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Refers to a property of a dynamical system that stays qualitatively the same under small perturbations of the system. Example: Anosov diffeomorphisms are structurally stable as their qualitative behavior does not change under small perturbations.

A fixed point in a dynamical system where the Jacobian matrix has no eigenvalues on the unit circle in the complex plane. Nearby trajectories diverge rapidly from the fixed point. Example: In the system $x_{n+1} = 2x_n$, the point $x = 0$ is a hyperbolic fixed point.

A generalization of uniform hyperbolicity where stable and unstable manifolds still exist but their rates of expansion and contraction may vary from point to point. Example: The Hénon map is a system exhibiting nonuniform hyperbolicity for certain parameter values.

A diffeomorphism of a compact manifold that is uniformly hyperbolic on the entire manifold. Points are stretched along one direction and compressed along another under its tangent map. Example: The Arnold's cat map on the torus is an Anosov diffeomorphism.

A tool for studying the behavior of a dynamical system by considering its intersection with a lower-dimensional slice (section) and the associated return map. Example: The Poincaré map of a periodic orbit in a continuous-time dynamical system provides a discrete map that can reveal the system's stability.

Refers to systems where the tangent space at each point can be split into stable and unstable directions, with distinct behavior (exponential shrinking and expanding respectively). Example: The Lorenz system exhibits hyperbolicity in certain parameter regimes.

States that for each hyperbolic fixed point or periodic point, there exist stable and unstable manifolds, which are as smooth as the map, tangent to the eigenspaces of the Jacobian at the point. Example: The stable manifold of a saddle point in a linear system is the set of points that approach the saddle point as time goes to infinity.

Orbits that repeat themselves after a fixed period. In hyperbolic systems, they often serve as the 'skeleton' around which the dynamics are organized. Example: In the logistic map, periodic orbits can be found for certain parameter values, and they play a crucial role in understanding the map's bifurcations.

A branch of mathematics that studies the statistical properties of dynamical systems that are invariant under time evolution. It relates the long-term average behavior of orbits to the structure of the space they occupy. Example: Ergodic theory deals with questions like whether the time average of a function along the orbits equals its space average.