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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]

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]