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Nonlinear Oscillators

Phase-space trajectories of the four oscillators, obtained by integrating their default Hamiltonian problems:

using GeometricIntegrators: integrate, Gauss
using CairoMakie

import GeometricProblems.DuffingOscillator as duffing
import GeometricProblems.LennardJonesOscillator as lj
import GeometricProblems.MorseOscillator as morse
import GeometricProblems.MathewsLakshmananOscillator as ml

models = [("Duffing", duffing), ("Lennard-Jones", lj), ("Morse", morse), ("Mathews–Lakshmanan", ml)]

fig = Figure(size = (800, 700))
for (i, (name, m)) in enumerate(models)
    row = (i - 1) ÷ 2 + 1
    col = (i - 1) % 2 + 1
    ax = Axis(fig[row, col]; xlabel = "q", ylabel = "p", title = name)
    sol = integrate(m.hodeproblem(), Gauss(2))
    lines!(ax, sol.q[:, 1], sol.p[:, 1])
end
fig
Example block output

Duffing Oscillator

GeometricProblems.DuffingOscillator — Module

Duffing Oscillator

The (undamped, unforced) Duffing oscillator is a one-dimensional anharmonic oscillator with a quartic potential. In this conservative form it is an autonomous Hamiltonian system with

\[H(q, p) = \frac{p^2}{2m} + \frac{\alpha}{2} q^2 + \frac{\beta}{4} q^4 ,\]

and Lagrangian

\[L(q, \dot{q}) = \frac{m}{2} \dot{q}^2 - \frac{\alpha}{2} q^2 - \frac{\beta}{4} q^4 ,\]

giving the equation of motion $m \ddot{q} + \alpha q + \beta q^3 = 0$.

System parameters:

  • m: mass
  • α: linear stiffness
  • β: coefficient of the cubic force (quartic potential) nonlinearity
source

Lennard-Jones Oscillator

GeometricProblems.LennardJonesOscillator — Module

Lennard-Jones Oscillator

A single particle oscillating in a one-dimensional Lennard-Jones potential,

\[V(q) = 4 \varepsilon \left[ \left( \frac{\sigma}{q} \right)^{12} - \left( \frac{\sigma}{q} \right)^{6} \right] ,\]

with Hamiltonian and Lagrangian

\[H(q, p) = \frac{p^2}{2m} + V(q) , \qquad L(q, \dot{q}) = \frac{m}{2} \dot{q}^2 - V(q) .\]

The potential has its minimum at $q = 2^{1/6}\sigma$; small oscillations about this point are bounded. Only $q > 0$ is physical.

System parameters:

  • m: mass
  • ε: depth of the potential well
  • σ: finite distance at which the potential is zero
source

Mathews-Lakshmanan Oscillator

GeometricProblems.MathewsLakshmananOscillator — Module

Mathews-Lakshmanan Oscillator

The Mathews-Lakshmanan oscillator is a nonlinear oscillator with a position-dependent effective mass, defined by the Lagrangian

\[L(q, \dot{q}) = \frac{1}{2} \frac{\dot{q}^2 - \omega^2 q^2}{1 + \lambda q^2} .\]

The conjugate momentum is $p = \dot{q} / (1 + \lambda q^2)$, and the corresponding Hamiltonian is

\[H(q, p) = \frac{1}{2} \big( 1 + \lambda q^2 \big) p^2 + \frac{1}{2} \frac{\omega^2 q^2}{1 + \lambda q^2} .\]

It exhibits amplitude-dependent (anisochronous for $\lambda \neq 0$) oscillations and is a standard example of a quadratic Liénard-type nonlinear oscillator (M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics, Springer, 2003).

System parameters:

  • ω: (bare) angular frequency
  • λ: nonlinearity parameter
source

Morse Oscillator

GeometricProblems.MorseOscillator — Module

Morse Oscillator

The Morse oscillator models an anharmonic bond with the potential

\[V(q) = D \left( 1 - e^{-a (q - r_0)} \right)^2 ,\]

with Hamiltonian and Lagrangian

\[H(q, p) = \frac{p^2}{2m} + V(q) , \qquad L(q, \dot{q}) = \frac{m}{2} \dot{q}^2 - V(q) .\]

Trajectories with energy below the dissociation energy $D$ are bound and oscillate anharmonically about the equilibrium $q = r_0$.

System parameters:

  • m: reduced mass
  • D: well depth (dissociation energy)
  • a: controls the width of the well
  • r₀: equilibrium position
source