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A phase space is a mathematical space with coordinates representing all possible states of a system. Each point in this space corresponds to one possible state of the system.

In phase space, a trajectory is a path that represents the state of the system evolving over time. It is determined by the initial conditions and the system's equations of motion.

A fixed point in a dynamical system is a point in phase space where the system does not change in time. For a continuous system, it is where the derivatives are zero.

Stability in dynamical systems refers to whether a system returns to a fixed point after a small perturbation. Stable if it returns, unstable if it diverges.

An attractor is a set of numerical values toward which a system tends to evolve. It can be a point, curve, manifold, or more complicated structure.

A limit cycle is a closed trajectory in phase space that represents a periodic orbit. Systems that exhibit this behavior are oscillatory.

Bifurcation is a qualitative change in the structure of a system's phase space, often as a parameter is varied. It marks the appearance of a new system behavior.

A Hamiltonian system is a dynamical system governed by Hamilton's equations. The Hamiltonian function represents the total energy of the system.

A Lyapunov function is a scalar function that decreases along system trajectories and is used to assess the stability of equilibrium points in phase space.

A Poincaré map is a specific type of recurrence plot used in the study of dynamical systems to analyze the behavior of orbits by looking at their intersections with a lower-dimensional subspace.

Ergodicity describes a system where, in the long term, every state is equally probable. Any one trajectory will visit every possible state given sufficient time.

A center manifold is an invariant manifold in a dynamical system containing trajectories that neither exponentially grow nor decay. They often separate regions of different behavior.

Floquet theory analyzes the stability of solutions to differential equations with periodic coefficients. It extends the concept of eigenvectors and eigenvalues to periodic systems.

A strange attractor is an attractor that has a fractal structure. This is typical in chaotic systems, where it exhibits sensitive dependence on initial conditions.

A homoclinic orbit is a trajectory in phase space that joins a saddle point to itself. It is a connection between the stable and unstable manifolds of the same point.

A heteroclinic orbit is a trajectory in phase space that connects two different saddle points. It represents a path from the unstable manifold of one point to the stable manifold of another.

Hamilton's equations describe the evolution of a physical system in Hamiltonian mechanics. It consists of two sets of equations: