Damped Pendulum Simulator
\(\ddot{\theta} + b\dot{\theta} + \dfrac{g}{L}\sin\theta = 0\)
Explore the motion and phase portrait of a damped nonlinear pendulum. Compare small-angle (linear) and large-angle (nonlinear) dynamics.
ExploreInteractive Dynamical Systems Tools
Explore advanced topics in chaos, nonlinear dynamics, and complex systems. Visualize pendulums, bifurcations, strange attractors, and oscillatory phenomena in real time.
This collection covers advanced dynamical systems topics: pendulum mechanics, Fourier analysis, ecological models, chaotic attractors, discrete dynamics, and phase-plane analysis of linear and nonlinear systems.
Phase Portraits
- 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 Wave Machine — collective wave patterns from tuned pendulum frequencies.
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.
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.
Tool Descriptions
Elastic (Spring) Pendulum
\(\mathcal{L} = \tfrac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) + mgr\cos\theta - \tfrac{1}{2}k(r-L_0)^2\)
Simulate a mass on a spring that can both stretch and swing. Explore energy transfer between radial and angular modes, autoparametric resonance, and quasi-periodic orbits.
ExploreCoupled Pendula with Spring
\(\ddot{\theta}_1 = -\dfrac{g}{L}\sin\theta_1 - \kappa(\theta_1 - \theta_2)\)
Simulate two pendulums coupled by a spring. Explore normal modes, energy transfer, and beat phenomena.
ExploreFourier Series Approximation
\(f(x) = \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^{N}\bigl(a_n\cos nx + b_n\sin nx\bigr)\)
See how partial sums of sine and cosine terms approximate periodic functions. Adjust the number of terms and observe convergence.
ExploreLotka-Volterra Model
\(\dfrac{dx}{dt} = \alpha x - \beta xy, \quad \dfrac{dy}{dt} = \delta xy - \gamma y\)
Simulate classic predator-prey population cycles in phase space. Observe periodic orbits and the relationship between predator and prey densities.
ExploreLotka-Volterra with Carrying Capacity
\(\dfrac{dx}{dt} = \alpha x\!\left(1 - \dfrac{x}{K}\right) - \beta xy\)
Extend the predator-prey model with logistic growth limits. Explore how carrying capacity stabilizes or changes system dynamics.
ExploreLorenz System
\(\dot{x} = \sigma(y-x),\quad \dot{y} = x(\rho-z)-y,\quad \dot{z} = xy - \beta z\)
Visualize the famous chaotic attractor and sensitive dependence on initial conditions. Explore how parameters \(\sigma\), \(\rho\), \(\beta\) shape the dynamics.
ExploreMulti-Pendulum Simulator
\(\dfrac{d}{dt}\dfrac{\partial \mathcal{L}}{\partial \dot{\theta}_i} - \dfrac{\partial \mathcal{L}}{\partial \theta_i} = 0\)
Simulate double and triple pendulums with chaotic trajectory visualization. Observe sensitivity to initial conditions in Lagrangian mechanics.
ExploreParticle in a Cubic Potential
\(V(x) = ax^3 + bx, \quad \ddot{x} = -V'(x)\)
Study motion and phase space trajectories in an asymmetric potential well. Explore bounded and unbounded orbits depending on initial energy.
ExploreParticle in a Double Well Potential
\(V(x) = ax^4 - bx^2, \quad \ddot{x} = -V'(x)\)
Explore bistability in a symmetric double well. Observe how initial energy determines which well the particle occupies.
ExploreBifurcation Diagram
\(\text{plot }\{(r,\,x_n)\}\text{ as }r\text{ varies}\)
Watch period-doubling cascades, the Feigenbaum constant \(\delta\approx4.669\), and chaos emerge as the parameter sweeps. Dots coloured by Lyapunov exponent \(\lambda\). Lyapunov exponent plotted separately below. Custom map support.
