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This paper demonstrates sufficient conditions for the existence of Kolmogorov-Arnol’d-Moser (KAM) tori in a singly thermostated, integrable Hamiltonian system with n degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2â€"which include the NosĂ© thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends theorem 3.2 of Legoll et al (2009 Nonlinearity 22 1673â€"94) to prove that a ‘typical’ singly thermostated, integrable, real-analytic Hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable Hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion. [ABSTRACT FROM AUTHOR]

This note studies 1 and 2 degrees of freedom Hamiltonian systems that are thermostated by a single-variable thermostat. It provides concrete examples of integrable Hamiltonians and single thermostats, including the Nos\'e--Hoover thermostat, where the existence of a horseshoe in the flow of the thermostated system is proven. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of a family of Hamiltonian systems that originate from Friedman–Lemaître–Robertson–Walker space-times with a coupled field and non-zero curvature. In four distinct cases, previously considered by Maciejewski, Przybylska, Stachowiak and Szydowski, it is shown that there are homoclinic connections to invariant submanifolds and the connections split. These results imply the non-existence of a real-analytic integral independent of the Hamiltonian. [ABSTRACT FROM AUTHOR]

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM (Kolmogrov-Arnol'd-Moser) tori in a singly thermostated, one degree-of-freedom Hamiltonian vector field. This result is applied to four important single thermostats in the literature, and it is shown that in each case, if the Hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends the results of Legoll, Luskin, and Moeckel [Arch. Ration. Mech. Anal. 184, 449 (2007)] and Legoll et al. [Nonlinearity 44, 1673 (2009)]. [ABSTRACT FROM AUTHOR]

Let $${H(q,p) = \frac{1}{2}{\parallel p \parallel}^2 + V(q)}$$ be an n-degree of freedom C mechanical Hamiltonian on $${T^{*}{\bf T}^n}$$ where $${r > 2n+2}$$ . When the metric $${{\parallel \cdot \parallel}}$$ is flat, the Nosé-thermostated system associated to H is shown to have a positive-measure set of invariant tori near the infinite temperature limit. This is shown to be true for all variable mass thermostats similar to Nosé's, too. These results complement results of Bulter (Nonlinearity 11(29):3454-3463, 2016), Legoll et al. (Arch Ration Mech Anal 184(3):449-463, 2007, Nonlinearity 22(7):1673-1694, 2009). [ABSTRACT FROM AUTHOR]