Phase coherence is not related to topology

Area of Science:

• Nonlinear dynamics
• Complex systems analysis

Background:

• Phase coherence is a key concept in nonlinear science, often linked to system topology.
• Existing definitions of phase and phase coherence lack universal acceptance.
• Understanding phase coherence is vital for studying synchronization phenomena.

Purpose of the Study:

• To challenge the notion that phase coherence is determined by attractor topology using counterexamples.
• To introduce a novel, topology-independent measure for phase coherence.
• To provide a quantitative method for assessing phase coherence without a priori phase definition.

Main Methods:

• Development of counterexamples to disprove the topological association of phase coherence.
• Utilizing recurrence plots to analyze trajectory similarities.
• Defining a phase-coherence measure based on the probability of recurrences between different trajectories.

Main Results:

• Demonstrated that phase coherence is independent of the attractor's topology.
• Introduced a new phase-coherence measure based on recurrence plot analysis.
• The proposed measure effectively distinguishes between phase-coherent and non-phase-coherent systems.

Conclusions:

• Phase coherence is a dynamical property, not a topological invariant.
• The recurrence plot-based measure offers a practical and generalizable method for quantifying phase coherence.
• This work advances the understanding and measurement of phase coherence in nonlinear systems.