Lagrange stability

Lagrange stability is a concept in the stability theory of dynamical systems, named after Joseph-Louis Lagrange.
For any point in the state space, formula_1 in a real continuous dynamical system formula_2, where formula_3 is formula_4, the motion formula_5 is said to be "positively Lagrange stable" if the positive semi-orbit formula_6 is compact. If the negative semi-orbit formula_7 is compact, then the motion is said to be "negatively Lagrange stable". The motion through formula_8 is said to be "Lagrange stable" if it is both positively and negatively Lagrange stable. If the state space formula_9 is the Euclidean space formula_10, then the above definitions are equivalent to formula_11 and formula_12 being bounded, respectively.
A dynamical system is said to be positively-/negatively-/Lagrange stable if "for each" formula_1, the motion formula_5 is positively-/negativey-/Lagrange stable, respectively.