Characteristics of Chaotic Systems:
- Deterministic systems.
- Nonlinearity.
- Aperiodic behaviour.
- Sensitivity to initial conditions.
- Stretching and folding.
- Infinite repulsive cycles.
- Emerging patterns.
- Adaptation.
Some explenations:IMPORTANT
- The rule/equation that govern a chotic system are deterministic and non-linear.
- A chotic system will be aperiodic, also the trajectory will never pass two times in a single point.
- Sensitivity to initial conditions, we have already seen.
- The combination of the two mechanisms: stretching and folding:
- Strethcing: the trajectory “streaches” form the steady state, there are some diverging mechanisms.
- Folding: prevents the trajectory from escaping the region where the strange attractor lives.
- “Chaos is a combination of two mechanisms: diverge and convergence”.
- Often chaos includes the presence of inifinite repulsive cycles, this is typical of the Rosell attractor (it’s not typical of the Lorenz system), but is a possible path to produce chaos.
- In general there are two properties to chaotic systems:
- Emerging patterns: like for strange attractors, those are emering patterns, a geometry we didn’t expect, we did not impose it in the system, it is self-organized.
- Adaptation: chaotic systems have infinite different kinds of behaviors inside them, so we can say that the system has a “richness” of dynamics inside, you can visit all parts of the phase space (more precisly of the region defined by the strange attractors), so the dynamic is very free to evolve, ergo the system is very adaptive, you cannot disturb very much the system (it will go back to its chaotic trajectory), it is sentive to even a small change, but the system cannot be easily destroyed, so it is higly adaptive.
