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Find the fixed points of a nonlinear two-dimensional system:

$$\dot{x} = \sin y$$ $$\dot{y} = x - x^3.$$

I know that $0 = x(1 - x²) \implies x = 0, 1, -1$. I am not sure what to do after this.

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    By fixed point, I think you mean the equilibrium points of the system, right? Then simply solve $\dot{x}=0$ and $\dot{y}=0$. Any solution would be an equilibrium. In your OP, there are infinite ones, such as $y=k\pi$ and $x=0,1,-1$. Of course, the stability of different equilibriums may be different. That would be another story.2012-12-07

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