Cubic planar flow with diagonal approach directions ($\dot{x}=x(x^2+y^2),\dot{y}=2y^3$)

For this cubic vector field, every non-axis trajectory has the first integral $I(x,y)=1/y-y/x^2$. Direct differentiation gives $\nabla I\cdot (\dot{x},\dot{y})=0$ where $xy\ne 0$. Equivalently the trajectories satisfy $x^2-y^2=C{x}^{2}y$. The axes are separate invariant trajectories, and the diagonals are the cases $C=0$. For each fixed finite $C$, approach to the origin forces $y^2/x^2=1-Cy\to 1$, so non-axis trajectories have diagonal limiting directions.

The origin is repelling in forward time. For $V=x^2+y^2$, $\dot{V}=2x^4+2x^2{y}^2+4y^4\ge V^2$, so every nonzero trajectory eventually exits any small disk. The zero Jacobian matrix at the origin leaves the linearized system constant, demonstrating why nonlinear terms are needed for this stability classification.