Analysis of Nonlinear Dynamic Systems

This Python code utilizes the sympy library to perform symbolic computations related to a given dynamic system described by two equations.

1. Symbol Definition: x, y, a, p, c1, c2 are defined as symbolic variables using sympy.
2. Dynamic Equations: Two dynamic equations, F1 and F2, representing a system of nonlinear equations, are defined based on the symbolic variables. These equations capture the dynamics of the system.
3. Jacobian Matrix: The code calculates the Jacobian matrix (J) of the system. The Jacobian matrix is a matrix of first-order partial derivatives that describes the linearization of the system around a given point.
4. Output Jacobian Matrix: The Jacobian matrix is printed to the console, providing insight into the linearization of the dynamic system at a particular state.
5. Determinant and Trace: The code then calculates the determinant and trace of the Jacobian matrix. These quantities provide information about the stability and behavior of the system near equilibrium points.
6. Output Results: Finally, the determinant and trace of the Jacobian matrix are printed to the console, with the expressions simplified for better readability.

In summary, the code analyzes the linearization properties of a given nonlinear dynamic system by calculating and examining its Jacobian matrix, determinant, and trace. This information is useful for understanding the stability characteristics of equilibrium points in the system.

import sympy as sp

# Define symbols
x, y, a, p, c1, c2 = sp.symbols('x, y, a, p, c1, c2')

# Define dynamic equations
F1 = x*(1-x)*(a*p*y-c1)
F2 = y*(1-y)*((1-a)*p*x-c2);

# Calculate Jacobian matrix
J = sp.Matrix([[sp.diff(F1, x), sp.diff(F1, y)], [sp.diff(F2, x), sp.diff(F2, y)]])

# Output Jacobian matrix
print("Jacobian Matrix:")
print(J)

# Calculate determinant and trace of Jacobian matrix
det_J = sp.simplify(J.det())
trace_J = sp.simplify(J.trace())

# Output results
print("Determinant of Jacobian Matrix (Simplified):", det_J)
print("Trace of Jacobian Matrix (Simplified):", trace_J)