Slow passage through a saddle-node bifurcation in discrete dynamical systems.

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Computational Modeling

Background:

• Realistic systems are often modeled using discrete, non-autonomous dynamical systems due to available time-series data.
• Autonomous systems with frozen parameters provide a simplified, continuous counterpart for analysis.
• Saddle-node bifurcations are critical points where system behavior can abruptly change.

Purpose of the Study:

• To investigate bifurcation delay phenomena in discrete non-autonomous systems.
• To analyze the impact of a slowly varying bifurcation parameter (sweep rate ϵ) on system dynamics.
• To contrast the behavior of discrete systems with their continuous autonomous counterparts.

Main Methods:

• Analysis of a discrete non-autonomous system with a time-varying bifurcation parameter.
• Mathematical modeling using super- and sub-solutions.
• Phase portrait analysis to understand bifurcation delay.

Main Results:

• A bifurcation delay, proportional to ϵ^(2/3), is observed when ϵ/Δt is O(1).
• When ϵ/Δt is o(1), discrete systems exhibit altered dynamics before the bifurcation point, unlike continuous systems.
• System dynamics are shown to be dependent on the time mesh size (Δt).

Conclusions:

• The ratio ϵ/Δt quantifies the discrete nature of the system's behavior.
• Bifurcation delay in discrete systems offers a window for proactive interventions.
• The discrete nature complicates analytical studies but provides more realistic modeling capabilities.