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The invariant curve caused by Neimark-Sacker bifurcation. (English) Zbl 1018.37030
It is considered the difference equation $x_{n+1}=f_{\mu}(x_n)=Ax_n+F_{\mu}(x_n),\quad x_n \in \mathbb{R}^m,\quad \mu \in \mathbb{R}$ where $$A$$ is the Jacobian matrix of $$f_{\mu}$$ evaluated at the fixed point $$x^*$$ (for sufficiently small $$\mu$$), having a pair of simple eigenvalues $$\lambda(\mu),\overline{\lambda}(\mu)$$ satisfying the conditions $$|\lambda(0)|=1; \lambda(0)^j\neq 1, j=1,2,3,4; \frac{d}{d\mu}|\lambda(\mu)|\bigm|_{\mu=0}\neq 0$$. The other eigenvalues have moduli less than 1 for $$\mu$$ sufficiently small. The Neimark-Sacker bifurcation: under some conditions on the nonlinear term $$F_{\mu}$$, the invariant curve can be created around the fixed point $$x^*$$ when the parameter $$\mu$$ is varied near zero. The author derives the formula to compute the stability conditions of the invariant curve and its explicit expression. These results are applied to the delayed logistic equation.

##### MSC:
 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 39A11 Stability of difference equations (MSC2000)