Program for Period-Doubling Bifurcation Diagram of Impulse Differential Equations - Equation Structure

Period-doubling bifurcation diagrams for impulse differential equations serve as crucial tools for studying chaotic phenomena in nonlinear systems. Such equations typically model dynamic systems subject to periodic impulse disturbances, including applications in biological rhythms, circuit systems, and mechanical vibrations. The core implementation approach generally involves these key steps: First, establishing a numerical solution framework for impulse differential equations, typically employing Runge-Kutta methods for integrating the continuous dynamics, while separately handling state transitions at discrete impulse moments. Through parameter scanning, control parameters (such as impulse strength or period) are systematically varied, with steady-state behavior recorded at each parameter value. For constructing period-doubling bifurcation diagrams, the critical procedure involves running long-term simulations at each parameter point, waiting for transient processes to dissipate, and then recording extreme values or sampling points of system state variables. These points form characteristic patterns as parameters change: transitioning from single-period to double-period, then to quadruple-period, and ultimately entering chaotic regions. Bifurcation diagrams clearly demonstrate system stability characteristics and pathways to chaos under parameter variations. In practical programming implementation, special attention must be paid to numerical integration step size selection, balancing computational accuracy with efficiency. Additionally, impulse moment handling requires careful implementation to ensure accurate execution of state variable jumps. Key programming considerations include implementing adaptive step-size control for the continuous phase and maintaining precise event detection for impulse timing.