Chaotic behaviour of a Hamiltonian with a quartic potential

Scattering and complete integrability in conformally invariant nonlinear theories

“…We also show that Minkowski time evolution for u tf + q 0 (f) = 0 is completely integrable on the space H. This contrasts with the results of Nikolaevskii and Shchur 6 (see also Savvidy 7 and Steeb et al 8 ). They embed the manifold of space-independent solutions of u tf + q 0 (f) = 0, where f = (f 1 ; f 2 ) and q(f) = f 2 1 f 2 2 , in the space of smooth solutions of the Yang-Mills equations.…”

“…X 0 )) n a ) ? : (8) Thus S n a = a if and only if (?U q (exp(? X 0 )) n a = a , and a is almost periodic for S if and only if a is almost periodic for ?U q (exp(?…”

Chaotic dynamics and two-field inflation

“…However, from the point of view of inflation, if the 2 term is missing the potential has a perfectly flat direction, and there is no guarantee that the inflaton will roll towards the minima of the potential. Moreover, Hamiltonian systems with terms like 2 2 are non-integrable, and Steeb et al summarize discussions of chaotic behaviour when the potential consists solely of a 2 2 term [25]. Thus, the chaotic properties of the system we consider does not depend on the specific form of equation ( 13) and, consequently, much of the discussion here is applicable to other models of two-field inflation.…”

“…and Müller [23] discuss a system which lacks the Φ 2 /2 term in the potential but is otherwise equivalent to our model in the absence of friction. In general, Hamiltonian systems with terms like Φ 2 Ψ 2 are non-integrable, and Steeb et al summarize discussions of chaotic behavior when the potential consists solely of a Φ 2 Ψ 2 term [24].…”

On the chaos of D-brane phase transitions

“…The chaotic nature of the resulting oscillations should come as no surprise in a nonlinear system such as this. Indeed the system with canonical kinetic terms in two fields X and Y , with an X 2 Y 2 potential is a well known partially chaotic system that has been studied in some depth [35][36][37]. This system (which we refer to as the 'XY system') is coincident with ours in the limit where the ρ 4 term in our potential is negligible and in the limit where the dynamics of the metric moduli and dilaton are associated with timescales much longer than those governing the oscillations we have been describing.…”