37Dxx Hyperbolic systems
This subtopic studies hyperbolic systems, including stable and unstable manifolds, chaotic behavior, and geometric mechanisms behind sensitive dependence on initial conditions.
Specific topics
37D05 Dynamical systems with hyperbolic orbits and sets
Overview
37D05 studies dynamical systems with hyperbolic orbits and sets in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Dynamical systems with hyperbolic orbits and sets
Useful Links
Key Ideas
- Core definitions and model classes for dynamical systems with hyperbolic orbits and sets
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D10 Invariant manifold theory for dynamical systems
Overview
37D10 studies invariant manifold theory for dynamical systems in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Invariant manifold theory for dynamical systems
Useful Links
Key Ideas
- Core definitions and model classes for invariant manifold theory for dynamical systems
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D15 Morse-Smale systems
Overview
37D15 studies morse-smale systems in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Morse-Smale systems
Useful Links
Key Ideas
- Core definitions and model classes for morse-smale systems
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Overview
37D20 studies uniformly hyperbolic systems (expanding, anosov, axiom a, etc.) in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Useful Links
Key Ideas
- Core definitions and model classes for uniformly hyperbolic systems (expanding, anosov, axiom a, etc.)
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Overview
37D25 studies nonuniformly hyperbolic systems (lyapunov exponents, pesin theory, etc.) in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Useful Links
Key Ideas
- Core definitions and model classes for nonuniformly hyperbolic systems (lyapunov exponents, pesin theory, etc.)
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D30 Partially hyperbolic systems and dominated splittings
Overview
37D30 studies partially hyperbolic systems and dominated splittings in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Partially hyperbolic systems and dominated splittings
Useful Links
Key Ideas
- Core definitions and model classes for partially hyperbolic systems and dominated splittings
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
Overview
37D35 studies thermodynamic formalism, variational principles, equilibrium states for dynamical systems in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
Useful Links
Key Ideas
- Core definitions and model classes for thermodynamic formalism, variational principles, equilibrium states for dynamical systems
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D40 Dynamical systems of geometric origin and hyperbolicity
Overview
37D40 studies dynamical systems of geometric origin and hyperbolicity in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Dynamical systems of geometric origin and hyperbolicity
Useful Links
Key Ideas
- Core definitions and model classes for dynamical systems of geometric origin and hyperbolicity
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Overview
37D45 studies strange attractors, chaotic dynamics of systems with hyperbolic behavior in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Useful Links
Key Ideas
- Core definitions and model classes for strange attractors, chaotic dynamics of systems with hyperbolic behavior
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks
37D50 Hyperbolic systems with singularities (billiards, etc.)
Overview
37D50 studies hyperbolic systems with singularities (billiards, etc.) in hyperbolic systems. Typical results characterize invariant structures, orbit complexity, and regularity/stability properties that determine global and asymptotic behavior of dynamical models.
Related Wikipedia Page
Wikipedia search: Hyperbolic systems with singularities (billiards, etc.)
Useful Links
Key Ideas
- Core definitions and model classes for hyperbolic systems with singularities (billiards, etc.)
- Invariant sets, recurrence, and long-time orbit behavior
- Rigorous techniques for stability, bifurcation, entropy, and structural properties
Typical Uses
Used to classify dynamical regimes, choose tools for proving recurrence or stability, and connect geometric, topological, and measure-theoretic viewpoints in long-time analysis.
Applications
- Qualitative modeling of deterministic time evolution in mathematics and physics
- Analysis of stability and transition phenomena in nonlinear systems
- Foundations for numerical and computational studies of complex dynamics
References
Recommended Textbooks