Horseshoes for singly thermostated hamiltonians

This note studies 1 and 2 degree of freedom hamiltonian systems that are thermostated by a single-variable thermostat. Under certain conditions on the hamiltonian and thermostat, the existence of a horseshoe in the flow of the thermostated system is proven.


Introduction
One of the core models of equilibrium statistical mechanics is an isolated mechanical system, modeled by a hamiltonian H, that is immersed in, and in equilibrium with, a heat bath B at the temperature T = 1/β. Nosé [26], based on earlier work of Andersen [2], created a dynamical model of the exchange of energy between heat bath and system. This consists of adding an extra degree of freedom s and rescaling momentum by s: n is the number of degrees of freedom of H, M is the "mass" of the thermostat and k is Boltzmann's constant. Solutions to Hamilton's equations for G model the evolution of the state of the infinitesimal system along with the exchange of energy with the heat bath. Hoover reduced Nosé's thermostat by eliminating the state variable s and rescaling time t [8]: The Nosé-Hoover thermostat for a 1 degree of freedom hamiltonian H can be put in the form: where 2 = 1/M (see Lemma 3.1 below). Hoover observed that this thermostat is ineffective in producing the statistics of the Gibbs-Boltzmann distribution from single orbits of the thermostated harmonic oscillator [8]. There are numerous extensions of the Nosé-Hoover thermostat; a sample of these works includes [19,18,24,22,9,25]. In [19,18], a class of two-variable thermostats is introduced that is formally similar to, and extends the Nosé-Hoover thermostat by controlling both momentum and configuration variables. In [24], a class of n-variable thermostats is introduced by making the simple observation that the Nosé hamiltonian (eq. 2) can itself be thermostated-hence n thermostats can be recursively chained together. [22] studies a few variants of recursively thermostats. In [9], a two-variable thermostat is used to control the first two non-trivial moments of momentum of the thermostated system. In [25], the notion of a chain of thermostats is extended to a network of thermostats which are coupled based on a graph. All of these extensions share a common feature: the state of the thermostat is n ≥ 2 dimensional.
On the other hand, there are numerous extensions of the Nosé-Hoover thermostat that model the exchange of energy with the heat bath using a single, additional thermostat variable (ξ in 2), the so-called single thermostats. A sample includes [36,35,28,31,32,33,34]. In [36], a "variable" mass thermostat is introduced, which precludes the Hooverian reduction (eq. 2). In [35], singly thermostated harmonic oscillators, where the thermostat controls a single moment of momentawhich might be thought of as a weighted average temperature-are shown to have a first-order averaged system that is integrable. In [28], a variant of the Nosé-Hoover thermostated harmonic oscillator is considered, where the total-as opposed to kinetic-energy is controlled. In [31], the linear friction of the Nosé-Hoover thermostat is replaced with a tanh-friction that saturates at large magnitudes of the thermostat state ξ. In [32,33], the Nosé-Hoover thermostated harmonic oscillator is investigated and regions of phase space are found where apparently chaotic dynamics exist and regions where invariant tori appear with various knot types. In [34], the same authors visit the variant of the Nosé-Hoover thermostated harmonic oscillator that thermostats total energy, and demonstrate (numerically) the existence of a horseshoe. The numerical evidence also seems to show that the knotted and linked tori that are found are due to secondary bifurcations of KAM tori from the first-order averaged system [34, fig. 2].
In addition to the extensions of the Nosé-Hoover thermostat, there are several notable recent studies of this thermostat itself including [20,21,23]. In [20], it is shown that the Nosé-Hoover thermostated harmonic oscillator enjoys KAM tori near the = 0 decoupled limit; [21] extends these results and shows that a Nosé-Hoover thermostated integrable system has a first-order averaged system that is also integrable. In [23], it is shown that the Nosé-Hoover thermostated harmonic oscillator is not integrable in the class of Darboux integrals, which implies nonintegrability in the class of polynomial integrals, but not necessarily in the class of real-analytic or smooth integrals.
