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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) [itex]\frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = F(x) [/itex]

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)[itex] \frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = G(x,t) [/itex]

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

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# Time dependent forcing and nonlinear systems

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