Global Attractors for Dissipative Flows on Degenerate Constraint Manifolds

TL;DR

This paper develops a framework for analyzing dissipative dynamical systems on degenerate constraint manifolds, establishing the existence of global attractors and describing their structure despite the lack of coercive Lyapunov functionals.

Contribution

It introduces a novel approach to attractor theory for systems with degenerate structures, using compatible functionals and quotient dynamics to handle null distributions.

Findings

Bounded trajectories are confined to invariant leaves of the foliation.

Existence of a compact global attractor saturated by null leaves.

Effective dimensional reduction occurs due to constraint-induced degeneracy.

Abstract

A class of dissipative dynamical systems evolving on smooth constraint hypersurfaces endowed with degenerate induced bilinear forms is studied. The intrinsic evolution is generated by constraint--preserving vector fields on manifolds whose tangent bundles admit a nontrivial null distribution associated with the degeneracy of the induced structure. In this indefinite setting, the absence of coercive Lyapunov functionals prevents the direct application of classical attractor theory developed for Riemannian phase spaces. Dissipation is instead characterized relative to functionals that are compatible with the null distribution and exhibit decay exclusively in directions transverse to the associated foliation. Under suitable involutivity and regularity assumptions on the null distribution, all bounded trajectories are shown to be asymptotically confined to invariant leaves of the corresponding foliation. Asymptotic compactness of the intrinsic evolution is then established without coercivity by reducing the dynamics to a projected semiflow on the quotient manifold determined by the characteristic distribution. In the presence of a bounded absorbing set and continuous semiflow structure, the intrinsic evolution admits a compact global attractor saturated by the null leaves, whose effective asymptotic dynamics are governed by a compact invariant subset of the reduced phase space. Furthermore, when the compatible functional satisfies a Morse--type nondegeneracy condition in directions transverse to the null distribution and the induced transversal linearization admits no center spectrum, the projected semiflow possesses no neutral directions. The resulting framework provides a mechanism by which constraint--induced degeneracy enforces effective dimensional reduction in dissipative geometric evolution systems.