1.1. Results: 2 degrees of freedom. The first result of the present note concerns the creation of transverse homoclinic points by the Nosé-Hoover thermostat and thermostats similar to it. To do this, I start with an integrable 2-degree of freedom hamiltonian system which enjoys a normally hyperbolic invariant 2-manifold foliated by periodic orbits and with coincident stable and unstable manifolds. When thermostated, the invariant manifold is "blown up" into a normally hyperbolic invariant 4-manifold that is foliated by isotropic 2-tori. Poincaré-Melnikov theory is applied to detect transverse intersections of the stable and unstable manifolds. This theory is perturbative in nature, so to obtain a system to which the theory is applicable, a few steps are needed. First, the 6-dimensional symplectic phase space is symplecticly reduced by a T 1 action. The resulting, parameterized family of 2-degree of freedom hamiltonians are nearly integrable and enjoy a normally hyperbolic invariant 2-manifold that is foliated by periodic orbits-i.e. each system enjoys a saddle-centre fixed point. Second, the systems are rescaled to simplify the parametric dependence, elucidate the nature of the saddle-centre and the near-integrable nature of the problem. This work is all done within a hamiltonian framework. Surprisingly, the nature of the thermostat equations appears to preclude the use of a hamiltonian formalism to compute the Poincaré-Melnikov function that detects the transverse homoclinic points. The penultimate step is a two-part rescaling that destroys the canonical nature of the differential equations but reveals both the near-integrability of said equations and makes computation of the Poincaré-Melnikov function transparent.
Let us state the theorem: Theorem 1.1. Let r ≥ 3, Ξ = R or T 1 and V : Ξ −→ R be a C r potential function that has a unique non-degenerate local maximum value of 0 at q = 0. Let The subspace N = {(q 1 , p 1 , q 2 , p 2 , s, S) | q 2 = 0, p 2 = 0} is a normally hyperbolic invariant manifold of the thermostated hamiltonian G = H + N T (eq. 1). The reduction of the level set p 1 = µ by the T 1 action reduces N ∩ {p 1 = µ} to a normally hyperbolic invariant manifoldN µ for the reduced hamiltonianĜ µ .
Note that f is the Fourier transform of the kinetic energy in the subsystem described by H (2) m ; since (0, 0) is a non-degenerate saddle for this subsystem, the kinetic energy decays exponentially as t −→ ±∞. This implies that f is realanalytic and has at most countably many zeros.
It needs to be emphasized that Theorem 1.1 does not simply apply to the Nosé-Hoover thermostat; it includes the logistic thermostat of [31], for example. Below, it is also shown that the results extend with only minor modifications to the variablemass thermostats like that in [36] (see Theorem 3.2 below).
In addition, I should point out that Theorem 1.1 admits a few straightforward extensions. One can take H (1) to be a purely kinetic hamiltonian on T * T n for any n ≥ 1; equally, it could be a bi-invariant metric hamiltonian on T * G for any compact Lie group G. Once momentum is fixed and the co-adjoint orbit of the momentum is symplecticly reduced by the G × G action on T * G, the reduced equations are essentially those in Theorem 1.1. Moreover, one can take H (2) m to be a sum of any number of decoupled 1-degree of freedom mechanical hamiltonians each with a potential satisfying the same condition. Finally, one can even take H (1) to be mechanical with a non-trivial potential. By applying averaging to the H (1) subsystem at high energy, it behaves like a purely kinetic system up to an error that is negligible for the purposes here.
1.2. Results: 1-degree of freedom. Much of the literature on single thermostats focuses on singly thermostated 1-degree of freedom hamiltonians. The relatively poor "thermalization" of the Nosé-Hoover thermostated harmonic oscillator led researchers to introduce more non-linearity into the thermostat. One way to do this, that has not been explored in the literature, is to make the thermostat's state compact. To see why this might be interesting, consider a thermostat friction, like the logistic thermostat, that is odd in ξ and which saturates at 1 when ξ = ∞. The planes ξ = ±∞ in the extended phase space possess straight-line connecting orbits that connect (q c , 0, −∞) to (q c , 0, +∞) when q c is a critical point of the potential energy. If we glue two copies of the extended phase space along copies of ξ = +∞ and ξ = −∞ withξ reversed in one copy, the result is a thermostated system with the thermostat friction depending periodically on the state. The planes at ξ = ±∞ are invariant manifolds, so they separate the two copies, but one can see that the reversal ofξ means that this invariance can be destroyed by a small perturbation. That is, the dynamics of the two systems can be made to intermingle.
