Time dependent forcing and nonlinear systems

[nickthequick]

Hi,

I'm trying to find a toy (i.e. analytic) example of a nonlinear system that has very different behavior for two different types of forcing:

1) $\frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = F(x)$

where u(x,t) is the dependent variable, N represents some nonlinear operator with only spatial derivatives and F is a forcing, independent of time.

2)$\frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = G(x,t)$


where G now represents a time dependent forcing. I also constrain (1) and (2) to impart the same total amount of momentum to the system in some given time/space interval that we are examining.


I can think of particular scenarios in fluid dynamics where very different behavior can be achieved under similar scenarios to those written above, but I'd like to first think about this for a simpler system, and am currently drawing a blank on examples.

Thanks!

Nick

 

[eljose79]

Hello Nick,

One toy example of a nonlinear system that exhibits different behavior for different types of forcing could be the Van der Pol oscillator. This is a simple model that describes the behavior of a nonlinear oscillator, and it can be written in the form of your equations (1) and (2).

In its simplest form, the Van der Pol oscillator is described by the following equations:

1) \frac{d^2x}{dt^2} - \mu (1-x^2)\frac{dx}{dt} + x = F(t)

2) \frac{d^2x}{dt^2} - \mu (1-x^2)\frac{dx}{dt} + x = G(t)

where x is the dependent variable, \mu is a parameter that controls the nonlinearity, and F(t) and G(t) are different types of forcing functions.

For equation (1), we can choose a forcing function F(t) that is a constant, such as F(t) = 1. This would represent a constant external force acting on the oscillator. In this case, the behavior of the oscillator would be stable and predictable.

However, for equation (2), we can choose a forcing function G(t) that is time-dependent, such as G(t) = \sin(t). This would represent a periodic external force acting on the oscillator. In this case, the behavior of the oscillator would become chaotic and unpredictable.

Both equations have the same amount of total momentum imparted to the system, but the different types of forcing lead to very different behaviors. This example can help illustrate the concept of how different types of forcing can affect the behavior of a nonlinear system.

I hope this helps in your search for a toy example. Best of luck!