Toward Analytical Chaos in Nonlinear Systems

2.3 Bifurcation and Stability Switching

The dynamical characteristics of equilibriums in nonlinear dynamical systems in Equation (2.4) are based on the given parameters. With varying parameters in dynamical systems, the corresponding dynamical behaviors will change qualitatively. The qualitative switching of dynamical behaviors in dynamical systems is called bifurcation and the corresponding parameter values are called bifurcation values. To understand the qualitative changes of dynamical behaviors of nonlinear systems with parameters in the neighborhood of equilibriums, the bifurcation theory for equilibrium of the nonlinear dynamical system in Equation (2.4) will be investigated. and will be adopted from now on. For no specific notice, .

Definition 2.19

Consider an n-dimensional, autonomous, nonlinear dynamical system in Equation (2.4) with an equilibrium point . Suppose there is a neighborhood of the equilibrium as , and in the neighborhood, Equation (2.24) holds. The linearized ...