To state the result of this note in this direction, let Σ be a 1-manifold, T * Σ = Σ × R be its cotangent bundle and H : T * Σ −→ R be a C r hamiltonian, r > 2. Let Ξ be a 1-manifold and P = T * Σ × Ξ be a trivial Ξ-bundle over T * Σ with projection maps The Poisson bracket {, } on T * Σ pulls back to P in a natural manner, as does the hamiltonian vector field X H = { , H}. A C r−1 vector field T on P is a thermostat for H if it satisfies the definition 2.2 below. Let s ∈ T * Σ be a saddle critical point for H. Assume that γ ⊂ T * Σ is a homoclinic connection for s that bounds the compact region r. Then N = {s} × Ξ is a normally hyperbolic invariant manifold for X H whose stable and unstable manifolds W ± (N ) contain Γ = γ × Ξ and Γ bounds the region R = r × Ξ. By invariant manifold theory [7], Y = X H + T possesses a normally hyperbolic invariant manifold N that is a graph over N and similarly the local stable (unstable) manifold W ± loc (N ) is a graph over W ± loc (N ). Let us say that the thermostat T is monotone there is a neighbourhood U of Γ = γ × Ξ such that dξ, T does not vanish on U . This implies that Y |N does not vanish for all in some deleted neighbourhood of 0. If T is monotone, then N is an orbit of Y ; when Ξ = T 1 , this orbit is periodic.
that has a saddle critical point s with homoclinic connection γ. If T is a C r−1 monotone thermostat for H that is topologically transverse (resp. transverse) at γ (definition 2.1), then for all = 0 sufficiently small, the stable and unstable manifolds of the periodic orbit N are topologically transverse (resp. transverse).
In particular, the thermostated vector field Y enjoys a horseshoe.

1.3.
Outline. This note is organized as follows: §2 gives a proof of Theorem 1.2; §3 gives a proof Theorem 1.1; §3.2 proves an extension of the latter to variable-mass thermostats; §4 concludes.

Horseshoes in a thermostated 1-degree of freedom hamiltonian
Let us use the notation and terminology in the paragraph preceding the statement of 1.2. Let ω = π * (dp ∧ dq) be the pullback of the canonical symplectic form on T * Σ to P and for each homoclinic connection γ to a saddle critical point s of H define a function M γ : Ξ −→ R, Definition 2.1. T is topologically transverse at γ if M γ changes sign; it is transverse at γ if it is topologically transverse and 0 is a regular value of M γ . If T is topologically transverse (resp. transverse) at each γ, then T is said to be topologically transverse (resp. transverse).
In [5], the following definition of a thermostat vector field for a hamiltonian H is introduced. It is intended to capture the idea that the extended dynamics on the extended phase space should preserve a Gibbs-Boltzmann type probability measure dµ β whose marginal over ξ should be the original Gibbs-Boltzmann measure. Moreover, the extended dynamics should heat the system at low energy and cool it at high energy-at least on average.
on P such that the following holds and The average values in condition 3 are orbit averages, taken over the orbits of the vector field Y 0 = X H . Proposition 2.1. If T is a C r−1 monotone thermostat that is (resp. topologically) transverse at γ, then for all = 0 sufficiently small, the first-return map f of the vector field Y = X H + T enjoys a horseshoe in a neighbourhood of γ × {0}.
Proof. First, let us show that the local stable and unstable manifolds of N , W ± loc (N ), split. The proof is a straightforward application of the Poincaré-Melnikov function: Next, let us show that there is a horseshoe. Let us fix > 0 such that the local stable and unstable manifolds, W ± loc (N ), split to first order, so the zeros of the Poincaré-Melnikov function M detect the splitting. Assume that M γ (ξ) changes sign at ξ = 0. Let ζ : T * Σ −→ R be a smooth function that vanishes at the critical points of H and ζ| γ has a non-degenerate zero.
