Nonlinear Oscillators

Numerical modeling of oscillators with nonlinear restoring force, damping, and external driving.

Conceptual context

A nonlinear oscillator does not necessarily repeat at the same rhythm independently of amplitude. Its $x$ changes the $F_r$ itself, so period, trajectory shape, and driven response may evolve. When $c$ and $F_0$ also act, the system exchanges energy with its surroundings and may display shifted resonance, amplitude jumps, multistability, or chaos. Computational physics follows this evolution by resolving how $v$ and $a$ change with $t$. The central issue is not merely how much the system oscillates, but which physical mechanism organizes each observed regime.

🟢 Essential level

The decisive feature is that $F_r$ is no longer strictly proportional to $x$.

Therefore, a large oscillation may behave differently from a small one. $\alpha$ determines whether the system hardens, raising its effective frequency, or softens, lowering it. $c$ removes $E$, while $F_0$ and $\Omega$ inject energy periodically. The response emerges from competition among these effects. The same excitation may produce different motions for different initial conditions, and a small parameter change may trigger an amplitude jump. Numerically, $x$, $v$, and $a$ are followed over many cycles. The transient is discarded, then the steady response is examined through time traces, phase space, or sampling once every $T_f$. A reliable interpretation also compares nearby initial states and repeats the integration with a smaller step, because numerical artifacts can imitate physical complexity.

An irregular-looking trace is not automatic evidence of chaos: it must first survive refinement of the numerical step and extension of the observation window before that label is justified.

🔵 Formal level

Basic formulas

The reference model is the Duffing oscillator, balancing inertia, viscous damping, linear restoration, cubic restoration, and sinusoidal excitation:

Duffing equation

The conservative part is summarized by:

Nonlinear restoring force

Its sign must be read using the chosen axis convention. The term governed by $k$ dominates near the origin, while the term governed by $\alpha$ becomes important as $x$ grows. The energy landscape follows from:

Duffing potential energy

The resulting quantity is $U$, which combines the quadratic and quartic contributions of displacement. The instantaneous kinetic-plus-potential total is:

Mechanical energy

Real student questions

Is every irregular signal chaotic? No. It may be a long transient, quasiperiodic superposition, or integration error.

Why does period depend on amplitude? Because the cubic term changes effective stiffness.

Can identical parameters produce several responses? Yes. Different initial conditions may approach different attractors.

How do I know whether the time step is adequate? Repeat with a smaller step and check convergence of amplitude, phase, and attractor structure.

Must energy be conserved? Only without damping and forcing.

How do driving and response frequencies differ? The first belongs to the excitation; the second emerges from the motion and may be equal, multiple, submultiple, or nonperiodic.

Final synthesis

A nonlinear oscillator turns amplitude, dissipation, and driving into rich dynamics. Reliable simulation requires a physical model, numerical control, and graphical interpretation. The goal is not merely to obtain a curve, but to identify the mechanism shaping it and the point at which the model stops being adequate.