TheExistence of Invariant Tori andQuasiperiodic Solutions of the Nosé–Hoover Oscillator

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x′ � y, y′ � − x − yz, and z′ � y2 − a, where a is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever a is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. #e discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.


Introduction
In 1984, Nosé constructed a system called the Nosé equations to model the interaction of a particle with a heat-bath [1]. Later, in 1986, Posch et al. [2] simplified the Nosé equations by omitting an inessential variable and replaced the residual "momentum" by a "friction coefficient" and then got the Nosé-Hoover oscillator with the following equations of motion: where q and p are the coordinate and momentum of the oscillator, respectively; ξ is a friction coefficient; and a is a coupling positive real parameter. Specially, for a > 0, by the linear change of variables, Nosé-Hoover oscillator is transformed into which is usually called Sprott A system since it was presented by Sprott [3] with a � 1 as case A in a list of nineteen distinct differential systems with quadratic nonlinearities and having chaotic behavior. In this paper, we are mainly interested in the behaviors of Nosé-Hoover equation for a > 0 small so that we focus on the equivalent form (3) in the sequel.
Nosé-Hoover oscillator possesses rich dynamic behaviors: periodic and quasiperiodic solutions, nested tori, and chaos, all of which have been observed through numerical simulations (see [3][4][5][6] and references therein), even this system without equilibrium points. In the last few years, researchers have devoted to the rigorous proofs of these phenomena and achieved some results. In [7,8], Legoll et al. discussed the "simple" Nosé-Hoover thermostated harmonic oscillator: By means of an averaging argument, they reduced the thermostated equations to a nondegenerate twist map to show the existence of KAM tori near the decoupled limit of M � ∞ and ξ � 0. Subsequently, Butler [9] complemented the result of Legoll et al. and proved the existence of invariant tori near the high-temperature limit T � ∞ with the thermostat mass M held constant. Although we also pay attention to the existence of invariant tori of Nosé-Hoover equation, our methods are totally different. Compared with the average method mentioned in previous papers, we directly start from the equation itself and give its Poincaré map. Recently, in [10], Messias and Reinol proved the existence of a linearly stable periodic orbit of Sprott A system, which bifurcates from a nonisolated zero-Hopf equilibrium point located at the origin for a > 0 small, by using the averaging method. Moreover, they showed numerically the existence of nested invariant tori surrounding this periodic orbit, just like in Figure 1. e present paper explores the existence of invariant tori and quasiperiodic solutions of (3), which is absent of rigorous proof up to now. It is well known that Moser's twist theorem is a powerful tool to detect the existence of invariant curve (see [11][12][13][14] and references therein), but the application of twist theorem on 3-dimensional systems to obtain invariant tori is few. e key point is how to transform a 3dimensional system into a 2-dimensional one reasonably. Different from the way of reduction dimension by averaging theory in [7], we use the equation itself to eliminate some variables and perform a series of transformations to obtain a 2-dimensional system about action-angle variables. Besides the invariant tori, we further discuss the solutions of the transformed 2-dimensional system starting from the invariant curve. ey are quasiperiodic, which corresponds to the quasiperiodic solutions of the original 3-dimensional system with the same frequencies.
Another difficulty brought by system (3) is the absence of area-preserving property or intersection property needed in the twist theorem since it is not a conservative system. Fortunately, we find that the original 3-dimensional system (3) is invariant under the transformation of coordinates (x, y, z) ⟶ (− x, − y, z), and the transformed 2-dimensional system after every transformation keeps reversible property. erefore, a twist theorem established in [15] by Liu and Song for reversible systems is valid. In the application of this twist theorem (given in Section 3, eorem 2), we need to expand the corresponding Poincaré map into the form like (32), which is a tedious work needing much calculation. Now, we state our main result in the following.

Theorem 1.
For a > 0 small enough, system (3) admits an infinite number of invariant tori, which tends to the origin as a ⟶ 0 and thus an infinite number of quasiperiodic solutions. The structure of the paper is as follows. In Section 2, we first briefly introduce some definitions and properties about reversible systems (one can refer to [16] for details). en, we transform equation (3) into system (14) by a serious of transformations, including cylindrical coordinate change, translation change, scale, and polar coordinate change. Subsequently, we give the expression of Poincaré mapping of (14) and also prove the main result by a twist theorem for reversible system in Section 3. In the last section, numerical simulations of quasiperiodic solutions are given to support our results.

