Chaos (dynamical system)


August 8, 2022

Chaos in dynamical systems are trajectories that exhibit irregular and complex behavior despite deterministic laws. In particular, it is also called deterministic chaos. One of the essential characteristics of chaos is that minute differences grow into huge differences in the future. This creates the inherent unpredictability of chaos, also known as the butterfly effect. Chaos is a nonlinear phenomenon that does not occur in linear systems. The earliest discovery of chaos was made by Henri Poincaré in his work on the three-body problem. Chaos in dissipative systems exists as a strange attractor. The butterfly-shaped attractor generated by the Lorentz equation is probably the most famous strange attractor in the world.



Dynamical systems are roughly divided into linear systems and nonlinear systems, but chaos cannot occur in linear systems. For chaos to occur, the system must have some nonlinearity.


Chaos takes an aperiodic orbit that is asymptotic to neither a fixed point, nor a periodic orbit, nor a quasi-periodic orbit.

Default Sensitivity

For a map f : X → X on the metric space, there exists some δ > 0 such that for any x ∈ X and ε > 0, d(f (x), f (y)) < ε and d( f is said to be initial value sensitive if there exists y ∈ X such that f n(x), f n(y)) > δ and a natural number n.


A stronger property than initial value sensitivity is the concept of scalability. A map f : X → X on the metric space satisfies d(f n(x), f n(y)) > δ with some δ >0 for any distinct x, y ∈ X. f is said to be expansive if n exists.

Phase transitivity

A continuous map f : X → X is said to be topologically transitive if it has dense orbits on X. Equivalently, we say that f is topologically transitive if there exists some n > 1 such that any non-empty open set U, V ⊂ X has f n(U) ∩ V ≠ ∅. .

Phase mixability

A property stronger than the phase transitivity is the phase mixing property. f is said to be topologically mixed if there exists a natural number N such that for all n > N, any non-empty open set U, V ⊂ X satisfies f n(U) ∩ V ≠ ∅. If a dynamical system is phase-mixed, it is clearly phase-transitive at the same time.


To eliminate undesirable examples, it is desirable to include in the definition of chaos that the space in which the trajectory or system is defined is bounded or compact. Unfavorable examples are systems such as x ax or x ↦ ax (where a is a positive constant), where initial sensitivity and phase transitivity are satisfied, but the orbit is exponential. is unsuitable for inclusion in chaos because it only increases monotonically to .



Kazuyuki Aihara (ed.), 1990, "Chaos - Fundamentals and Applications of Chaos Theory" 1st Edition, Science Publishing ISBN 4-7819-0592-7 Toru Koda "Introduction to Chaos" Motomasa Komuro, Takashi Matsumoto, CHUA, Leon O. “Capturing Chaos with Electronic Circuits” Masayoshi Inoue, Hiroki Hata, 1999, "Fundamentals and Developments of Chaos Science: Towards Understanding Complex Systems" First Edition, Kyoritsu Shuppan ISBN 4-320-03323-X Toshio Niwa, 1999, "Can mathematics elucidate the world? Chaos and pre-established harmony" reprint, Chuokoron Shinsha ISBN 4-12-101475-8 Kazuyuki Aihara, 1993, "Chaos - a completely new creation"