Dynamical Systems

This collection covers advanced dynamical systems topics: pendulum mechanics, mass-spring oscillators, Fourier analysis, ecological models, chaotic attractors, discrete dynamics, and phase-plane analysis of linear and nonlinear systems.

Phase Portraits

• Direction Field — plot the slope field of a first-order ODE \(\dot{y}=f(x,y)\) and overlay solution curves from any clicked initial point.
• Phase Line — equilibria and stability for autonomous first-order ODEs \(\dot{y}=f(y)\): identify stable, unstable, and semi-stable fixed points from the sign of \(f\).
• Linear Phase Portraits — classify equilibria of 2×2 linear systems \(\dot{\mathbf{x}}=A\mathbf{x}\): spirals, centers, nodes, and saddle points.
• Nonlinear Phase Portraits — plot direction fields and trajectories for arbitrary planar systems \(\dot{x}=f(x,y),\ \dot{y}=g(x,y)\), including limit cycles.

Pendulum Systems

• Damped Pendulum — explore motion and phase portraits of a damped nonlinear pendulum.
• Elastic Pendulum — simulate a mass on a spring that can both stretch and swing, with coupled modes and resonance.
• Coupled Pendula — two pendulums coupled by a spring exhibiting normal modes and energy transfer.
• Multi-Pendulum — double and triple pendulums with chaotic dynamics.
• Pendulum Phase Array — apparent wave patterns from \(N\) independent pendulums with tuned frequencies.
• Swinging Atwood Machine — a counterweight coupled to a swinging bob; integrable for \(M/m=3\), chaotic otherwise.

Mass-Spring Systems

• Mass-Spring-Damping — simulate \(m\ddot{x} + c\dot{x} + kx = 0\) across underdamped, critically damped, and overdamped regimes.
• Multiple DOF Mass Spring — coupled mass-spring systems and normal modes of vibration.
• Forced Vibrations — transient and steady-state behavior, including resonance, for \(m\ddot{x} + c\dot{x} + kx = F(t)\).

Fourier & Ecology

• Fourier Series — approximate periodic functions with partial sums of sines and cosines.
• Lotka-Volterra — predator-prey population cycles and periodic orbits.
• Lotka-Volterra with Carrying Capacity — ecological model with logistic growth limits.

Chaotic Systems & Nonlinear Oscillators

• Lorenz System — the famous chaotic attractor and sensitive dependence on initial conditions.
• Rössler System — 3D single-scroll chaotic attractor with a period-doubling route to chaos.
• Malkus Waterwheel — mechanical analogue of the Lorenz system: a tilted wheel of leaky buckets whose angular velocity reverses chaotically.
• Van der Pol Oscillator — limit cycles and relaxation oscillations.
• Duffing Oscillator — nonlinear forced oscillator with double-well potential and chaos.
• Poincaré Sections — high-resolution stroboscopic and hyperplane sections for Duffing, Hénon–Heiles, Chirikov standard map, and the double pendulum, integrated with adaptive Tsit5 in a Web Worker.

Potentials, Bifurcations & Discrete Maps

• Cubic Potential — motion in asymmetric potential wells.
• Double Well Potential — bistability in symmetric double wells.
• Bifurcation Diagram — period-doubling cascades toward chaos, with Lyapunov exponents.
• Bifurcation Family — saddle-node, transcritical, and pitchfork bifurcations with coupled phase-line and (r, x*) diagrams.
• Logistic Map — cobweb diagrams and the full period-doubling route to chaos.
• Cobweb Diagram — visualize fixed-point iteration, convergence, and chaos for general maps.

Accessibility

All tools meet WCAG 2.1 AA standards: keyboard navigation, screen reader support via MathJax assistive MathML, high-contrast mode, visible focus indicators, and ARIA labels throughout.