Arnold Tongues Calculations

This script calculates and visualizes the Arnold tongues, a concept from dynamical systems theory. It examines the behavior of a system across varying parameters K and Omega.

Overview

The script iterates over ranges of K (amplitude parameter) and Omega (frequency parameter), computing a value W at each iteration. It compares W against predefined Win values to determine the start and end points of the Arnold tongues in the parameter space.

Parameters

• Niter: The number of iterations for each calculation.
• t0: The initial value of t.
• Win: A list of window values for comparison against W.
• epsilon: The tolerance used for comparing W to Win values.
• K: The range of amplitude parameter values.
• Omega: The range of frequency parameter values.

Outputs

• Kb, Ob: Arrays marking the start points of the Arnold tongues in the K-Omega parameter space.
• Ke, Oe: Arrays marking the end points of the Arnold tongues in the K-Omega parameter space.

Tutorial: quasip Function

The quasip function simulates a dynamical system based on a set of parameters. It explores different behaviors of the system under various conditions.

Parameters

• Map (int): Determines the type of system to simulate. Set to 0 for differential equation mode, 1 for map mode.
• alpha, beta, gamma (float): Parameters influencing the system's behavior in differential equation mode.
• A, B (float): Amplitude parameters used in the differential equation.
• Omega (float): Frequency parameter for the system.
• Picture (int): Determines the type of plot. 1 for a 1D map, 2 for theta_dot vs theta, and 3 for a 2D torus plot.
• k, b (float): Parameters used in map mode. k = coupling constant

Returns

• Pendulum (list of tuples): Contains positions (b1, b2) for plotting the pendulum's orientation.
• Circle_map (list of tuples): Contains positions for plotting the 1D map if Picture == 1.
• Theta_dot (list of tuples): Contains positions for plotting theta_dot vs theta if Picture == 2.
• Torus (list of tuples): Contains line segments for the 2D torus plot if Picture == 3.

Example

Pendulum, Circle_map, Theta_dot, Torus = quasip_test (Map, alpha, beta, gamma, A, B, Omega, Picture, k, b)