ODE system. Limit cycle; Hopf bifurcation.

In summary, the two-dimensional system of ODEs has a limit-cycle solution for certain values of the parameter a. At the critical value of a=0, a stable limit cycle appears due to a Hopf bifurcation. The nature of the bifurcation is that for negative a, paths spiral into (0,0) and then towards the stable limit cycle, while for positive a, paths spiral towards the limit cycle. The equilibrium point at (0,0) is an attractor spiral for negative a, a repellor spiral for positive a, and a center for a=0.
  • #1
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Homework Statement



The following two-dimensional system of ODEs possesses a limit-cycle solution for certain values of the parameter [itex]a[/itex]. What is the nature of the Hopf bifurcation that occurs at the critical value of [itex]a[/itex] and state what the critical value is.

Homework Equations



[itex]\dot{x}=-y+x(a+x^2+(3/2)y^2)[/itex]

[itex]\dot{y}=x+y(a+x^2+(3/2)y^2)[/itex]

The Attempt at a Solution



By setting each equation to zero, i found the only equilibrium point to be [itex](0,0)[/itex].

For the Jacobian matrix at [itex](0,0)[/itex], I have:

[itex]J(0,0)=\left( \begin{array}{cc}
a & -1\\
1 & a\end{array} \right)[/itex]

So:

[itex]\tau=2a[/itex]

[itex]\delta=a^2+1>0[/itex]

[itex]\bigtriangleup=4a^2-4a^2-4=-4<0[/itex]

which gives:

[itex]a<0[/itex]: [itex](0,0)[/itex] is an attractor spiral.

[itex]a>0[/itex]: [itex](0,0)[/itex] is a repellor spiral.

[itex]a=0[/itex]: [itex](0,0)[/itex] is a center.

Does this mean that paths spiral into [itex](0,0)[/itex] for negative [itex]a[/itex], and then spiral out towards a stable limit cycle. And, for positive [itex]a[/itex], beyond the limit cycle, paths spiral in towards it.

How do I find the critical value?
 
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  • #2
Is it just 0? The nature of the Hopf bifurcation that occurs at the critical value of a is that a stable limit cycle appears when a passes through 0. The critical value is a=0.
 

Related to ODE system. Limit cycle; Hopf bifurcation.

1. What is an ODE system?

An ODE (Ordinary Differential Equation) system is a mathematical model used to describe the behavior of a dynamical system over time. It consists of one or more differential equations that relate the rate of change of a variable to its current value.

2. What is a limit cycle in an ODE system?

A limit cycle is a periodic solution in an ODE system where the state of the system repeats itself after a certain period of time. This means that the variables in the system will oscillate between certain values, creating a cyclical behavior.

3. How does a limit cycle arise in an ODE system?

A limit cycle can arise in an ODE system when the system has a stable equilibrium point and the behavior of the system is dependent on a parameter. As the parameter changes, the system can undergo a bifurcation, resulting in the emergence of a limit cycle.

4. What is a Hopf bifurcation in an ODE system?

A Hopf bifurcation is a type of bifurcation that can occur in an ODE system where a stable equilibrium point turns into an unstable limit cycle. This means that as the parameter of the system changes, the system transitions from a steady state to a cyclical behavior.

5. How can Hopf bifurcation be predicted in an ODE system?

Hopf bifurcation can be predicted by analyzing the eigenvalues of the Jacobian matrix of the system at the equilibrium point. If the eigenvalues cross the imaginary axis, it indicates the possibility of a Hopf bifurcation, which can be further confirmed through numerical simulations.

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