Hidden vs Self-Excited

The topological theory behind hidden attractors.

The classification of attractors into hidden and self-excited was formalized by Leonov and Kuznetsov in the early 2010s. This distinction is not just semantic; it fundamentally changes how we must search for chaotic behavior in engineering systems.

Self-Excited Attractors

A self-excited attractor has a basin of attraction that intersects with a small neighborhood of an unstable equilibrium point.

Most classic chaotic systems fall into this category:

  • The Lorenz system
  • The Rössler system
  • The standard Chua circuit parameters

Why they are easy to find: If you start a simulation very close to the unstable equilibrium, the trajectory will naturally be repelled outward, eventually falling into the attractor. You can “excite” the attractor simply by perturbing the equilibrium.

Hidden Attractors

A hidden attractor has a basin of attraction that does not intersect with any open neighborhood of an equilibrium point.

This occurs in systems where:

  1. All equilibria are strictly stable (point attractors).
  2. The system has no equilibria at all.
  3. The system has infinite equilibria (like a line or plane of equilibria).

Why they are dangerous: Standard computational methods (like simulating from near an equilibrium point) will never find a hidden attractor. If a hidden attractor exists in a physical engineering system (e.g., an aircraft control system or an electronic circuit), standard stability analysis might conclude the system is completely safe and stable, while in reality, a large perturbation could knock the system into a disastrous, hidden chaotic state.

Searching for Hidden Attractors

Because we cannot rely on unstable equilibria to guide our search, locating hidden attractors requires specialized techniques:

  1. Massive Grid Sweeps: Blindly sampling the entire phase space. Very computationally expensive, especially for fractional systems.
  2. Analytical Continuation: Finding a self-excited attractor in a known parameter regime, and slowly sweeping parameters until the equilibrium becomes stable, tracking whether the attractor survives the bifurcation. This is the basis of the Robustness Overlay workflow.
  3. Describing Function Method: Using frequency-domain control theory to predict the approximate location of hidden limit cycles, and seeding initial conditions there. This is used in the Integer Lur’e workflow.