For all , η > 0 sufficiently small, such a neighbourhood and return map exist by virtue of the monotone condition. By construction, f preserves an area form since ϕ t preserves the volume form dµ β ; it has a hyperbolic fixed point s = s+O( ) and the local stable and unstable manifolds of s , W ± loc (s ), coincide with W ± loc (N ) ∩ Z η . Therefore, these manifolds split. If T is transverse, then they split transversely and an application of the Birkhoff-Smale homoclinic theorem implies the result. Otherwise, if T is only topologically transverse, the work of Burns & Weiss implies the result [4].
Remark 2.1. The reason that r > 2 in Proposition 2.1 is because Burns & Weiss prove that if the diffeomorphism f (= f above) is C 1 , then it possesses an invariant set Λ which factors onto a full shift on two symbols. If the diffeomorphism is area preserving and C r for r > 1, then one can appeal to a theorem of Katok [14] which states that in such a case, the diffeomorphism f actually possesses a horseshoe. Hidden in the proof of Proposition 2.1 is the fact that the return time to the crosssection is O(1/ ). Proof. Let γ be a homoclinic connection for H. Then, in the case that where c(γ) > 0. Since B 1 is odd and not identically 0, the first case of the lemma is proven; the second case is similar.
Example 2.1. A simple example of a monotone thermostat vector field that satisfies the hypotheses of lemma 2.1 is this: Let F : Ξ −→ R be a non-constant, smooth function and define T by This defines a thermostat vector field that is separable. It is monotone at a saddle connection γ if T > max γ {p · H p }, which holds for all T sufficiently large. When , the thermostat behaves, for ξ ∼ 0, similar to the Nosé-Hoover thermostat where F (ξ) = 1 2 ξ 2 (eq. 2).
3. Horseshoes in a thermostated 2-degree of freedom hamiltonian The previous section dealt with the creation of transverse intersections on stable and unstable manifolds in a single thermostated system using the reduced dynamics. In this section, thermostated hamiltonians are studied using the hamiltonian formalism on the extended phase space. The following lemma establishes the general connection between the reduced thermostat vector field Y when the thermostat vector field T is separable and takes a particularly simple form.
Lemma 3.1. Let T = −ρF (ξ)∂ ρ + (−T + ρ · H ρ )∂ ξ be a C r−1 thermostat vector field for the C r hamiltonian H. Then, for = 0, the vector field Y = X H + T is a reduction of the hamiltonian vector field X G , where S = ξ and F (S) = F (ξ).
Proof. Let H i be the partial derivative of H with respect to the i-th variable.
One can eliminate the differential equation for s and arrive at a system of differential equations described by the vector field Y .
As noted in the introduction, the Nosé-Hoover thermostat occupies a privileged position in the literature on thermostat dynamics. Let us formulate a definition which captures this centrality.  Proof. In each case, F (S) = 1 2 S 2 + O(S 4 ).

Elementary thermostats of order 2: split homoclinic connections.
In this section, it will be assumed that all hamiltonians are C r for r ≥ 3. The results that are proven below do hold for r ≥ 2, but the details are somewhat more cumbersome. In addition, the following assumptions are made: (1) H = H m (q 1 , p 1 , q 2 , p 2 ) is a C r hamiltonian with q 1 an angle variable defined mod2π and is a normally hyperbolic invariant manifold for the thermostated hamiltonian vector field G = G 1 (eq. 8). Moreover, if N 0 = {P ∈ N | p 1 = 0}, then N 0 is foliated by 2-dimensional tori which degenerate to a family of circles parameterized by p 1 .
The proof of the lemma is straightforward.
Let p 1 = µ = 0 be fixed. Since p 1 is a first integral of the thermostated vector field X G , we can symplecticly reduce the thermostat phase space by fixing p 1 = µ and ignoring the cyclic variable q 1 . Since the hamiltonian G is invariant under the symplectic automorphism that maps (q 1 , p 1 ) −→ (−q 1 , −p 1 ) and fixes the other coordinates, it can be assumed without loss of generality that µ > 0.
If x is an object on the thermostat phase space P that is T 1 invariant, thenx µ denotes the reduced object. In particular,P µ is the reduced phase space which is symplectomorphic to T * (Ξ × R + ).
