Summary of Unit Seven

Phase Space:Two and Three-Dimensional

Dynamical Systems

Introduction to Dynamical Systems and Chaos

http://www.complexityexplorer.org

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Two-Dimensional Differential Equations

●

● Main example: Lotka-Volterra equations● Basic model of predator-prey interaction

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

The Phase Plane

● Plot R and F against each other● Similar to a phase line for 1D equations● Shows how R and F are related

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

The Phase Plane

● Often useful to show several different solutions on the same phase plane

● Gives a view of the qualitative features of the differential equations

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Determinism in the Phase Plane

● The dynamical system is deterministic.● The initial condition and the differential

equation specifies a unique path through the phase plane.

● As a result, curves on the phase plane can not cross.

● If they did cross, it would violate determinism.

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

No Chaos in 2D Differential Equations

● The fact that curves cannot cross limits the possible long-term behaviors of two-dimensional differential equations.

● There can be stable and unstable fixed points, orbits can tend toward infinity, and there can be limit cycles, attracting cyclic behavior.

● Poincaré–Bendixson theorem: bounded, aperioidc orbits are not possible for two-dimensional differential equations.

● Thus, 2D differential equations can not be chaotic.

(There are some nice analytic techniques for 2D differential equations, but they are beyond the scope of this course. They are covered in most Diff EQ textbooks.)

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Two-Dimensional Iterated Functions

● ● Example: Hénon Map●

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Two-Dimensional Iterated Functions

●

● Two-dimensional iterated functions are similar to one-dimensional iterated functions.

● Their orbits can be periodic.● Their orbits can be chaotic: bounded,

aperiodic, with sensitive dependence on initial conditions.

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Three-Dimensional Differential Equations

●

● Example: Lorenz equations● Simplified model of atmospheric convection

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Three-Dimensional Differential Equations

● Solutions are x(t), y(t), and z(t).

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Phase Space

● Instead of a phase plane, we have (3d) phase space.

David P. Feldman Introduction to Dynamical Systemsand Chaos

http://www.complexityexplorer.org

Phase Space

● Curves in phase space cannot intersect.● But because the space is three-dimensional,

curves can go over or under each other.● This means that 3D differential equations are

capable of more complicated behaviors than 2D differential equations.

● 3D differential equations can be chaotic.● We will explore chaos in 3D differential equations

and 2D iterated functions in the next unit.