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Newhouse sinks in the self-similar bifurcation structure
1Laser and Plasma Technology Division, Bhabha Atomic Research Centre, Mumbai 400 085, India.
Summary
Numerical analysis reveals that periodically driven Toda oscillators exhibit self-similar bifurcation structures. Higher-order Newhouse sinks emerge with smaller basins, intertwined with lower-order sinks, creating complex dynamics.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Mathematical Physics
Background:
- Periodically driven systems, such as the Toda oscillator, exhibit complex dynamics.
- Newhouse sinks and saddles are critical elements in understanding bifurcations in dynamical systems.
- Subharmonic resonance regions and homoclinic tangencies are key phenomena in nonlinear dynamics.
Purpose of the Study:
- To numerically analyze the dynamics of a periodically driven Toda oscillator.
- To investigate the birth and behavior of primary and secondary Newhouse orbits.
- To elucidate the self-similar bifurcation structure in the parameter space.
Main Methods:
- Numerical analysis of the periodically driven Toda oscillator.
- Sweeping the control parameter in the parameter space near homoclinic tangency.
- Observing period n-tupling and period doubling processes.
Main Results:
- Primary Newhouse orbits (sinks and saddles) emerge sequentially in subharmonic resonance regions.
- First-order secondary Newhouse sinks are born via period n-tupling around primary sinks.
- Higher-order secondary Newhouse sinks arise recurrently, forming a self-similar structure with progressively smaller, intertwined basins.
Conclusions:
- The dynamics of the periodically driven Toda oscillator display a self-similar, hierarchical structure of bifurcations.
- Higher-order secondary Newhouse sinks possess smaller basins of attraction, leading to complex intertwined basin structures.
- Understanding these bifurcations is crucial for predicting the long-term behavior of such nonlinear systems.