronicity, and also appears in coupled Van der Pol oscillators.

  • X. CONCLUSION: The four-dimensional HJB solution changes sharply with coupling, from near-limit-cycle phase-based control to deep transient amplitude collapse before reaching anti-phase.At stronger coupling, one oscillator’s amplitude is reduced substantially and then rebuilt.
  • X. CONCLUSION: Amplitude collapse can help even for isochronous dynamics by reducing synchronizing-coupling influence, while nonisochronicity adds a small-amplitude phase-velocity surplus.The two effects provide distinct geometric and energetic benefits.
  • X. CONCLUSION: The tested phase-reduced baselines and direct gradient-based and stochastic optimizers do not recover the lower-cost deep off-cycle HJB trajectory.This indicates that the mechanism is difficult to access from the considered direct-search initializations.
  • X. CONCLUSION: Coupled Van der Pol results indicate that optimal-control exploitation of off-cycle dynamics is not specific to Stuart-Landau-like oscillators.The broader implication is that strong coupling may make far-from-cycle state-space regions useful for oscillator control.
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