Stabilization and destabilization of periodic solution of Stuart-Landau equation

Bogaevskaya V.G., Kashchenko I.S.

Yaroslavl State University, Russia , Russia

We study model Stuart-Landau equation that has qubic nonlinearity: $\dot{z} = \sigma z+\gamma|z|^2z.$ Here $z(t)$ is complex-valued function; $\sigma$ and $\gamma$ are complex parameters. Using the assumption that the equation has periodic solution we researched two issues: if this cycle is unstable we tried to stabilize it by delay feedback and if it's stable we solved the problem of destabilisation by the same way. We show that it's always possible to destabilize stable periodic orbit using delay feedback $K(z(t-T) -z(t))$. Necessary and sufficient conditions for control’s parameters were found for this type of feedback. Stabilization problem is significant that only one multiplier of unstable cycle located outside unit circle. We found analytically conditions for parameters of delay feedback, necessary and sufficient conditions for coefficients of initial problem that the cycle became stable. Also we researched the influence of several delay feedback summands of this type to possibility of stabilization unstable orbit in some cases.