Taking a look in Wikipedia, we found
$${\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}$$
as part of the definition of dynamical systems.
I am new in this area, so I ask:
- This is the same that the so called "invariance solution" property?
- This "invariance solution" property is about autonomous system, isn't? If yes, how to propose non-autonomous dynamical systems?
These two questions are about the same ideia: in a dynamical system, I can put any point in a trajectory and the solution for that initial condition will have the same orbit. Am I right?
For instance, what about $dx/dt=t$? The orbit will change if I start in other point of the trajectory… Is this a dynamical system?
Thank you so much.
Best Answer
The link you posted is about time-invariance of the dynamical system, meaning that the behavior of the system at some time $t$ is the same as at some other time $t′$.
For instance, the dynamical system $\dot{x}(t)=ax(t)$ where $a\in\mathbb{R}$ is time invariant. If you start the system at time $t_0$ from the state $x(t_0)=x_0$ and if you start the system at time $t_1\ne t_0$ from the state $x(t_1)=x_0$, then the two trajectories will be translated versions of each other.
The dynamical system $\dot{x}(t)=tx(t)$ is not time-invariant, but time-varying. In this case, the initial time will matter.
The composition formula $${\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}$$ works for both time-varying and time-invariant systems (with no input).
Finally, the example you gave is indeed a dynamical system whose state is given by $x(t)=x(t_0)+(t-t_0)^2/2$. In this case, the system has an input and this is not captured by the $\Phi$ formula you gave which is for systems with no input. In fact, the solution to that system is given by
$$x(t)=\Psi(t,s)x(s)+\int_s^t\Psi(t,\theta)\theta d\theta,$$
where $\Phi(t,s,x)=\Psi(t,s)x=x$, $t\ge s$, and where $\Psi(t,s)$ is the so-called state transition matrix (see e.g. https://en.wikipedia.org/wiki/State-transition_matrix). Finally, since the system is time-varying, the trajectory will not only depend on the initial condition but also on the initial time.
If you want to know more about dynamical systems, you may look at "Nonlinear Dynamics and Chaos" by Steven Strogatz. It is usually a good starting point.