Bifurcation structures of periodically forced oscillators.

Area of Science:

• Nonlinear Dynamics
• Theoretical Physics
• Chemical Systems

Background:

• Periodically forced oscillators exhibit complex behaviors, including subharmonic entrainment.
• Arnold tongues (entrainment bands) are key features in the frequency-amplitude plane.
• Understanding the termination and global structure of these tongues is crucial.

Purpose of the Study:

• To theoretically investigate bifurcation structures in periodically forced oscillators.
• To analyze the termination of Arnold tongues at higher forcing amplitudes.
• To explore the formation of global bifurcation structures and their relation to resonances.

Main Methods:

• Application of the method of multiple time scales for small forcing amplitudes.
• Derivation of solutions on the invariant torus, Arnold tongue widths, and Liapunov exponents.
• Study of normal forms for periodically forced Hopf bifurcations at moderate to large amplitudes.

Main Results:

• Subharmonic entrainment occurs in v-shaped Arnold tongues for small forcing amplitudes.
• These tongues terminate at higher forcing amplitudes, forming global bifurcation structures.
• Complicated bifurcation structures arise from strong resonances in moderate to large amplitude regimes.

Conclusions:

• The study provides a theoretical framework for understanding complex bifurcation phenomena in forced oscillators.
• The termination of Arnold tongues and associated global structures are explained.
• Predicted phenomena align with observations in model and experimental chemical systems.