# Chaotic system w/ initial condition

1. Apr 9, 2012

### Brown Arrow

so i have been studying chaotic system in class, and i just want to know if we change the initial conditions of a chaotic system can it become non-chaotic?

I think yes because, chaotic system is sensitive to initial condition hence it would have an effect on the chaotic behavior.

I'm I right? i have a feeling I'm wrong.

:/ I'm contradicting my self

2. Apr 9, 2012

### Q_Goest

Hi Arrow. I'm no expert on chaotic systems but isn't a pencil balanced on its point a chaotic system?
Ref:
RIGB.org
The Frontiers of Science
If so, then what happens if the initial conditions of the perfectly vertical pencil were to be changed?

3. Apr 9, 2012

### Brown Arrow

thanks for the reply Q Goest.

umm yes that is true it will no longer be chaotic.

guess i did not specify the system :/

I had a damped driven pendulum and Duffing Oscillator,

I ran some plotting (in python) for it and changed the initial condition. what once was a chaotic system became non-chaotic after the change in initial condition.

so is it safe to assume that chaotic system can become non-chaotic depending on the initial condition? from the reference you gave Q Goest i think the answer is yes.

4. Apr 9, 2012

### Filip Larsen

In short, for a (dissipative) dynamic system with a given fixed set of parameters there can in general be one or more attractors (with at least one of these being a chaotic attractor a.k.a. strange attractor if the system is to be chaotic) each with an associated basin of attraction. If the system in addition to the chaotic attractor(s) has a non-chaotic attractor (say, a fix point) then there obviously must be some initial conditions, namely those in the basic of attraction for this non-chaotic attractor, that will lead to a non-chaotic trajectory.

For the Duffing Oscillator (Duffing's Equation) I believe there are parameters for which the system has both chaotic and non-chaotic trajectories, and in those cases you will get chaotic or non-chaotic trajectory depending on the initial conditions. For instance, it looks like there should be both a chaotic and non-chaotic attractor for k = 0.2 and B = 1.2 (liftet from Ueda's parameter map for Duffing's Equation as it is shown in [1]).

[1] Nonlinear Dynamics and Chaos, Thompson and Steward, Wiley, 2002.