Two-dimensional discrete-time laser model with chaos and bifurcations

We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in ${\mathbb{R}}_{+}^2$. It is shown that for all $a$, $b$, $c$ and $p$, model has boundary fixed point $P_{0y}(0,\frac{p}{c})$, and the unique positive fixed point $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$ if $p>\frac{bc}{a}$. Further, local dynamical characteristics with topological classifications for the fixed points $P_{0y}(0,\frac{p}{c})$ and $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$ have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point $P_{0y}(0,\frac{p}{c})$, and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$. Moreover, it is also explored that for the fixed point $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$, laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period $-2$, $-3$, $-4$, $-5$, $-8$ and $-9$. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model.