It follows from the lemma that, up to a rescaling of time, the orbits of the reduced thermostated vector field XĜ µ coincide with those of XĜ β,µ +R β .
The proof is a straightforward calculation. Note that I have resisted the temptation to putĜ (1) into Birkhoff normal form because, although the remainder term on the left is improved, the coupling term involving u inĜ (2) becomes slightly less transparent.
The penultimate step to obtain a simple normal form in a neighbourhood of the saddle-centre singularity is a non-symplectic change of variables. A scale parameter > 0 is introduced via the change of variables and energy in the first subsystem: And, momentum and energy are rescaled in the second subsystem: Of course, the two systems are coupled and the two change of variables & energy cannot be decoupled as such. So, we must apply the change of variables to the hamiltonian vector field XĜ β,µ +R β , which produces a non-canonical vector field. Lemma 3.5. Let p 2 = mµ 2 P 2 , (u, U ) = (w, W ) and set m = ( /µ) 2 , β = mµ 2 = 2 , µ = 1/(αγ). (18) Then, the hamiltonian differential equations of the reduced, rescaled thermostated hamiltonian βĜ µ =Ĝ β,µ +R β are transformed tȯ From lemma 3.4 it follows that the parameter γ = 4 2M/µ 2 . So, we can treat , α and γ as independent positive parameters in (eq. 19) while b is a fixed constant. Note that the non-linear terms in the right-hand side ofẇ originate from the remainder term R β -the rescaling is unable to remove these terms. Recall that the planeN 0,µ = {(w, W, q 2 , P 2 ) | q 2 = P 2 = 0} is a normally hyperbolic invariant manifold for all non-negative values of the parameters. When = 0, the two pairs of differential equations decouple into a (non-linear, hamiltonian) oscillator in the (w, W ) plane and a mechanical hamiltonian in the (q 2 , P 2 ) plane with a saddle at (0, 0) and a saddle connecting orbit Γ(t) = (q 2 (t), P 2 (t)) (this was the motivation to fix β in terms of mµ 2 ). The hamiltonian function H 1 (q 2 , P 2 ) (eq. 11), which will be denoted by H (2) , is a constant of motion and H (2) • Γ ≡ 0 by the hypotheses on V . On the other hand, when > 0, the local stable and unstable manifolds ofN 0,µ ,Ŵ ± ,loc , are graphs overŴ ± 0,loc ⊃N 0,µ × Γ(R ± ). In general, these invariant manifolds do not coincide, but intersect transversely along homoclinic orbits. If Ω ± :Ŵ ± 0,loc −→Ŵ ± ,loc is a parameterization of the local stable and unstable manifolds that is C 2 in all variables and Ω ± 0 is the identity map, then d = H (2) • Ω + − H (2) • Ω − = M + O( 2 ) measures the distance between the invariant manifolds. The Poincaré-Melnikov function M :Ŵ ± 0,loc −→ R measures the lowest order difference in the invariant manifolds. If M vanishes at a point P and dM P = 0, then the implicit function theorem implies that the zero set of M is a surface in a neighbourhood of P . In addition, the implicit function theorem implies that the zero set of d is an O( ) perturbation of the zero set of M near P . SinceŴ ± ,loc are codimension-1 submanifolds, the zero set of d is where they intersect, and if dM P = 0, they intersect transversely near P .
The calculation and result in Proposition 3.2 is similar to that in [17, §V.3.9]. In the latter work, Kozlov studies the effect of a sinusoidal, time-dependent forcing of an ideal stationary planar perfect fluid flow. If the unperturbed systems has a saddle fixed point, the 1 1 2 -degree of freedom forced system has hyperbolic periodic orbits and the Poincaré-Melnikov integral for the specific perturbation leads to a Fourier transform.
