Mathematical definition for Transient phase?

Iancar29

Hi guys , going through corrections for my final year project on Chaos Theory and my supervisor wants me to put in a mathematical definition of transient.
Can anyone here help me with one ?

I know what it is in laymens terms : The portion of a trajectory/ set of points that lead onto an attractor or chaotic pattern.

With marked example here...


Any help would be much appreciated

bnt

Now, I haven't done any chaos theory at all, and it's been a while since I did university maths, but what I see in the graphic is a single transient - the spike above the x-axis - and then the area you marked in blue looks like a transient response undergoing positive feedback. I can see the amplitude increasing as higher-level harmonics die away, leaving a single sine wave. (The response is usually more interesting than the transient itself.)

In mathematical terms a transient response is a solution to a differential equation that describes the system. Your transient response (before the chaotic response starts) is a periodic waveform, which tells me it's a second order differential equation. I don't know how much detail your lecturer wants beyond that kind of thing, but uf you want equations, have a look at e.g. this.

PS: this happens in reality e.g. a string on a guitar can be modelled by a 2nd order equation, with damping and mass as first and second order variables: pluck an electric guitar string while standing close to the amplifier, the result is a periodic waveform with positive feedback, and the result can be chaos if you're doing it right.