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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.

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