IN THIS PAPER, WE ADVANCE THE STUDY OF THE LYAPUNOV STABILITY AND INSTABILITY OF EQUILIBRIUM SOLUTIONS OF HAMILTONIAN FLOWS, WHICH IS ONE OF THE OLDEST PROBLEMS IN MATHEMATICAL PHYSICS. MORE PRECISELY, IN THIS WORK WE STUDY THE NONLINEAR STABILITY IN THE LYAPUNOV SENSE OF ONE EQUILIBRIUM SOLUTION IN AUTONOMOUS HAMILTONIAN SYSTEMS WITH N-DEGREES OF FREEDOM, ASSUMING THE EXISTENCE OF TWO VECTORS OF RESONANCE, BOTH OF ORDER FOUR, WITH INTERACTION IN ONE FREQUENCY. WE PROVIDE CONDITIONS TO OBTAIN A TYPE OF FORMAL STABILITY, CALLED LIE STABILITY. SUBSEQUENTLY, WE GUARANTEE SOME SUFFICIENT CONDITIONS TO OBTAIN EXPONENTIAL STABILITY IN THE SENSE OF NEKHOROSHEV FOR LIE STABLE SYSTEMS WITH THREE AND FOUR DEGREES OF FREEDOM. IN ADDITION, WE GIVE SUFFICIENT CONDITIONS FOR THE INSTABILITY IN THE SENSE OF LYAPUNOV. WE APPLY SOME OF OUR RESULTS IN THE SPATIAL SATELLITE PROBLEM AT ONE OF ITS EQUILIBRIUM POINTS, WHICH IS A NOVELTY IN THIS PROBLEM.

IN THIS WORK, WE ADVANCE IN THE STUDY OF THE LYAPUNOV STABILITY AND INSTABILITY OF EQUILIBRIUM SOLUTIONS OF HAMILTONIAN FLOWS. MORE PRECISELY, WE STUDY THE NONLINEAR STABILITY IN THE LYAPUNOV SENSE OF EQUILIBRIUM SOLUTIONS IN AUTONOMOUS HAMILTONIAN SYSTEMS WITH N-DEGREES OF FREEDOM, ASSUMING THE EXISTENCE OF TWO RESONANCE VECTORS K1 AND K2 WITHOUT INTERACTION (|K1|?|K2|). WE PROVIDE CONDITIONS TO OBTAIN A TYPE OF FORMAL STABILITY, CALLED LIE STABILITY. IN PARTICULAR, WE NEED TO NORMALIZE THE HAMIL TONIAN FUNCTION TO ANY ARBITRARY ORDER, AND OUR RESULTS TAKE INTO ACCOUNT THE SIGN OF THE COMPONENTS OF THE RESONANCE VECTORS. SUBSEQUENTLY, WE GUARANTEE SOME SUFFICIENT CONDITIONS TO OBTAIN EXPONENTIAL STABILITY IN THE SENSE OF NEKHOROSHEV FOR LIE STABLE SYSTEMS. IN ADDITION, WE GIVE SUFFICIENT CONDITIONS FOR THE INSTABILITY IN THE LYAPUNOV SENSE OF THE FULL SYSTEM. FOR THIS, IT IS NECESSARY TO NORMALIZE THE HAMILTONIAN FUNCTION TO AN ADEQUATE ORDER, AND ASSUMING THAT THE COMPONENTS OF AT LEAST ONE RESONANCE VECTOR CHANGE OF SIGN.

WE CONSIDER THE 2-BODY PROBLEM IN THE SPHERE S2. THIS PROBLEM IS MODELED BY A HAMILTONIAN SYSTEM WITH 4 DEGREES OF FREEDOM AND, FOLLOWING THE APPROACH GIVEN IN [4], ALLOWS US TO REDUCE THE STUDY TO A SYSTEM OF 2 DEGREES OF FREEDOM. IN THIS WORK WE WILL USE THEORETICAL TOOLS SUCH AS NORMAL FORMS AND SOME NONLINEAR STABILITY RESULTS ON HAMILTONIAN SYSTEMS FOR DEMONSTRATING A SERIES OF RESULTS THAT WILL CORRESPOND TO THE OPEN PROBLEMS PROPOSED IN [4] RELATED TO THE NONLINEAR STABILITY OF THE RELATIVE EQUILIBRIA. MOREOVER, WE STUDY THE EXISTENCE OF HAMILTONIAN PITCHFORK AND CENTER-SADDLE BIFURCATIONS.