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I learnt that any autonomous system can be re-written how a non-autonomous system in this thread When can a non-autonomous system NOT be re-written as an autonomous system? So I thought how change the portrait phase of this systems for an given system? For instance when the system autonomous is in two dimension the system non-autonomous will be in three dimension. I thin perhaps that is the same solution but it just lengthens in the three variable that appears. It is really? can someone give me an example when i can to see the phase portrait? Thanks.

Zacarias89.
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    This is the other way round. We turn a non-autonomous system into an autonomous one by increasing its dimension by one. The added dimension is not that interesting when it is time as it will just evolve according to time itself. So, in the end you can just look at the phase portrait in the original dimensions. Note, however, that the phase portrait will now depend on the initial time and the initial conditions, and not only the initial condition since the system is non-autonomous. – KBS Mar 11 '22 at 04:13
  • Hello @KBS. Thanks for the answer. is really This is the other way round. In the thread I was thinking in for instance $(\dot{x},\dot{y}) = (x-y,x+y)$ which is a linear focus, so when I put this system in non-autonomous what is the phase portrait perhaps spirals? – Zacarias89. Mar 11 '22 at 22:08
  • I am sorry but I do not really understand what you want to say. You do not need to formulate the system you give in non-autonomous form (which does not really mean much here). If the trajectories are spiraling out, this is because the eigenvalues of the system have a positive real part with a non-zero imaginary part. The real part is responsible for the divergence and the imaginary part for the oscillations. In polar coordinates, the system writes $\dot{\rho}=\rho$ and $\dot{\theta}=1$, which describes a spiral. – KBS Mar 11 '22 at 22:28
  • Hi, I was thinking in put the system $(\dot{x},\dot{y})=(x−y,x+y)$, if this were possible or made sense in non-autonomous system, then see what happens with phase portrait. But i don´t know if this make sense. – Zacarias89. Mar 11 '22 at 22:39
  • This system is autonomous. You cannot turn it into a non-autonomous system without changing it; i.e. adding some time-varying components. An autonomous system is a particular case of a non-autonomous system, a degenerate form that does not depend on time. I am not sure you really understand what autonomous and non-autonomous really mean. I guess you should back to the definitions. – KBS Mar 11 '22 at 22:45
  • Yes KBS, I agree that the system is autonomous and that is precisely my question how change the phase portrait when I adding other variable for instance $t$ so this system will be non-autonomous. – Zacarias89. Mar 12 '22 at 14:30
  • Then, in this case, you will have to more explicit on how you add "t" in the system. Results will be completely different depending on how this variable enters the system. – KBS Mar 13 '22 at 14:11

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