CDS - Bifurcation • Structural Stability • Bifurcation Value • Bifurcation Diagram

Remember:

Let’s consider a non-linear system $f$ that also depends on some parameter $k$, so:$\dot{x}=f(x,k)$We have that the system equations depends on the parameter $k$, so:

• The steady states, their existance, the Jacobian matrix of the linearized system as well as the stability of the steady states, can all depend on this parameter $k$.

Definition ‘Structural Stability’: A system is structurally stable if the flow in the phase space does not vary when the system’s parameters change.

• The structural stabilty is related to the stability of the steady states: if all steady states are hyperbolic, then the system is structurally stable.
• A change in the geometry of the phase space may occur only when a steady state is non-hyperbolic:
• ~Ex.: steady states that disappear when changing the parameters.
• ==To have structural instability we need that chagning the parameter we change the sing of the eigenvalues==, so for a certain value of the parameter, we have a change of sing, so for $k^{\ast }:{\lambda }_1\text{ or }{\lambda }_2=0$, we say that we have a “bifurcation” ($k$ is the parameter)

Consider a dynamic system:$\dot{x}=f(x,\mu )$Where: $x\in {\mathbb{R}}^n$ and $\mu \in {\mathbb{R}}^m$, $\mu$ are the parameters.

• Definition ‘Bifurcation’:
  Then we define the bifurcation as a drastic change of the flow of a dynamic system, occurring when some parameters $(\mu )$ change.
• Definition ‘Bifurcation Value’:
  A value ${\mu }_0$ such that the flow of the system is not structurally stable is a bifurcation value if the system.
• Definition ‘Bifurcation Diagram’:
  The bifurcation diagram reporst the values visited or approached asymptotically, such as steady states, by the system as a function of a bifurcation parameter in the system, thus it is drawn in the plane $(\mu ,x)$.

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• ~Ex.: steady states that disappear when changing the parameters.
• If in a system all ss are hyperbolic ⇒ the system is structurally stable.
• To have structural instability we need that chagning the parameter we change the sing of the eigenvalues, at the value of the parameter, for which we have the change of sing, so for ${\mu }^{\ast }:{\lambda }_1\text{ or }{\lambda }_2=0$, we have a “bifurcation” ($\mu$ is the parameter)

• $f$ is a non-linear function.