The splitting of invariant manifolds in the hamiltonian setting is a well-studied problem. Koltsova and Lerman [15,16] prove that for a generic hamiltonian system with a saddle-centre critical point, at all nearby positive energy levels, there is a hyperbolic periodic orbit with 4 transverse homoclinic orbits. This can be explained in a simple way: in appropriate coordinates like those used in the proof of Proposition 3.2, the Poincaré-Melnikov function is, to lowest order, an indefinite quadratic form in (w, W ) which vanishes at the origin (the saddle-centre). The 4 transverse homoclinics originate from the 4 half-lines emanating from the origin where the quadratic vanishes. In the present case, the zeros of the Poincaré-Melnikov function occur on 2 half-lines emanating from the origin of the (w, W ) plane; it follows that the splitting studied here is not generic in the sense of Koltsova and Lerman, and that the reduced hamiltonians are not generic.
On the other hand, Churchill, Pecelli and Rod [6] study the stability of the periodic orbits in the (q 2 , P 2 ) plane which limit onto the separatrix Γ from energy levels below 0. They show that, under mild conditions satisfied by the Hénon-Heiles hamiltonian for example, there will be an infinite sequence of intervals of energy converging to 0 where the periodic orbits are alternately elliptic and hyperbolic. However, in the current problem, one sees that aside from = 0, there are appear to be no such continuous family of periodic orbits.

3.2.
Variable-mass thermostats. Let us extend definition 3.1 and remove the "elementary" aspect: we will allow the function F to be weighted by a variable mass, viz. In place of assumption 4 of section 3.1, it will be assumed henceforth (4) The thermostat N T (s, S) = Ω(s)F (S)+T ln s is a variable-mass thermostat of order 2. Thermostats like these have been studied in the literature [12,13,36] in different guises. For example, one can obtain this form from Winkler's thermostat, which rescales momenta by p/s e (e = 2 in Winkler's case), by the transformation (s, S) = (s 1/e 1 , es 1/r 1 S 1 ) where 1/e + 1/r = 1 and 1 < e, r < ∞. In the (s 1 , S 1 ) variables, the thermostat rescales momentum by p/s 1 and the effective thermostat temperature is T 1 = T /e. The Nosé-Hoover thermostat N (s, S) = 1 2 S 2 + T ln(s) is transformed to N (s 1 , S 1 ) = Ω(s 1 )F (S 1 ) + T 1 ln s 1 where Ω(s 1 ) = e 2 s 2/r 1 and F (S 1 ) = 1 2 S 2 1 (so Ω(s 1 ) = 4s 1 for Winkler's thermostat).
Remark 3.2. The proof of the theorem is straightforward and follows essentially the same arguments as above. Let us explain the differences between (eq. 29) and (eq. 19). In (eq. 27), one can rewrite the equation for α 2 as b 1 = 2α 4 µβ − 1 2 − c 1 µβ 1 2 , which explains the appearance of the (αW ) 2 term in the equation foṙ W (eq. 29). It also explains why the term that multiplies b in theẇ equation has a factor of γ vs. γ 3 and the appearance of the wW term in the same equation. It should also be noted that when a 1 = 0, the mass-like term 1/(2b 1 ) tends to 0 as β, 1/µ −→ 0 and α > 0 is fixed. This is similar to the constant mass case where M −→ 0 under the same conditions.

Conclusion
It has been shown that the Nosé-Hoover thermostat and closely-related thermostats create transverse homoclinic points near the saddle-centre equilibria of the reduced differential equations when applied to a separable system that is a sum of a 1-dimensional ideal gas and a planar pendulum. Several extensions of this example have been described. Several problems remain, including: (1) Extend the present results to thermostats like those in [9,11,35] where the thermostat N depends on p, s, S and/or the lowest-order term in N is S 4 (or higher); (2) Extend the present results to multi-variable thermostats, such as those discussed in the introduction; (3) Demonstrate the existence of transverse homoclinic points in systems like the thermostated harmonic oscillator; (4) Demonstrate the existence of Arnol'd diffusion in Nosé-Hoover thermostated n ≥ 2-degree of freedom systems [3]. The last problem motivated the present paper. However, it turns out that proving the existence of transverse homoclinic points in a neighbourhood of the periodic orbits of mixed type is already a sufficiently rich problem. The third problem also partially motivated the present paper: in contrast, though, its solution will involve exponentially small splitting and the more delicate calculations and error estimates that are involved.