Slidingdominated

Slidingdominated is a term used in the study of nonsmooth dynamical systems to describe a regime in which the evolution of a system is governed primarily by sliding along a constraint or switching surface, rather than by motion freely off the surface. In this context, trajectories spend a large portion of time constrained to or near a boundary that separates different dynamical modes.

Mathematically, consider a piecewise-smooth system ẋ = f_i(x) in regions S_i separated by a switching surface H = {x : h(x) = 0}. The sliding region occurs where the vector fields on either side point toward the surface. In Filippov’s framework, the sliding (or Filippov) vector field f_s is defined as a convex combination of the side fields chosen so that ḣ = ∇h(x)·f_s(x) = 0, forcing the motion to evolve along the surface h(x) = 0. A dynamics is termed sliding-dominated when trajectories spend the majority of their evolution on the surface, so the reduced dynamics on the surface govern long-term behavior.

Applications and implications include mechanical systems with dry friction, electrical circuits with piecewise-linear characteristics, and control systems employing sliding mode control. The regime is valued for robustness to disturbances due to the corrective nature of sliding motion, but it can introduce chattering, modeling challenges, and numerical stiffness.

Relating to broader concepts, slidingdominated dynamics are closely connected to sliding mode control, Filippov systems, and nonsmooth dynamical systems theory, and they contrast with crossing- or transversal-dominated regimes where trajectories frequently leave the surface rather than adhere to it.

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