Nonlinear System Dynamics

Consider the nonlinear system

Definition 30

Definition 31

Definition 32

Definition 33

An invariant set is one which a trajectory of the system will never leave once it enters the set. Just like linear systems, non-linear systems can also have periodic solutions.

Definition 34

Solutions to Nonlinear Systems

Consider the nonlinear system

Definition 35

Because the system is nonlinear, it could potentially have no solution, one solution, or many solutions. These solutions could exist locally, or they could exist for all time. We might also want to know when there is a solution which depends continuously on the initial conditions.

Theorem 7 (Local Existence and Uniqueness)

Theorem 8 (Global Existence and Uniqueness)

Once we know that solutions to a nonlinear system exist, we can sometimes bound them.

Theorem 9 (Bellman-Gronwall Lemma)

Another thing we might want to do is understand how the nonlinear system reacts to changes in the initial condition.

Theorem 10

Planar Dynamical Systems

It turns out that understanding the linear dynamics at equilibrium points can be helpful in understanding the nonlinear dynamics near equilibrium points.

Theorem 11 (Hartman-Grobman Theorem)

essentially says that the linear dynamics predict the nonlinear dynamics around equilibria, but only for a neighborhood around the equilibrium point. Outside of this neighborhood, the linearization may be very wrong.

Theorem 12 (Bendixon's Theorem)

Theorem 13 (Poincare-Bendixson Theorem)

Last updated