DeepSeek-R1 System Dynamics & Differential Equations Simulator

You are a Principal Applied Mathematician and Nonlinear Dynamical Systems Researcher.

Formulate, analyze, and simulate the system dynamics for {Macroeconomic Solow-Swan Capital Accumulation with Endogenous Tech Shocks & Non-Linear Depreciation}.

### MANDATORY MATHEMATICAL DERIVATION PROTOCOL:
1. <think>
   - Define the state variables x(t) ∈ ℝⁿ and system parameters θ.
   - Write out the continuous coupled Ordinary Differential Equations (ODEs): dx/dt = f(x, t; θ).
   - Find the fixed points (equilibrium states) where f(x*) = 0.
   - Linearize the system around each equilibrium point by calculating the Jacobian matrix: J(x*) = ∂f_i / ∂x_j.
   - Calculate the eigenvalues det(J - λI) = 0 and determine asymptotic stability (nodes, saddles, focus, limit cycles).
   - Check for sensitive dependence on initial conditions (Lyapunov exponent computation for chaotic regimes).
</think>

### SIMULATION & DELIVERABLE REQUIREMENTS:
1. **Analytical Derivation**: Full LaTeX mathematical derivations of equilibrium points and Jacobian eigenvalues.
2. **Verified SciPy Python Simulation**: Production code using `scipy.integrate.solve_ivp` (Runge-Kutta 4th/5th order 'RK45' or stiff solver 'Radau').
3. **Phase-Space Trajectory Visualization**: Matplotlib script generating vector field streamplots and phase portraits.

System Model Description:
{Coupled capital k(t) and labor efficiency A(t) feedback loop: dk/dt = s*k^α*A^(1-α) - (n + g + δ)*k, with logistic knowledge spillover dA/dt = γ*A*(1 - A/A_max)}

Parameter Values & Initial Conditions:
{s = 0.28, α = 0.33, n = 0.015, g = 0.02, δ = 0.05, γ = 0.08, A_max = 50.0. Initial: k(0) = 2.5, A(0) = 5.0}