ExplorePendulum Wave Machine
\(L_k = \dfrac{g\,T^2}{4\pi^2(n_0+k)^2}, \quad k = 0,\ldots,N{-}1\)
\(N\) pendulums tuned so pendulum \(k\) completes exactly \((n_0+k)\) oscillations in period \(T\). Collective motion produces snake, butterfly, in-phase, and wave patterns. Features a 3D perspective view, horizontal displacement wave strip, real-time pattern detection, dark/light/high-contrast modes, and a preset library.
ExploreVan der Pol Oscillator
\(\dfrac{d^2x}{dt^2} - \mu(1 - x^2)\dfrac{dx}{dt} + x = 0\)
Explore limit cycles and relaxation oscillations of the Van der Pol oscillator. Visualize phase portraits and time series with RK4/SDIRK2 hybrid solver for stiff regimes.
ExploreDuffing Oscillator
\(\dfrac{d^2x}{dt^2} + \delta\dfrac{dx}{dt} + \alpha x + \beta x^3 = \gamma\cos(\omega t)\)
Explore the Duffing oscillator — a nonlinear second-order ODE with double-well potential, hardening/softening spring regimes, and chaotic forcing. Visualize phase portraits, time series, and potential energy with an adaptive RK45 solver.
ExploreLogistic Map Explorer
\(x_{n+1} = r\,x_n(1-x_n)\)
Cobweb diagrams for the logistic map, tent map, and sine map. Explore the full period-doubling cascade — from stable fixed points through period-2, 4, 8 cycles to chaos. Eight quick-preset parameter values, real-time Lyapunov exponent, regime indicator, and step-by-step orbit animation.
ExploreCobweb Diagram
\(x_{n+1} = f(x_n)\)
Visualize cobweb diagrams for fixed-point iteration and discrete dynamical systems. Explore convergence, periodic orbits, and chaos with interactive map selection and parameter control.
ExploreRössler System
\(\dot{x} = -y - z,\quad \dot{y} = x + ay,\quad \dot{z} = b + z(x - c)\)
Explore the Rössler attractor — a 3D chaotic system with a single-scroll topology and a clean period-doubling route to chaos. Sweep the parameter \(c\) to watch periodic orbits split into period-2, period-4 cycles and finally the strange attractor.
ExploreBifurcation Family Explorer
\(\dot{x} = r + x^2,\ \ rx - x^2,\ \ rx - x^3,\ \ rx + x^3 - x^5\)
Coupled phase line and \((r,x^*)\) bifurcation diagram for the four canonical 1D bifurcations: saddle-node, transcritical, supercritical and subcritical pitchfork. Observe fixed-point creation, exchange of stability, symmetry breaking, and hysteresis.
ExploreMalkus Waterwheel
\(\dot{a}_1 = \omega b_1 - K a_1 + q_1,\ \ \dot{b}_1 = -\omega a_1 - K b_1,\ \ I\dot{\omega} = \pi g R\, b_1 - \nu\omega\)
Mechanical analogue of the Lorenz system: a tilted wheel of leaky buckets whose angular velocity reverses chaotically. Animated rotating wheel with leaky buckets, time-series, and a 3D phase-space view of the butterfly attractor.
ExploreLinear Phase Portraits
\(\dot{\mathbf{x}} = A\mathbf{x},\quad A\in\mathbb{R}^{2\times 2}\)
Classify equilibria of 2×2 linear systems — spirals, centers, nodes, and saddle points — by specifying the matrix \(A\) and reading off eigenvalues. Click anywhere on the canvas to integrate a trajectory from that initial condition.
ExploreNonlinear Phase Portraits
\(\dot{x} = f(x,y),\quad \dot{y} = g(x,y)\)
Plot direction fields and trajectories for arbitrary planar nonlinear systems. Visualize fixed points, limit cycles, and separatrices with RK4 and SDIRK2 hybrid solvers for stiff regimes. Click the canvas to integrate from any initial condition.
Explore