Direction Field
\(\dfrac{dy}{dx} = f(x,y)\)
Visualize slope fields and overlay solution curves for first-order ODEs. Enter any function \(f(x,y)\) and see the direction field update in real time.
ExploreInteractive Differential Equations Tools
Explore key concepts in differential equations through interactive JavaScript visualizations. Adjust parameters in real time and observe solution behavior.
This collection covers a broad range of topics in ordinary differential equations (ODEs), from first-order equations to nonlinear systems, chaotic dynamics, and fixed-point iteration.
First-Order Equations
- Direction Field — plot slope fields for \(\dfrac{dy}{dx} = f(x,y)\) and overlay solution curves.
- Phase Line Plot — analyze equilibrium points and stability for autonomous ODEs \(\dfrac{dy}{dt} = f(y)\).
- Newton's Law of Cooling — solve \(\dfrac{dT}{dt} = -k(T - T_a)\) with a visual heat-transfer simulation.
Second-Order & Higher-Order Equations
- Mass-Spring-Damping — simulate \(m\ddot{x} + c\dot{x} + kx = 0\) across underdamped, critically damped, and overdamped regimes.
- Forced Vibrations — transient and steady-state behavior including resonance via \(m\ddot{x} + c\dot{x} + kx = F(t)\).
- Laplace Transforms — visualize \(\mathcal{L}\{f(t)\}(s)=\int_0^\infty f(t)e^{-st}\,dt\) as a signed area integral with shifting theorems.
- Convolution — watch \(g(t-\tau)\) slide over \(f(\tau)\) and see \(h(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau\) build in real time.
- Fourier Series — approximate periodic functions with partial sums of sines and cosines.
- Pendulums — single damped, elastic (spring), coupled, and multi-pendulum systems.
Systems & Nonlinear Dynamics
- Phase Portraits — classify fixed points of \(\mathbf{x}' = A\mathbf{x}\) and explore nonlinear vector fields.
- Lotka-Volterra — predator-prey cycles and logistic carrying-capacity extensions.
- Lorenz System — visualize the chaotic attractor and sensitive dependence on initial conditions.
- Potential Wells — cubic \(V(x)=ax^3+bx\) and double well \(V(x)=ax^4-bx^2\) potentials.
Discrete Dynamics
- Bifurcation Diagram — sweep the parameter and watch period-doubling emerge. Colour-coded by Lyapunov exponent, with Feigenbaum constant \(\delta\approx4.669\) annotated. Lyapunov exponent plotted below.
Pendulum Wave Machine
- Pendulum Wave — \(N\) pendulums with lengths \(L_k = gT^2/[4\pi^2(n_0+k)^2]\) tuned so pendulum \(k\) completes exactly \((n_0+k)\) oscillations in period \(T\). Collective motion produces snake, butterfly, and in-phase patterns. 3D perspective view, wave strip, pattern detection, and preset library.
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
Phase Line Plot
\(\dfrac{dy}{dt} = f(y)\)
Analyze equilibrium points and stability for autonomous first-order ODEs. Identify stable, unstable, and semi-stable fixed points on the phase line.
ExploreMass-Spring-Damping Simulator
\(m\ddot{x} + c\dot{x} + kx = 0\)
Simulate underdamped, critically damped, and overdamped motion. Observe displacement and phase-plane trajectories in real time.
ExploreMultiple Degree of Freedom Mass Spring
\(M\ddot{\mathbf{x}} + C\dot{\mathbf{x}} + K\mathbf{x} = \mathbf{0}\)
Simulate coupled mass-spring systems and explore normal modes of vibration. Visualize how energy transfers between coupled oscillators.
ExploreDamped 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.
ExploreElastic (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.
ExploreForced Vibrations and Resonance
\(m\ddot{x} + c\dot{x} + kx = F_0\cos(\omega t)\)
Observe transient and steady-state behavior, including resonance. Explore how driving frequency affects amplitude response.
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.
ExploreLaplace Transform Visualizer
\(\mathcal{L}\{f(t)\}(s) = \displaystyle\int_0^{\infty} f(t)\,e^{-st}\,dt = F(s)\)
Visualize the Laplace transform as a signed area integral. Watch the weighted integrand accumulate, track convergence, and explore the first and second shifting theorems.
ExploreConvolution Visualizer
\((f*g)(t) = \displaystyle\int_0^{t} f(\tau)\,g(t-\tau)\,d\tau\)
Watch \(g(t-\tau)\) slide over \(f(\tau)\) as \(t\) grows. The overlapping area accumulates into \(h(t)\), tracing out the exact convolution in real time.
ExploreLinear Phase Portraits
\(\mathbf{x}' = A\mathbf{x}, \quad A \in \mathbb{R}^{2\times 2}\)
Visualize trajectories for 2D linear systems based on eigenvalue classification — nodes, spirals, saddles, and centers.
ExploreNonlinear Phase Portraits
\(x' = f(x,y), \quad y' = g(x,y)\)
Explore vector fields and nullclines for nonlinear autonomous systems. Identify fixed points and their local stability via linearization.
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.
ExploreNewton's Law of Cooling
\(\dfrac{dT}{dt} = -k(T - T_a)\)
Simulate heat transfer with realistic object visuals, jet colormap temperature display, steam effects, and real-time numerical solution.
ExploreBifurcation Diagram New
\(\text{plot }\{(r,\,x_n)\}\text{ as }r\text{ varies}\)
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 New
\(L_k = \dfrac{g\,T^2}{4\pi^2(n_0+k)^2}, \quad k = 0,\ldots,N{-}1\)
\(N\) 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.
Explore