systems.builtin.deterministic.continuous.NonlinearChainSystem

systems.builtin.deterministic.continuous.NonlinearChainSystem(*args, **kwargs)

Chain of five coupled nonlinear oscillators - first-order formulation.

Physical System:

A one-dimensional chain of five oscillators where each element influences its neighbors through nonlinear coupling.

Each oscillator has: - Linear restoring force (spring-like: -kx) - Linear damping (viscous: -cx) - Nonlinear coupling to neighbors via sin(x_j - x_i) - Only the first oscillator receives external control

State Space:

State: x = [x₁, x₂, x₃, x₄, x₅] - x₁: State of oscillator 1 [rad or m] - x₂: State of oscillator 2 [rad or m] - x₃: State of oscillator 3 [rad or m] - x₄: State of oscillator 4 [rad or m] - x₅: State of oscillator 5 [rad or m]

Control: u = [u] - u: External force/torque applied only to first oscillator - Influence propagates to other oscillators through coupling

Output: y = [x₁, x₃, x₅] - Sparse observation: only odd-numbered oscillators measured

Dynamics:

The equations of motion form a nearest-neighbor coupling structure:

Oscillator 1 (left boundary, receives control): dx₁/dt = -k·x₁ - c·x₁ + α·sin(x₂ - x₁) + u

Oscillator 2 (interior, coupled to neighbors): dx₂/dt = -k·x₂ - c·x₂ + α·sin(x₁ - x₂) + α·sin(x₃ - x₂)

Oscillator 3 (interior, coupled to neighbors): dx₃/dt = -k·x₃ - c·x₃ + α·sin(x₂ - x₃) + α·sin(x₄ - x₃)

Oscillator 4 (interior, coupled to neighbors): dx₄/dt = -k·x₄ - c·x₄ + α·sin(x₃ - x₄) + α·sin(x₅ - x₄)

Oscillator 5 (right boundary, no control): dx₅/dt = -k·x₅ - c·x₅ + α·sin(x₄ - x₅)

Physical interpretation: - Linear terms (-kx, -cx): individual oscillator wants to return to zero - Nonlinear coupling α·sin(x_j - x_i): synchronization force * When x_j > x_i: positive force on i (speeds it up) * When x_j < x_i: negative force on i (slows it down) * Maximum coupling at π/2 phase difference - Control u propagates through chain via coupling

Parameters:

k : float, default=1.0 Linear stiffness/restoring coefficient [1/s]. Higher k → stronger individual oscillator dynamics, weaker relative coupling influence. c : float, default=0.1 Damping coefficient [1/s]. Higher c → faster energy dissipation. Damps out transients and oscillations. alpha : float, default=0.1 Nonlinear coupling strength. Controls interaction between neighbors: - α = 0: Uncoupled oscillators - Small α: Weak coupling, local behavior dominates - Large α: Strong coupling, collective behavior emerges - α > k+c: Coupling-dominated dynamics, synchronization possible

Equilibrium:

Synchronous equilibrium (all at origin): x_eq = [0, 0, 0, 0, 0] (all oscillators aligned at zero) u_eq = 0 (no external force)

This equilibrium is stable due to damping. Other synchronized states (all x_i equal) are also equilibria for u=0.

See Also:

CoupledOscillatorSystem : Smaller coupled system with different structure VanDerPolOscillator : Single nonlinear oscillator with limit cycle Lorenz : Another system with complex nonlinear dynamics