CDS - Recap of Linear Systems

Remember:

Given the LINEAR system:$\dot{x}=Ax$Where:

• $x\in {\mathbb{R}}^n$
• $A\in {\mathbb{R}}^{n\times n}$

We have seen that:

• The nullclines are linear in the phase space.
• There exists at most one isolated steady state, if more than one steady state exists, then they are inifinite, as we have seen in some previous examples.
• Drawing the geometry of the phase space is possible in all cases.
• The stability of steady states depends on the eigenvalues of $A$:
  • If all eigenvalues of $A$ have negative real part, then the steady state is asymptotic stable.
  • If at least one eigenvalue have positive real part, then the steady state is unstable.
  • If at least one eigenvalue have zero real part, there is some marginality and the steady state is marginally stable (or “neutrally stable”) along some direction.

We can classify the stady states as:

• Real eigenvalues:
  • Stable nodes (negative eig.)
  • Unstable nodes (positive eig.)
  • Lines or planes of steady states (one or more null eig.)
  • Saddles (at least one positive and one negative)
• Imaginary eigenvalues or complex conjugate eigenvalues:
  • Stable spirals (negative real part eig.)
  • Unstable spirals (positive real part eig.)
  • Centers (null real part)

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• Recap, basically it exaplains the plot of $\tau$ and $\Delta$, we have seen previously