Calculate the eigenvalues of the Jacobian matrix.Ĥ. Calculate the Jacobian matrix of the system at the equilibrium point.ģ. Find an equilibrium point of the system you are interested in.Ģ. Linear stability analysis of discrete-time nonlinear systemsġ. Figure 5.7.2 shows a schematic summary of these classifications of equilibrium points for two-dimensional cases. Such equilibrium points are called a stable or unstable spiral focus or a neutral center, depending on their stabilities. Finally, if the eigenvalues are complex conjugates, oscillatory dynamics are going on around the equilibrium points. Once the dynamic is given, it is the task of mathematical dynamical systems theory to investigate the patterns of how states change in the long run. But actually, proving that the point is truly neutral requires more advanced nonlinear analysis, which is beyond the scope of this textbook. If\(|λd| = 1\), it indicates that the system may be neutral (also called Lyapunov stable), which means that the system’s state neither diverges away from nor converges to the equilibrium point. Such equilibrium points are called saddle points, where nearby trajectories are attracted to the equilibrium point in some directions but are repelled in other directions. Sometimes, an unstable equilibrium point may come with other eigenvalues that show stability. The Lefschetz Center for Dynamical Systems at Brown University promotes research in dynamical systems interpreted in its broadest sense as the study of evolving systems, including partial differential and functional equations, stochastic processes and finite-dimensional systems. )| 1\), the equilibrium point is unstable any small perturbation added to the system’s state grows exponentially and, eventually, the system’s state moves away from the equilibrium point.
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