CDS - Theorem 'Existence of Solutions'

Remember:

Theorem ‘Existence of Solutions’: Given an initial condition $x(t_0)=x_0$, and under the hypotesis that $f(x)$ is differentiable, a solution of equation $\dot{x}=f(x)$ exists and this solution is unique.

• This theorem implied that a solution in the phase space can’t intersect itself. It it intersect itself once it will intersect itself infinite times.
• Solutions are continuos on $x$ in the phase space.
• Steady states are constant solutions: fixed points of the phase space.

Unique solution in the phase space means that given a certain $x_0,\bar{t}$ we obtain a single point, as we will see that is always the case.

---

• So when desining the geometry of the phase space, we will have that ==a solution in the phase space intersect itself infinite times or not even once.==, still under the hypotesis that $f$ is a differentiable function.not-sure-about-this