37Exx Low-dimensional dynamical systems
This subtopic studies low-dimensional dynamical systems, where the geometry of phase space allows particularly rich classification and visualization of behavior.
Specific topics
37E05 Dynamical systems involving maps of the interval
Overview
37E05 studies dynamical systems involving maps of the interval in low-dimensional dynamical 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 involving maps of the interval
Useful Links
Key Ideas
- Core definitions and model classes for dynamical systems involving maps of the interval
- 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
37E10 Dynamical systems involving maps of the circle
Overview
37E10 studies dynamical systems involving maps of the circle in low-dimensional dynamical 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 involving maps of the circle
Useful Links
Key Ideas
- Core definitions and model classes for dynamical systems involving maps of the circle
- 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
37E15 Combinatorial dynamics (type of periodic orbits)
Overview
37E15 studies combinatorial dynamics (type of periodic orbits) in low-dimensional dynamical 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: Combinatorial dynamics (type of periodic orbits)
Useful Links
Key Ideas
- Core definitions and model classes for combinatorial dynamics (type of periodic orbits)
- 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
37E20 Universality and renormalization of dynamical systems
Overview
37E20 studies universality and renormalization of dynamical systems in low-dimensional dynamical 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: Universality and renormalization of dynamical systems
Useful Links
Key Ideas
- Core definitions and model classes for universality and renormalization of 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
37E25 Dynamical systems involving maps of trees and graphs
Overview
37E25 studies dynamical systems involving maps of trees and graphs in low-dimensional dynamical 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 involving maps of trees and graphs
Useful Links
Key Ideas
- Core definitions and model classes for dynamical systems involving maps of trees and graphs
- 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
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
Overview
37E30 studies dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces in low-dimensional dynamical 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 involving homeomorphisms and diffeomorphisms of planes and surfaces
Useful Links
Key Ideas
- Core definitions and model classes for dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
- 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
37E35 Flows on surfaces
Overview
37E35 studies flows on surfaces in low-dimensional dynamical 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: Flows on surfaces
Useful Links
Key Ideas
- Core definitions and model classes for flows on surfaces
- 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
37E40 Dynamical aspects of twist maps
Overview
37E40 studies dynamical aspects of twist maps in low-dimensional dynamical 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 aspects of twist maps
Useful Links
Key Ideas
- Core definitions and model classes for dynamical aspects of twist maps
- 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
37E45 Rotation numbers and vectors
Overview
37E45 studies rotation numbers and vectors in low-dimensional dynamical 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: Rotation numbers and vectors
Useful Links
Key Ideas
- Core definitions and model classes for rotation numbers and vectors
- 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