a gradient system
Picture a ball always rolling straight downhill on a landscape, taking the steepest descent at every step. It can never circle back up to where it started, because every move strictly lowers its height. A gradient system is the dynamical version of this picture: the motion is driven directly by the slope of a single height function, so the system can only ever go 'downhill' and never loops.
Formally, a system is a gradient system if it can be written as x' = -grad V(x), where V is a scalar potential function and grad V is its gradient (the vector pointing in the direction of steepest increase). The minus sign makes the state move in the direction of steepest decrease of V. Along any trajectory the potential satisfies V' = (grad V) · x' = -(grad V) · (grad V) = -|grad V|^2, which is less than or equal to zero — V is non-increasing always, and strictly decreases except at equilibria, which are exactly the critical points of V (where grad V = 0). So V is a built-in Lyapunov function, handed to you for free.
The consequence is sweeping: a gradient system can have no closed orbits and no limit cycles at all, because returning to a starting point would require V to come back to its starting value, yet V only ever decreases. Equilibria are the maxima, minima, and saddles of the landscape: minima are asymptotically stable (valleys you roll into), maxima are unstable (hilltops), saddles are unstable. Gradient systems appear in optimization (gradient descent is exactly this flow), in physics as overdamped motion, and as the cleanest contrast to Hamiltonian systems, which instead conserve their energy and circle forever.
With V(x,y) = x^2 + 2y^2, the gradient system is x' = -2x, y' = -4y; every trajectory slides straight downhill to the single minimum at the origin, and no closed orbit can exist anywhere.
The potential V can only decrease along the flow, so the system descends to a minimum and never circles.
Not every system is a gradient system — only special ones can be written as x' = -grad V (a symmetry condition on the Jacobian must hold). Where it does apply, the immediate payoff is that closed orbits and limit cycles are impossible.