An Exactly Soluble Three-Dimensional Toy Model with a Saddle Separatrix

We study an autonomous three-dimensional system defined by x˙=αx(1−x2)\dot{x}=\alpha x(1-x^2)x˙=αx(1−x2), y˙=−y(β+κx2+ηy2)\dot{y}=-y(\beta+\kappa x^2+\eta y^2)y˙=−y(β+κx2+ηy2), z˙=−δz+ϵy2\dot{z}=-\delta z+\epsilon y^2z˙=−δz+ϵy2. We prove that this system admits closed-form solutions for all coordinates: x(t)x(t)x(t) is given explicitly, y(t)y(t)y(t) via a Bernoulli linearization, and z(t)z(t)z(t) via variation of constants. Consequently, we establish global convergence to one of three equilibria: a hyperbolic saddle at the origin and two sinks at (±1,0,0)(\pm 1, 0, 0)(±1,0,0). The stable manifold of the saddle is exactly the plane {x=0}\{x=0\}{x=0}, which acts as a separatrix between the basins of attraction. The parameters α\alphaα and β\betaβ are chosen to match the logarithmic rates of the Jonah triangle multipliers (λu=451/343\lambda_u=451/343λu=451/343, λs=−143/343\lambda_s=-143/343λs=−143/343), though this correspondence is a parameter convention rather than a dynamical derivation; notably, the negative sign of the discrete multiplier is lost in the continuous flow. This model serves as a rigorous, analytically solvable benchmark for testing edge-tracking algorithms in dynamical systems, without claiming physical relevance to Navier-Stokes or fluid turbulence.