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Stable manifold consists of all points that approach a fixed point or a periodic orbit asymptotically as time goes to infinity. They outline the behavior of the system near attractors.

The center-stable manifold is an invariant manifold that combines the center and stable manifolds, composing of trajectories that neither diverge nor converge rapidly but eventually approach a critical point asymptotically.

A heteroclinic orbit is a path in the phase space that connects two different saddle points. Such orbits indicate possible transition paths between different system states.

Global invariant manifold refers to the structure in phase space that characterizes the behavior of trajectories over extended regions, not just close to equilibrium points. It provides insights into the global dynamics of a system.

Local invariant manifold pertains to the behavior of trajectories in the immediate vicinity of a critical point. It helps in understanding the local dynamics and stability of a system's solution.

The strong stable manifold is a subset of the stable manifold consisting of points that converge to a fixed point at an exponential rate, governed by the strongest (most negative) eigenvalues.

Center manifold is the set of points with dynamics that neither converge nor diverge rapidly, often associated with eigenvalues with zero or purely imaginary real parts. It captures the neutral behavior of a system.

Unstable manifold comprises all points that diverge from a fixed point or a periodic orbit when tracked backwards in time. They highlight the regions from where the system repels.

A homoclinic orbit is a trajectory in the phase space of a dynamical system that joins a saddle point to itself. It plays a significant role in the occurrence of complex dynamics such as chaos.