Mandelbrot set in coupled logistic maps and in an electronic experiment

Area of Science:

• Nonlinear dynamics
• Complex systems
• Chaos theory

Background:

• Complex analytic iterative maps exhibit rich dynamical behaviors.
• Understanding the physical realization of these maps is crucial for nonlinear dynamics research.
• The Feigenbaum universality class describes critical phenomena in chaotic systems.

Purpose of the Study:

• To propose a method for constructing physical systems with dynamics of complex analytic iterative maps.
• To explore the potential for coupled dissipative nonlinear systems to exhibit complex map dynamics.
• To experimentally validate the concept using an electronic system.

Main Methods:

• Transformation of a complex quadratic map into coupled real one-dimensional quadratic maps.
• Design and implementation of a coupled dissipative nonlinear electronic system.
• Experimental observation and analysis of dynamical phenomena, including the Mandelbrot set.

Main Results:

• A physical system was constructed exhibiting dynamical characteristics of complex analytic maps.
• The electronic system demonstrated phenomena related to the Feigenbaum universality class.
• The Mandelbrot set was successfully generated in the parameter plane of the physical system.

Conclusions:

• The proposed approach enables the physical realization of complex analytic map dynamics.
• Coupled dissipative nonlinear systems can effectively emulate complex iterative map behavior.
• The experimental results confirm the feasibility and provide a physical basis for studying complex dynamics.