Description
Devaney's definition of chaos: a continuous map f: X → X on a metric space is chaotic if (1) topologically transitive: for any open U, V there exists k with f^k(U) ∩ V nonempty; (2) dense periodic points: Per(f) is dense; (3) sensitive dependence: ∃δ > 0 such that for every x and neighborhood N, ∃y ∈ N, n ≥ 0 with d(f^n(x), f^n(y)) > δ. Banks et al. (1992): (1) and (2) imply (3), so sensitive dependence is redundant. Accessible definition avoiding measure theory.
Source: Wikipedia; Devaney, An Introduction to Chaotic Dynamical Systems; Banks et al.
Dependency Flowchart
graph TD
DefTrans["Def: Topological transitivity\nf^k(U) ∩ V ≠ ∅ for open U,V"]
DefDense["Def: Dense periodic points\nPer(f) dense in X"]
DefSens["Def: Sensitive dependence\n∃δ, ∀x,N ∃y∈N,n: d(f^n(x),f^n(y))>δ"]
DefChaos["Def: Devaney chaos\nTransitive + dense periodic + sensitive"]
ThmBanks["Thm: Banks et al. 1992\nTransitive + dense periodic ⇒ sensitive"]
ThmRedund["Thm: Sensitivity redundant\nTwo conditions suffice"]
ExLogistic["Ex: Logistic map μ=4\nChaotic on [0,1]"]
ExShift["Ex: Shift map σ\nChaotic on Σ"]
DefTrans --> DefChaos
DefDense --> DefChaos
DefSens --> DefChaos
DefTrans --> ThmBanks
DefDense --> ThmBanks
ThmBanks --> ThmRedund
DefChaos --> ExLogistic
DefChaos --> ExShift
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef example fill:#16a085,color:#fff,stroke:#0e6655
class DefTrans,DefDense,DefSens,DefChaos definition
class ThmBanks,ThmRedund theorem
class ExLogistic,ExShift example
Color Scheme
Blue Definitions
Teal Theorems
Green Examples
Process Statistics
- Nodes: 9
- Edges: 10
- Definitions: 4
- Theorems: 2