Stability of periodic solutions of singular pertubed Stuart-Landau equation

Kashchenko A.

Yaroslavl State University , Russia

It is considered singular perturbed Stuart-Landau equation $$ \varepsilon\dot{u}=(1+(-1+ic)|u|^2)u+\gamma e^{i\varphi}(u(t-1)-u). $$ Here $\varepsilon$ is small positive parameter ($0<\varepsilon\ll1$). We study existence and stability of periodic solutions $$ u_{R,\Lambda}=R \exp(i\Lambda t) $$ of given equation. Here $R, \Lambda$ are real constants and $R>0$. Let $L(c,\gamma,\varphi)$ be set of points $(\omega, ho^2)$ those belong to ellipse $$ ( ho^2-1+\gamma\cos\varphi)^2+(\omega-c ho^2+\gamma\sin\varphi)^2=\gamma^2 $$ and to half plane $ ho^2 > 0$. Then for every point $(\omega_0, ho_0^2)$ on ellipse $L(c,\gamma,\varphi)$ and for every integer number $n$ given equation has solution $R\exp(i\Lambda t)$. Real values $R=R_n(\varepsilon)$ and $\Lambda=\Lambda_n(\varepsilon)$ are presented below. $$ R_n(\varepsilon)= ho_0+o(1), \qquad \Lambda_n(\varepsilon)=\omega_0/\varepsilon+\Omega+2\pi n+o(1) \quad (\varepsilon o0). $$ Here $\Omega=\Omega(\omega_0, ho_0,\varepsilon)$ is some function with values in $[0,2\pi).$ Necessary and sufficient conditions of stability this solution are found. Location of stable and unstable solutions on ellipses is studied. Particularly it is shown, that\ 1) there is at least one point with stable solutions and at least one point with unstable solutions on ellipse $L(c,\gamma,\varphi)$ at any parameters values ;\ 2) region of stable periodic solutions on ellipse $L(0,\gamma,\varphi)$ is simply connected.