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On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals

Yirmeyahu Kaminski, Pierre Lochak

TL;DR

This work develops a geometric framework for the equilibrium locus of a parameterized, analytic dynamical system with $k$ independent first integrals. It proves that the equilibrium set $E=f^{-1}(0)$ is a smooth manifold of dimension $m+k$, and that the projection to the parameter space $\Lambda$ yields an equilibrium bundle $E\to\Lambda$ equipped with a canonical (flat) connection, enabling parallel transport and holonomy analysis. The authors connect this connection to a Riemannian submersion under a suitable metric, show transversality with integral manifolds, and construct curve liftings that transport equilibria across parameter values. They further relate eigenvalue variations along fiber loops to monodromy in the fundamental groups, providing a topological lens on stability changes, and illustrate with circular cell networks and ribosome-flow models, hinting at control-theoretic applications for maneuvering equilibria via parameters.

Abstract

For a family of dynamical systems with $k > 0$ independent first integrals evolving in a compact region of an Euclidean space, we study the equilibrium locus. We show that under mild and generic conditions, it is a smooth manifold that can be viewed as the total space of a certain fiber bundle and that this bundle comes equipped with a natural connection. We then proceed to show parallel transport for this connection does exist and explore some of its properties. In particular, we elucidate how one can to some extent measure the variation of the system eigenvalues restricted to a given fiber.

On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals

TL;DR

This work develops a geometric framework for the equilibrium locus of a parameterized, analytic dynamical system with independent first integrals. It proves that the equilibrium set is a smooth manifold of dimension , and that the projection to the parameter space yields an equilibrium bundle equipped with a canonical (flat) connection, enabling parallel transport and holonomy analysis. The authors connect this connection to a Riemannian submersion under a suitable metric, show transversality with integral manifolds, and construct curve liftings that transport equilibria across parameter values. They further relate eigenvalue variations along fiber loops to monodromy in the fundamental groups, providing a topological lens on stability changes, and illustrate with circular cell networks and ribosome-flow models, hinting at control-theoretic applications for maneuvering equilibria via parameters.

Abstract

For a family of dynamical systems with independent first integrals evolving in a compact region of an Euclidean space, we study the equilibrium locus. We show that under mild and generic conditions, it is a smooth manifold that can be viewed as the total space of a certain fiber bundle and that this bundle comes equipped with a natural connection. We then proceed to show parallel transport for this connection does exist and explore some of its properties. In particular, we elucidate how one can to some extent measure the variation of the system eigenvalues restricted to a given fiber.

Paper Structure

This paper contains 12 sections, 18 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Definitions and first properties
  3. On the nature of $\Lambda_0$
  4. Fibers over the parameter space
  5. Parallel transport of the fibers
  6. Connection
  7. Transversality between fibers and integral manifolds
  8. Lifting curves
  9. The natural connection
  10. The natural connection via a Riemannian submersion
  11. The eigenvalues along the fibers
  12. Examples and Applications

Key Result

Proposition 1

In the setting described by assumption hyp-indep-first-int above, the Jacobian matrix $\frac{\partial f}{\partial x}$ has rank at most $n-k$ on E.

Theorems & Definitions (33)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • ...and 23 more