Phase Space Lab — 2D Dynamical Systems Simulator

Click anywhere on phase plane to release trajectory

System: 2D Autonomous

Equilibria: 0

Trajectories: 0

Integrator: RK4

Step: 0.01

Domain: [-5.0, 5.0] × [-5.0, 5.0]

Dynamical System

dx/dt, dy/dt

dx/dt = f(x,y) Horizontal Velocity

dy/dt = g(x,y) Vertical Velocity

Supports x, y, pi, e, +, -, *, /, ^, implicit multiplication, sin, cos, tan, exp, log, sqrt, abs, sinh, cosh, tanh, floor, ceil, sign, min(a,b), max(a,b).

Display Toggles

Vector Field Arrowheads Nullclines Equilibria Trajectories Coordinate Grid

Density 24x24

Arrow Scale 1.0x

RK4 Numerical Integration

Step Size (h): 0.010

Fwd Time (+T): 20s

Bwd Time (-T): 20s

Time Animation Stopped

Detected Equilibria

Searching phase plane for fixed points...

Initial Conditions

Click the phase plane to add trajectories

Classical Dynamical Systems Gallery

Select any classic 2D nonlinear system to immediately simulate its vector field, nullclines, equilibrium points, and limit cycles.

Jacobian Linearization Analysis

Detailed linear stability spectrum around selected equilibrium point

No Selection

Fixed Point (x*, y*)

(0.000, 0.000)

Stability Classification

Select Equilibrium

Jacobian Matrix J = ∂(f,g)/∂(x,y)

[

+0.0000 +0.0000 +0.0000 +0.0000

]

Trace τ = Tr(J)

0.0000

Determinant Δ = det(J)

0.0000

Discriminant (τ² - 4Δ)

0.0000

Eigenvalues λ₁, λ₂

λ₁ = 0.0000

λ₂ = 0.0000

Eigenvectors

v1 = --

v2 = --

Local Linearization

u_dot = J u

Trace–Determinant Diagram

Bifurcation and stability classification plane (τ vs Δ)

Diagnostic Guide:

• Δ > 0, τ < 0: stable node or spiral

• Δ > 0, τ > 0: unstable node or spiral

• Δ < 0: saddle point

• Δ = τ²/4: repeated eigenvalues; Δ = 0: non-hyperbolic.

Theory & Mathematics of 2D Phase Space

A 2D Autonomous Dynamical System is governed by a set of ordinary differential equations without explicit time dependence:

dx/dt = f(x, y), dy/dt = g(x, y)

1. Vector Field & Velocity

At every point (x, y) in state space, the function vector F(x, y) = [f(x,y), g(x,y)]ᵀ specifies the instantaneous velocity of motion.

2. Phase Trajectories (RK4)

Solutions (x(t), y(t)) trace curves in phase space called trajectories. Integrations run both forward in time (+T) and backward in time (-T) using 4th-order Runge–Kutta.

3. Nullclines

Curves where motion is strictly vertical (f(x,y)=0) or horizontal (g(x,y)=0). Their intersections mark equilibrium points.

4. Fixed Points & Stability

Equilibria satisfy f(x*, y*) = 0 and g(x*, y*) = 0. Linearization via the Jacobian matrix determines local behavior.