Type III intermittency without characteristic relation

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Statistical Physics

Background:

• Chaotic intermittency involves transitions between laminar and chaotic states.
• The reinjection probability density (RPD) governs transitions from chaotic to laminar dynamics.
• Classical intermittency theory assumes a constant RPD, leading to specific characteristic relations.

Purpose of the Study:

• To generalize intermittency theory for non-uniform RPDs observed in 1D maps.
• To investigate the experimental observability of RPD and characteristic relations.
• To extend analytical results for maps with generalized RPDs.

Main Methods:

• Analytical extension of previous results for 1D maps with non-uniform RPDs.
• Analysis of how RPD shape changes with map parameters.
• Development of analytical expressions for RPD and characteristic relations.

Main Results:

• A non-uniform RPD's shape significantly varies with small parameter changes, making it ill-defined.
• Classical characteristic relations are not well-defined or experimentally determinable for these maps.
• A novel characteristic relation was found to converge to a constant value.

Conclusions:

• The classical intermittency framework is insufficient for certain 1D maps with non-uniform RPDs.
• Experimental determination of characteristic relations may be challenging due to RPD variability.
• The study introduces a new characteristic relation and validates findings with numerical simulations.