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Fractalization route to strange nonchaotic dynamics
Sandip Datta1, Ramakrishna Ramaswamy, Awadhesh Prasad
1School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
Summary
This study investigates the fractalization of tori in nonlinear dynamical systems, revealing how smooth tori become strange nonchaotic attractors (SNAs) as forcing amplitude increases. We identify unstable sets and link attractor merging crises to torus fractalization, explaining SNA formation.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Fractal Geometry
Background:
- Strange nonchaotic attractors (SNAs) form in quasiperiodically driven nonlinear systems.
- Torus fractalization is a key mechanism in SNA formation, where smooth tori become fractal as forcing amplitude increases.
- Lyapunov exponents remain nonpositive during this process.
Purpose of the Study:
- To investigate the fractalization route for SNA formation.
- To identify unstable sets within SNA's using approximation techniques.
- To understand the role of attractor merging crises in torus fractalization.
Main Methods:
- Studied torus fractalization by approximating quasiperiodic drives with periodic forcing of increasing period.
- Utilized techniques by Kim et al. to identify unstable sets in SNAs.
- Analyzed attractor merging crises in the periodically forced system.
Main Results:
- Identified an unstable set embedded within the attractor.
- Observed a cascade of attractor merging crises in the approximated system.
- Established a link between these crises and the fractalization of tori.
Conclusions:
- Attractor merging crises in periodically forced systems serve as an analogue to the process causing torus fractalization in quasiperiodic systems.
- This process explains the formation of strange nonchaotic attractors (SNAs).
- The fractalization route provides a mechanism for understanding SNA generation.