systems.builtin.deterministic.continuous.ControlledVanDerPolOscillator

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

Van der Pol oscillator - self-excited nonlinear oscillator with limit cycle and forcing term

Physical System:

Originally meant to model electronic oscillator circuits. The system exhibits self-sustained oscillations.

The key feature is nonlinear damping: - Near origin: negative damping (pumps energy in) - Far from origin: positive damping (dissipates energy) - Result: stable limit cycle (periodic orbit)

State Space:

State: x = [x, y] - x: Primary variable [V or dimensionless] * In electrical circuit: voltage or current * In general: oscillating quantity

- y: Derivative-related variable [V/s or dimensionless]
  * y ≈ ẋ for μ → 0
  * Not exactly velocity for μ > 0 (includes nonlinear term)

Control: u = [u] - u: External forcing/input [V or dimensionless] - Can perturb the natural oscillation - Can be used for synchronization or frequency control

Output: y_out = [x] - Measures only x (the oscillating variable) - Partial observation (y not directly measured)

Dynamics:

The Van der Pol equation in standard form:

ẋ = y
ẏ = μ(1 - x²)y - x + u

Or as a second-order ODE: ẍ - μ(1 - x²)ẋ + x = u

First equation: Simply defines y ≈ ẋ

Second equation: - μ(1 - x²)y: Nonlinear damping (Van der Pol term) * When |x| < 1: (1 - x²) > 0 → negative damping (adds energy) * When |x| > 1: (1 - x²) < 0 → positive damping (removes energy) * Balance creates stable limit cycle

• -x: Linear restoring force (like harmonic oscillator)
  • Provides natural frequency ω₀ ≈ 1
• u: External forcing/control

Parameters:

mu : float, default=1.0 Nonlinearity parameter [dimensionless]. Controls strength of nonlinear damping and oscillation shape:

- **μ → 0**: Nearly sinusoidal (harmonic oscillator)
  * Period T ≈ 2π
  * Smooth, sinusoidal limit cycle

- **μ = 1**: Standard Van der Pol
  * Period T ≈ 6.7
  * Mildly distorted sinusoid

- **μ >> 1**: Relaxation oscillations
  * Period T ≈ (3 - 2ln(2))μ ≈ 1.614μ
  * Sharp "fast" and "slow" phases
  * Almost discontinuous (spikes and plateaus)

Behavior Regimes:

1. Small μ (μ < 0.1): Harmonic-like - Nearly sinusoidal oscillations - Frequency ≈ 1 rad/s - Smooth limit cycle - Weak nonlinearity

2. Moderate μ (0.1 < μ < 3): Nonlinear oscillations - Visible waveform distortion - Frequency slightly reduced - Standard Van der Pol behavior

3. Large μ (μ > 3): Relaxation oscillations - Two-timescale dynamics - Fast jumps between slow plateaus - Very non-sinusoidal - Period proportional to μ

Equilibrium:

Origin (unstable): x_eq = [0, 0] u_eq = 0

For u = 0, the origin is: - Unstable focus (spiral): trajectories spiral outward - All trajectories (except origin) approach the limit cycle - Eigenvalues: λ = μ/2 ± i√(4-μ²)/2 * Real part positive (unstable) * Imaginary part gives oscillation frequency

Limit Cycle:

For u = 0, the system has a unique stable limit cycle:

Properties: - Globally attracting (except from origin) - Isolated (no nearby periodic orbits) - Amplitude ≈ 2 for all μ (approximately) - Period depends on μ: * μ → 0: T → 2π (harmonic) * μ = 1: T ≈ 6.7 * μ >> 1: T ≈ 1.614μ

Basin of attraction: Entire plane except origin - Any non-zero initial condition → limit cycle - Time to converge depends on distance from cycle

Relaxation Oscillations (μ >> 1):

For large μ, the system exhibits relaxation oscillations:

Mechanism: 1. Slow phase: x grows slowly along stable manifold 2. Jump: At x ≈ 1, rapid transition (fast manifold) 3. Slow phase: x decreases slowly along stable manifold 4. Jump: At x ≈ -1, rapid transition back 5. Repeat

Characteristics: - Distinct timescales (ε = 1/μ is small parameter) - Almost piecewise linear trajectory - Useful model for on-off systems (heart beats, neurons)

See Also:

DuffingOscillator : Another nonlinear oscillator (can be chaotic) Lorenz : 3D system that exhibits chaos NonlinearChainSystem : Multiple coupled oscillators