Deriving a Hopf Bifurcation – Perturbation Method vs. Jacobian Matrix

bifurcationdynamical systemsjacobiannonlinear dynamicsperturbation-theory

When deriving a Hopf bifurcation of a dynamical system, the usual process is:

  1. Find a fixed point $(x_0, y_0)$
  2. Perturb the system about the fixed point $(x_0+\tilde{x}, y_0+\tilde{y})$
  3. Linearize, neglecting terms quadratic or higher in the perturbation
  4. Find the solution of the linearized system

For example, starting with
$$ \frac{dx}{dt} = −y+(a−x^2−y^2)x $$
$$ \frac{dy}{dt} = x+(a−x^2−y^2)y $$
we deduce a fixed point at $(x,y) = (0,0)$, perturb the solution by substituting $(x,y)$ for $(0+\tilde{x}, 0+\tilde{y})$, and linearize the result to obtain:

$$ \frac{d\tilde{x}}{dt} = a\tilde{x}− \tilde{y} $$
$$ \frac{d\tilde{y}}{dt} = \tilde{x}+a\tilde{y} $$

The matrix associated with this linear system is:
$$
\begin{bmatrix}
a & -1 \\
1 & a \\
\end{bmatrix}
$$

The resulting eigenvalue problem yields eigenvalues of $s = a \pm i$. However, another method to deduce the Hopf bifurcation here is a little more straightforward:

  1. Find the Jacobian matrix $J(x,y)$
  2. Evaluate $J(x,y)$ at the fixed point $(x_0, y_0)$
  3. Find the eigenvalues of $J(x_0, y_0)$

Indeed, the Jacobian matrix of our example, evaluated at $(0,0)$, is
$$
\begin{bmatrix}
a & -1 \\
1 & a \\
\end{bmatrix}
$$

This is the same matrix derived from perturbing the system, and of course its eigenvalues will be the same. My question is: Why go through the machinery of perturbing the solution, when directly finding and evaluating the Jacobian produces the same result? In what cases will/won't it produce the same result?

Best Answer

Both methods, the perturbation approach and the Jacobian approach, share a common underlying concept, which is to linearize the differential equation in the vicinity of the equilibrium point and analyze the trajectory behavior in that region. The perturbation approach is typically employed when the linear component of the differential equation is easily identifiable. Also, one may consider different kinds of perturbations, at different rates, which could be useful to study certain families of solutions of the differential equation.

On the other hand, the Jacobian approach is more commonly used as it provides a means to linearize a system of differential equations through differentiation alone, making it valuable even in cases where the linear components are not immediately apparent. This is particularly useful when dealing with systems containing non-polynomial terms that complicate the identification of the linear parts upon initial inspection.

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