Homoclinic orbit: starts and ends in the same
unstable equilibrium point.

Heteroclinic orbit: links several unstable
equilibrium points.

8. Lagrangian dynamics

= constraint force_{j}

9. Linear systems

Eigenvalues:

Eigenvalues λ

Type of point

Stability

2 real with opposite signs

saddle point

unstable

2 real and positive

repulsive node

unstable

2 real and negative

attractive node

stable

2 complex with positive real part

repulsive focus

unstable

2 complex with negative real part

attractive focus

stable

2 imaginary

center

stable

1 real, positive

improper node

unstable

1 real, negative

improper node

stable

10. Nonlinear systems

Jacobian matrix:

At each equilibrium point, it is the matrix of the linear
approximation to the system near that point.

11. Limit cycles and population dynamics

Limit cycle: Isolated cycle in phase space.

Two-species systems

12. Chaotic systems

Positive limit set:
= where
curve
goes at
→∞

Negative limit set:
= where
curve
goes at
→−∞

Divergence:

Poincaré-Bendixson theorem. In a system with only
two state variables, if
or
exist, they must be one of the following three cases:

equilibrium point;

cycle;

homoclinic or heteroclinic orbit.

With 3 or more state variables, a limit set that does not
belong to any of those three classes is called a strange
attractor.

Bendixson criterium. In a dynamical system with only
two state variables, if the divergence is always positive or
always negative in a simply-connected region of the phase
space, then there are no cycles or orbits in that
region.