Weak chaos, anomalous diffusion, and weak ergodicity breaking in systems with delay.

Area of Science:

• Nonlinear dynamics
• Complex systems analysis
• Chaos theory

Background:

• Standard delay systems with linear and nonlinear terms are foundational in modeling complex phenomena.
• Understanding the impact of delay on system behavior, including chaos and diffusion, is crucial.
• Previous research has explored chaos and diffusion in delay systems, but specific nonlinearities and delay modulations require further investigation.

Purpose of the Study:

• To investigate the emergence of weak chaos, subdiffusion, and ergodicity breaking in standard delay systems with specific nonlinearities.
• To analyze the effect of large constant delay times on observable anomalous behavior.
• To explore how periodic modulation of delay influences chaotic dynamics, solution types, and anomalous diffusion.

Main Methods:

• Mathematical modeling of standard delay systems.
• Analysis of system dynamics under different nonlinearity conditions.
• Investigation of system behavior in the limit of large constant delays.
• Application of periodic modulation to delay times.
• Characterization of chaotic phases, solution types, and diffusion patterns.

Main Results:

• Specific nonlinearities lead to weak chaos, asymptotic subdiffusion, and weak ergodicity breaking.
• Large constant delays can mask anomalous behavior due to long crossover times.
• Periodic delay modulation significantly reduces the dimension of chaotic phases.
• Novel solutions and anomalous diffusion emerge at short times with modulated delays.
• Nonhyperbolic fixed points in function space are identified as the cause of anomalous behavior.

Conclusions:

• The choice of nonlinearity critically influences the dynamics of delay systems, leading to weak chaos and subdiffusion.
• Delay modulation is a key factor in controlling anomalous diffusion and system complexity.
• Nonhyperbolic fixed points are fundamental to understanding anomalous behavior in these systems.