Question

For the Rössler system

x' = - y - z
y' = x + Ay
z' = B + z(x -C)

with A = B + 1/4, find a value of C for which one of the fixed points has a linearization with one real negative eigenvalue, and two complex eigenvalues with positive real parts.

Answer 1

The student needs to find a value of C for which the Rössler system's fixed point has a specific eigenvalue configuration, requiring calculations involving the Jacobian matrix and characteristic polynomial.

Step-by-step explanation:

The student is dealing with a dynamical system, specifically the Rössler system, and is seeking a value of C that results in a specific eigenvalue configuration for a fixed point. A linearization of a fixed point can be analyzed using the Jacobian matrix to find its eigenvalues.


The condition that must be satisfied is one negative real eigenvalue and two complex eigenvalues with positive real parts, which characterize a saddle-focus in dynamical systems theory. To find C satisfying these conditions, typically one would compute the Jacobian matrix at the fixed point, and then determine the eigenvalues by solving the characteristic polynomial, typically a cubic equation in this context.

To provide an example based on the provided formulas and information, we can consider an approach similar to that used in the solution of a quadratic equation at² + bt + c = 0. The real parts of the eigenvalues are found by using the real coefficients from the linearization and the complex parts follow from the discriminant of the characteristic polynomial when the quadratic formula is applied.