Page 657 - Mathematical Techniques of Fractional Order Systems

628 Mathematical Techniques of Fractional Order Systems

21.2 GENERAL INFORMATION ON CHAOTIC DYNAMIC
SYSTEMS

21.2.1 Concept of the Deterministic Chaos
The theory of chaos appeared in the early 1960s in meteorology and rapidly
spread to several areas of research. There are several mathematical defini-
tions of chaos in the literature, but so far there is no universal mathematical
definition of chaos. The habitual definition considers chaos as aperiodic
long-term behavior in a deterministic system that exhibits sensitive depen-
dence on initial conditions and some recurrence properties. Mathematically,
the most recognized chaos definition is that of (Devaney, 1989), given as:

Definition: For a consider function F, defined from and to a real interval I,
is said to be constituted of a chaotic dynamics if:
1. F is topologically transitive;
2. The sum of the periodic points of F is dense in I;
3. F has sensitive dependence on initial conditions.
In a nonchaotic deterministic system, neighboring initial conditions lead to
neighboring evolutions. Contrary, a chaotic system is a deterministic system fully
determined by initial state and subject to a law of evolution which can be
extremely simple and regular, but its evolution is extremely complicated and per-
fectly irregular. This gives it unpredictable behavior over time which is known as
the butterfly effect developed by (Poincare ´, 1890) as: “It so happens that small
differences in the initial state of the system can lead to very large differences in
its final state. A small error in the former could then produce an enormous one in
the latter. Prediction becomes impossible, and the system appears to behave ran-
domly.” Moreover, these systems have a very high sensitivity to the initial condi-
tions. Thus, chaotic behavior is usually defined as a deterministic low-
dimensional dynamics with high sensitivity to initial conditions and some recur-
rence properties. The concept of recurrence goes back to (Poincare ´,1890).

This sensitivity to the initial conditions explains the fact that, for a chaotic
system, a minimal modification of the initial conditions can lead to
unpredictable results over the long term. The degree of sensitivity to initial condi-
tions quantifies the chaotic nature of the system. The presence of this chaos prop-
erty is detected on the basis of the Lypunov exponent indicator or on the basis of
analysis of the quantification of recurrences. The recurrence analysis methods for
the chaotic dynamical system are based on both the recurrence plot and the recur-
rence quantification analysis.
21.2.1.1 The Recurrence Plot

Definition: A recurrence plot (RP) is an advanced technique of nonlinear
data analysis. It is a visualization (or a graph) of a square matrix, in which