Homoclinic orbit: starts and ends in the same
unstable equilibrium point.
Heteroclinic orbit: links several unstable
8. Lagrangian dynamics
= constraint forcej
9. Linear systems
Type of point
2 real with opposite signs
2 real and positive
2 real and negative
2 complex with positive real part
2 complex with negative real part
1 real, positive
1 real, negative
10. Nonlinear systems
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.
12. Chaotic systems
Positive limit set:
Negative limit set:
Poincaré-Bendixson theorem. In a system with only
two state variables, if
exist, they must be one of the following three cases:
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
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