Some Facts on Reversible
Systems. Before performing transformations on system (3), we first give some definitions and facts related to reversible systems ( [12,16]).
is called reversible with respect to G, if with DG denoting the Jacobian of G.

Complexity
Usually, for the two-dimensional system, we are interested in the special involution G(x, y) � (− x, y) and G(x, y) � (x, − y) for z � (x, y) ∈ R 2 , under which condition (6) is transformed into an obvious form for G: and for G: Definition 2. Let T(·, t) be an invertible transformation of Ω for every fixed t, and suppose that G is an involution with Returning to the special example G(x, y) � (x, − y) from above, condition (9) requires where T � (T 1 , T 2 ).
Next lemma is useful to show that a transformed version of a reversible system is again reversible. (5) is reversible with respect to G. If a transformation T(·, t): Ω ⟶ R n is G-invariant, then the transformed system, i.e., the system satisfied by (z(t) � T(z(t), t)), is also reversible with respect to G.

Lemma 1. Assume that
Finally, reversible systems lead to reversible Poincaré maps, in the following sense.
Definition 3. Let f: Ω ⟶ R n be a homeomorphism onto its image, and let G be an involution.
We also note that system (21) is reversible under the In this 2-dimensional system (21), I plays the role of action variable, φ is the angle variable, and θ denotes the time variable. We will focus on the Poincaré mapping at θ � π in the sequel.

Proof of Theorem 1
e key of the whole proof is to figure out the Poincaré mapping induced by (21). Since it involves a lot of tedious calculations by Taylor expansions and numerous notations, we give a sketch of the proof.
Step 1. Observing that g 3 and g 4 defined in (22) are real analytic, π-periodic with respect to θ and 2π-periodic with respect to φ, we do the Taylor expansion of order 3 of (21) at ε � 0. en, there are To derive an expression for the corresponding Poincaré map, we set for the solution (I(θ), φ(θ)) of (14) with (I(0), φ(0)) � (I 0 , φ 0 ). Integrating (23) and comparing with (25), we see that and 4 Complexity where We then represent F 1 and F 2 according to the order of ε and simplify the coefficients of each item in the next lemma.
e Poincaré time-π-map of (21) is well defined for ε small enough, and by Lemma 3, it has the following form: In order to complete the proof of eorem 1, we will show that, for ε small enough, the Poincaré map P has an invariant closed curve in the annulus which surrounds the point (0, 0).
According to Moser's twist theorem [17], the twist map should have twist term, which is independent of initial point and tiny amount ε. However, the twist term in Poincaré map (31) disappears, and the disturbance terms are at least the first-order term of ε. erefore, we consider the theorem established by Liu and Song for reversible map, which matches the form of (31). We state their result below, and one can refer to [15] for details.
Let A � [a, b] × S 1 (S 1 � (R/2πZ)) denote a cylinder with universal cover [a, b] × R. It will be assumed that a map f: A ⟶ R × S 1 has a lift, which can be expressed in the following form: Moreover, f is reversible with respect to the involution (θ, r) ⟶ (− θ, r).

Complexity
In addition, we assume that there is a function I: A ⟶ R satisfying

(35)
Then, there exist δ > 0 and Δ > 0 such that if ε < Δ and The map f has an invariant curve in A. e constants Δ and δ are dependent on a, b, a, b, l 1 , l 2 and I. In particular, δ is independent of ε. Remark 1. If the signs of l 1 and (zl 1 /zr) are inverse, the theorem is still valid. en, we turn to verify the conditions in eorem 2 for the Poincaré map P.
Consequently, by eorem 2 and Remark 1, the Poincaré map P has an invariant closed curve for small enough ε. In other words, there is an embedding ψ: T π (R/πZ) ⟶ A � [0, ε] × S 1 of a circle, which is invariant under the map P. Furthermore, on this invariant curve, the map P is conjugated to a rotation with number ω * :   Complexity