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Example Systems
Simple Pendulum
\begin{cases} \dot{x} = y \\ \dot{y} = -\sin(x) \end{cases}
Domain: [-6.25, 6.25] × [-5, 5]
Van der Pol Oscillator
\begin{cases} \dot{x} = y \\ \dot{y} = (1 - x^2) y - x \end{cases}
Domain: [-3.75, 3.75] × [-3, 3]
Lotka-Volterra (Predator-Prey)
\begin{cases} \dot{x} = x - 0.1 x y \\ \dot{y} = 0.075 x y - 1.5 y \end{cases}
Domain: [0, 50] × [0, 30]
Duffing Oscillator
\begin{cases} \dot{x} = y \\ \dot{y} = x - x^3 - 0.15 y \end{cases}
Domain: [-2, 2] × [-2, 2]
Simple Nonlinear
\begin{cases} \dot{x} = y - y^3 - 0.25x \\ \dot{y} = x - y - xy \end{cases}
Domain: [-5, 5] × [-5, 5]
Limit Cycle (Polar)
\begin{cases} \dot{x} = -y + x(1 - x^2 - y^2) \\ \dot{y} = x + y(1 - x^2 - y^2) \end{cases}
Domain: [-2.5, 2.5] × [-2, 2]
Escape Equation
\begin{cases} \dot{x} = y \\ \dot{y} = -0.1y - x(1-x) \end{cases}
Domain: [-2.5, 2.5] × [-2, 2]
Nonlinear Focus
\begin{cases} \dot{x} = -x + y + xy \\ \dot{y} = -x - y + y^2 \end{cases}
Domain: [-3, 3] × [-3, 3]
Competitive Lotka-Volterra
\begin{cases} \dot{x} = x(1 - x) - 0.5 x y \\ \dot{y} = y(0.5 x - 0.8) \end{cases}
Domain: [0, 4] × [0, 3]
Competitive LV: Coexistence
\begin{cases} \dot{x} = x(1 - x - 0.5 y) \\ \dot{y} = y(1 - y - 0.6 x) \end{cases}
Domain: [0, 2] × [0, 2]
Competitive LV: Species 1 Wins
\begin{cases} \dot{x} = x(1 - x - 0.4 y) \\ \dot{y} = y(1 - y - 1.5 x) \end{cases}
Domain: [0, 2] × [-1, 1]
Competitive LV: Species 2 Wins
\begin{cases} \dot{x} = x(1 - x - 1.5 y) \\ \dot{y} = y(1 - y - 0.4 x) \end{cases}
Domain: [-1, 1] × [0, 2]
Competitive LV: Both Extinct
\begin{cases} \dot{x} = x(-0.2 - x - 0.5 y) \\ \dot{y} = y(-0.2 - y - 0.5 x) \end{cases}
Domain: [-1, 1] × [-1, 1]
FitzHugh-Nagumo Model
\begin{cases} \dot{x} = x - \frac{x^3}{3} - y \\ \dot{y} = 0.08 (x + 0.7 - 0.8 y) \end{cases}
Domain: [-2.5, 2.5] × [-1, 2.5]
Brusselator
\begin{cases} \dot{x} = 1 + x^2 y - 4 x \\ \dot{y} = 3 x - x^2 y \end{cases}
Domain: [0, 4] × [0, 5]
Selkov Glycolysis Model
\begin{cases} \dot{x} = -x + 0.08 y + x^2 y \\ \dot{y} = 0.6 - 0.08 y - x^2 y \end{cases}
Domain: [0, 2] × [0, 3]
Hopf Bifurcation
\begin{cases} \dot{x} = y(x - 1) \\ \dot{y} = -x + y(1 - 2x) \end{cases}
Domain: [-2, 2] × [-2, 2]
Unstable Limit Cycle
\begin{cases} \dot{x} = y + x(x^2 + y^2 - 1) \\ \dot{y} = -x + y(x^2 + y^2 - 1) \end{cases}
Domain: [-2.0, 2.0] × [-1.5, 1.5]