Consider a differential equation given by $x'=f(x,t)$ Depending on the parameter $ t $, that is, not autonomous. I want to understand the behavior of the solutions to these equations. For example: Consider a family of ODE's of the type: $$ x' = x - x^{3} - b\sin\left(\,{2\pi t}\,\right) $$
In order to understand the phase diagram I considered the case where $ b $ is null, thus $ b\sin (2 \pi t) $ translates $ x-x ^ 3 $ on the vertical axis. If I take $ x '= 0 $, we have $ x-x ^ 3-b\sin (2 \pi t) = 0 $ and I can find the equilibrium points. I would like to know what happens when I take small $ | b | $, or when I increase that module.
One way to study the solutions is to plot this ODE graph and analyze the bifurcation points.
Can anyone help me know if this ODE family has a periodic solution when $ | b | $ is small or large? Can I draw the line $ y = x $ and find the points of intersection with the graph $ x-x ^ 3-b\sin (2 \pi t)$ ?
Could someone show me details of whether this family has periodic solutions?
The function $f(x)=x-x^3$ is surjective on the real line. Thus it is always possible to find $x_b>1$ with $f(x_b)=-2|b|$, $f(x)<-2|b|$ for $x>x_b$. By symmetry $f(-x_b)=2|b|$ etc.
This gives the interval $[-x_b,x_b]$ as a trapping region, as $x'\le -|b|$ for $x\ge x_b$ and $x'\ge|b|$ for $x\le-x_b$.
If $\phi(t;t_0,x_0)$ is the flow of the ODE, then the map $x\mapsto\phi(1;0,x)$ has the image of $[-x_b,x_b]$ inside itself, which implies that a fixed point $x_*$ exists. It is now easy to prove that $\phi(t;0,x_*)$ is a periodic solution.
If $|b|<\frac1{3\sqrt3}$ is small enough, then other lines with $|f(x)|=2|b|$ can be identified around the roots $0,\pm1$ of $f$. This gives rise to periodic solutions oscillating around these roots.
Numerical experimentation suggests that the pattern of 3 periodic solutions is not only true for small $b$.
See Is this periodic solution unique? (ODE) (and links) for a similar problem.