Dynamical Systems

A Dynamical System is a triplet $(T, M, \Phi)$ where $T$ is the time domain, $M$ is the Phase Space (manifold of possible states), and $\Phi: T \times M \to M$ is an evolution function satisfying:

1. $\Phi(0, x) = x$ (Identity)
2. $\Phi(t, \Phi(s, x)) = \Phi(t+s, x)$ (Group property)

Discrete Dynamics: Iterated Maps

In a discrete system, the state is updated by a map $f: M \to M$: $x_{n+1} = f(x_n)$

The sequence $\{x_n\}_{n=0}^\infty$ is called the Orbit of $x_0$.

def logistic_map(x, r):
  return r * x * (1 - x)

def iterate(f, x0, n):
  curr = x0
  for _ in range(n):
      curr = f(curr)
  return curr

# Test logistic map at r=2.5 (Stable fixed point)
r = 2.5
f = lambda x: logistic_map(x, r)
print(f"Iteration 10: {iterate(f, 0.5, 10):.3f}")
print(f"Iteration 50: {iterate(f, 0.5, 50):.3f}")
print("At r=2.5, the system settles to a single point.")

Fixed Points and Stability

A fixed point $x^*$ satisfies $f(x^*) = x^*$.

• A point is Lyapunov Stable if for every $\epsilon > 0$, there exists $\delta > 0$ such that $d(x_0, x^*) < \delta \implies d(x_n, x^*) < \epsilon$ for all $n > 0$.
• For a differentiable map $f: \mathbb{R} \to \mathbb{R}$, $x^*$ is Asymptotically Stable if $|f'(x^*)| < 1$.

def check_stability(f_prime, x_star):
  derivative_val = f_prime(x_star)
  is_stable = abs(derivative_val) < 1
  return is_stable, derivative_val

# For logistic map, f'(x) = r(1 - 2x)
# Fixed point x* = 1 - 1/r
r = 3.2
x_star = 1 - 1/r
f_prime = lambda x: r * (1 - 2 * x)

print(f"Is fixed point stable at r=3.2? {check_stability(f_prime, x_star)}")
print("At r=3.2, |f'(x*)| > 1, so the fixed point is unstable (leads to limit cycles).")

Continuous Dynamics: Flow

Continuous systems are defined by a vector field $\mathbf{V}: M \to TM$ and an autonomous differential equation: $\dot{\mathbf{x}} = \frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x})$

The solution through $x_0$ is the Integral Curve or Flow Line. Stability is determined by the eigenvalues of the Jacobian Matrix $J = \nabla \mathbf{f}$. If all eigenvalues have negative real parts, the equilibrium is stable.