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Feedback Integrators Revisited

Juho Bae, Dong Eui Chang

TL;DR

This work strengthens the theoretical foundation of Feedback Integrators by proving non-asymptotic preservation of first integrals within a discretized time horizon, independent of asymptotic limits. It derives an optimal fixed gain for Euler integration and introduces an Hessian-based adaptive gain scheme that obviates the need for a priori Lipschitz bounds while guaranteeing positive invariance of the first-integral sublevel set. Through numerical experiments on a free rigid body, the Kepler problem, and a perturbed Kepler problem, the adaptive gain consistently yields superior accuracy with comparable cost to fixed gains and outperforms unity gain, and often exceeds standard benchmarks like Störmer-Verlet at small step sizes. The results indicate that Feedback Integrators can be practically reliable and broadly applicable, offering robust structure-preserving integration across diverse dynamical systems on manifolds.

Abstract

We revisit the notion of Feedback Integrators introduced by D. E. Chang in 2016. Feedback integrators allow for numerically integrating dynamical systems on manifold while preserving the first integrals of the system. However, its performance was stated and proved in an asymptotic manner, which left a gap between its empirical success and theoretical understandings. In response, we prove preservation of first integrals over entire integration region up to arbitrarily small deviation under Feedback Integrator framework. Furthermore, we propose an adaptive gain selection scheme that significantly improves the performance. Numerical demonstrations are conducted on free rigid body motion in SO(3), the Kepler problem, and a perturbed Kepler problem with rotational symmetry. All demonstration codes are available at: https://github.com/johnbae1901/Feedback-Integrator.

Feedback Integrators Revisited

TL;DR

This work strengthens the theoretical foundation of Feedback Integrators by proving non-asymptotic preservation of first integrals within a discretized time horizon, independent of asymptotic limits. It derives an optimal fixed gain for Euler integration and introduces an Hessian-based adaptive gain scheme that obviates the need for a priori Lipschitz bounds while guaranteeing positive invariance of the first-integral sublevel set. Through numerical experiments on a free rigid body, the Kepler problem, and a perturbed Kepler problem, the adaptive gain consistently yields superior accuracy with comparable cost to fixed gains and outperforms unity gain, and often exceeds standard benchmarks like Störmer-Verlet at small step sizes. The results indicate that Feedback Integrators can be practically reliable and broadly applicable, offering robust structure-preserving integration across diverse dynamical systems on manifolds.

Abstract

We revisit the notion of Feedback Integrators introduced by D. E. Chang in 2016. Feedback integrators allow for numerically integrating dynamical systems on manifold while preserving the first integrals of the system. However, its performance was stated and proved in an asymptotic manner, which left a gap between its empirical success and theoretical understandings. In response, we prove preservation of first integrals over entire integration region up to arbitrarily small deviation under Feedback Integrator framework. Furthermore, we propose an adaptive gain selection scheme that significantly improves the performance. Numerical demonstrations are conducted on free rigid body motion in SO(3), the Kepler problem, and a perturbed Kepler problem with rotational symmetry. All demonstration codes are available at: https://github.com/johnbae1901/Feedback-Integrator.

Paper Structure

This paper contains 10 sections, 5 theorems, 53 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preservation of First Integrals
  4. Gain Selection for Feedback Integrators
  5. Adaptive Gain Selection under Euler's Method
  6. Numerical Demonstration
  7. Free Rigid Body Motion in SO(3)
  8. The Kepler Problem
  9. Perturbed Kepler Problem
  10. Conclusion

Key Result

Theorem 1

Suppose that a one--step method of order $p$ is applied to numerically integrate eq:conti_system, denoted as follows. Suppose that the vector field $f$ is $C^{p}$ and the function $V$ is $C^{p+1}$, and the assumptions (A1)-(A3) hold. Then there exists $h_{0}>0$ such that for all $h \in \left(0, h_0\right)$, the discrete system eq:feedback_discrete_system has a compact, uniformly asymptotically st

Figures (6)

  • Figure 1: Accuracy results of free rigid body motion in $\mathrm{SO}(3)$. Integration with Euler's method on $[0, 1000]$. (a) maximum $V(x_k)$ and deviation of first integrals along the trajectories. (b) CPU time dedicated for each integration scheme.
  • Figure 2: Trajectories of the body angular velocities of free rigid body motion in $\mathrm{SO}(3)$. Integration on $[0, 1000]$ with $h = 10^{-4}$. Feedback integrators are implemented with Euler's method.
  • Figure 3: Accuracy results of the Kepler problem. Integration with Euler's method over $1000$ periods with $T = 70.2481$. (a) maximum $V(x_k)$ and deviation of first integrals along the trajectories. (b) CPU time dedicated for each integration scheme.
  • Figure 4: Trajectories of the Kepler Problem. Integration over $1000$ periods with $T = 70.2481$ and $h = 10^{-3}$. Feedback integrators are implemented with Euler's method.
  • Figure 5: Accuracy results of perturbed Kepler problem. Integration with Euler's method over $[0, 200]$. (a) maximum $V(x_k)$ and deviation of first integrals along the trajectories. (b) CPU time dedicated for each integration scheme.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 1: Theorem 5.2, chang2016feedback
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Definition 1: Sublevel set and Hessian bounds
  • Theorem 3
  • proof
  • Remark 3
  • Remark 4
  • ...and 5 more