58Fxx Dynamical systems on manifolds
This subtopic introduces the core ideas in dynamical systems on manifolds, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
58F05 (Global theory of) Hamiltonian flows on manifolds
Overview
(Global theory of) Hamiltonian flows on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F06 Completely integrable systems on manifolds
Overview
Completely integrable systems on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F07 Completely integrable systems and methods of complexification on manifolds
Overview
Completely integrable systems and methods of complexification on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F08 Strange attractors on manifolds
Overview
Strange attractors on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F09 Morse-Smale systems on manifolds
Overview
Morse-Smale systems on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F10 Anosov systems on manifolds
Overview
Anosov systems on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F11 Ergodic theory on manifolds
Overview
Ergodic theory on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F12 Structure of hyperbolic sets on manifolds
Overview
Structure of hyperbolic sets on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F13 Strange attractors on manifolds (MSC2000)
Overview
Strange attractors on manifolds (MSC2000). This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F14 Bifurcation on manifolds
Overview
Bifurcation on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F15 Expanding maps; hyperbolicity; structural stability on manifolds
Overview
Expanding maps; hyperbolicity; structural stability on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F17 Geodesic flows on manifolds
Overview
Geodesic flows on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F18 $G$-spaces with Riemannian metrics
Overview
$G$-spaces with Riemannian metrics. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F19 Explosions and implosions in global analysis
Overview
Explosions and implosions in global analysis. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F20 Hyperbolic flows and flows with heterodimensional cycles on manifolds
Overview
Hyperbolic flows and flows with heterodimensional cycles on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F21 Persistence and stability of solutions of ODE and iteration theory
Overview
Persistence and stability of solutions of ODE and iteration theory. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F22 Periodic orbits on manifolds
Overview
Periodic orbits on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F23 Quasi-periodic flows and tori on manifolds
Overview
Quasi-periodic flows and tori on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F25 Normal forms on manifolds
Overview
Normal forms on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F30 Homoclinic and heteroclinic orbits on manifolds and horseshoes
Overview
Homoclinic and heteroclinic orbits on manifolds and horseshoes. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F35 Discrete dynamical systems on manifolds
Overview
Discrete dynamical systems on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F36 Dimension theory, Lyapunov exponents, entropy
Overview
Dimension theory, Lyapunov exponents, entropy. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F37 Topological and differentiable equivalence of flows and diffeomorphisms on manifolds
Overview
Topological and differentiable equivalence of flows and diffeomorphisms on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F38 Ergodic theory, symbolic dynamics and coding
Overview
Ergodic theory, symbolic dynamics and coding. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F39 One-dimensional maps on manifolds
Overview
One-dimensional maps on manifolds. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F40 Real-analytic and Nash manifolds (MSC2000)
Overview
Real-analytic and Nash manifolds (MSC2000). This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks
58F99 None of the above
Overview
None of the above. This topic treats dynamical systems on manifolds, including smooth, topological, and ergodic dynamics with geometric structure.
Related Wikipedia Page
Wikipedia: Dynamical system
Useful Links
Key Ideas
- flows and diffeomorphisms on manifolds
- stability, chaos, and bifurcation behavior
- invariant measures and ergodic features
Typical Uses
Used to analyze long-term behavior of geometric and physical systems with manifold-valued state spaces.
Applications
- Celestial and mechanical systems
- Chaotic dynamics
- Geometric control and modeling
References
Recommended Textbooks