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Transcritical bifurcations in nonintegrable Hamiltonian systems
1Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany.
Summary
Transcritical bifurcations in Hamiltonian systems are common, especially for systems with librating orbits. This study details their properties and semiclassical approximations for quantum systems.
Area of Science:
- Mathematical Physics
- Dynamical Systems Theory
- Quantum Chaos
Background:
- Nonintegrable Hamiltonian systems exhibit complex dynamics, including bifurcations of periodic orbits.
- Understanding these bifurcations is crucial for characterizing system stability and behavior.
Purpose of the Study:
- To investigate transcritical bifurcations of periodic orbits in 2D nonintegrable Hamiltonian systems.
- To establish existence criteria and analyze properties of these bifurcations using symplectic maps.
- To explore their occurrence in generalized Hénon-Heiles Hamiltonians and their semiclassical implications.
Main Methods:
- Utilizing a mathematical description of transcritical bifurcations in families of symplectic maps.
- Performing numerical simulations on generalized Hénon-Heiles Hamiltonians.
- Deriving normal forms for transcritical and isochronous pitchfork bifurcations.
- Developing uniform approximations for semiclassical trace formulas.
Main Results:
- Transcritical bifurcations are typical for Hamiltonians with straight-line librating orbits, even without discrete symmetries.
- Isochronous pitchfork bifurcations are identified as exceptional cases.
- Excellent agreement was found between semiclassical and quantum mechanical computations of the density of states.
Conclusions:
- Transcritical bifurcations play a significant role in the dynamics of nonintegrable Hamiltonian systems.
- The derived semiclassical approximations accurately predict quantum mechanical properties like the density of states.
- This work provides a framework for analyzing complex dynamics in quantum chaos.
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