Given an autonomous ordinary differential equation $\dot{x}=f(x)$ where $f(x)$ is at least continuously differentiable. The equilibria of the system can be calculated by letting $f(x)=0$. But when $f(x)$ is nonlinear, it can be very difficult to calculate the equilibria. Now rather than knowing the exact equilibria, I want to know the number of equilibria in this system. I think there must be some theorems regarding this question.
So:
- Could you mention some theorems regarding the number of equilibria of a dynamical system?
- Is there a theorem guaranteeing that there is only on equilibrium of a dynamical system?
This question is rather general, I hope you can give some hints.
I don't know any theorem like the ones you've required. By the way, whenever possible you can qualitatively study the function $f$ by exploiting its regularity, so by applying existence of zeros' theorem, computing the limits at the boundary of the domain and by supporting your study with the one of the critical points of $f$.
Sometimes it may be helpful to compare graphically the pieces that compose the function $f$ and see if they intersect, for example in the case:
$f(x) = \frac{1}{x^4+1} - (x^4-\frac{1}{2})^2$
you may see when $\frac{1}{x^6+1} = (x^4-\frac{1}{2})^2$
by drawing a qualitative graph of both the functions and check if there are intersections and in case they are, how many.