# The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator

Authors: Yanmin Niu; Xiong Li
Journal: Complexity
 (2020)

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## Abstract

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator,x′=y,y′=−x−yz,andz′=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. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.


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## Body

## 1. 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:(1)q′=p,p′=−q−pξ,ξ′=ap2−1,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,(2)q=xa,p=ya,ξ=z.Nosé–Hoover oscillator is transformed into(3)x′=y,y′=−x−yz,z′=y2−a,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 [3–6] 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:(4)q′=ρ,ρ′=−q−ξρ,ξ′=ρ2−TM.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 ofM=∞ 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.Figure 1
A periodic orbit of system (3) near the origin for a=10−4 surrounded by invariant tori.The 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 [11–14] and references therein), but the application of twist theorem on 3-dimensional systems to obtain invariant tori is few. The key point is how to transform a 3-dimensional 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. They 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. Therefore, 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, Theorem 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.
Fora>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 Section2, we first briefly introduce some definitions and properties about reversible systems (one can refer to [16] for details). Then, 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.


## 2. Transformations on System (3)

### 2.1. Some Facts on Reversible Systems

Before performing transformations on system (3), we first give some definitions and facts related to reversible systems ([12, 16]).Definition 1.
LetΩ∈ℝn be open and Z=Zz,t:Ω×ℝ⟶ℝn be continuous and 2π-periodic in the time variable t. Moreover, suppose G:ℝn⟶ℝn is an involution (i.e., G is a C1-diffeomorphism such that G2=idℝn) satisfying GΩ=Ω. The system(5)z′=Zz,t,is called reversible with respect to G, if(6)DGGzZGz,−t=−Zz,t,z∈Ω,t∈ℝ,with DG denoting the Jacobian of G.
Usually, for the two-dimensional system, we are interested in the special involutionGx,y=−x,y and G¯x,y=x,−y for z=x,y∈ℝ2, under which condition (6) is transformed into an obvious form for G:(7)Z1−x,y,−t=Z1x,y,t,Z2−x,y,−t=−Z2x,y,t,and for G¯:(8)Z1x,−y,−t=−Z1x,y,t,Z2x,−y,−t=Z2x,y,t,with Z=Z1,Z2.Definition 2.
LetT⋅,t be an invertible transformation of Ω for every fixed t, and suppose that G is an involution with GΩ=Ω. Then, T is called G-invariant if(9)GTz,t=TGz,−t,z∈Ω,t∈ℝ.
Returning to the special exampleG¯x,y=x,−y from above, condition (9) requires(10)T1x,−y,−t=T1x,y,t,T2x,−y,−t=−T2x,y,t,where T=T1,T2.
Next lemma is useful to show that a transformed version of a reversible system is again reversible.Lemma 1.
Assume that (5) is reversible with respect to G. If a transformation T⋅,t:Ω⟶ℝn is G-invariant, then the transformed system, i.e., the system satisfied by z¯t=Tzt,t, is also reversible with respect to G.
Finally, reversible systems lead to reversible Poincaré maps, in the following sense.Definition 3.
Letf:Ω⟶ℝn be a homeomorphism onto its image, and let G be an involution. Then, f is called reversible with respect to G on a set D⊂Ω∩fΩ with GD=D if(11)f−1=G∘f∘G,inD.Lemma 2.
Suppose the Cauchy problems associated with (5) are uniquely solvable and let the time 2π-map P and its inverse P−1 be defined. Then, P is reversible with respect to G, provided that (5) is reversible with respect to G.


### 2.2. Transformed System

Obviously, the equations of motion (3) are invariant under the transformation of coordinates x,y,z⟶−x,−y,z. After a rescaling of time t, equation (3) is firstly written into(12)x′=−y,y′=x+yz,z′=a−y2.Passing to cylindrical coordinates(13)x=rcosθ,y=rsinθ,z=z,via a short calculation (12) becomes(14)r′=rzsin2θ,θ′=zsinθcosθ+1,z′=a−r2sin2θ.Now, we introduce a small quantity by the change in variables(15)r=ερ,z=ετ,with ε=a>0. Then, system (14) is rewritten into(16)ρ′=ερτsin2θ,θ′=ετsinθcosθ+1,τ′=ε1−ρ2sin2θ.Noticingετsinθcosθ+1≠0 for ε small enough and taking θ as the independent variable, we get(17)dρdθ=ερτsin2θετsinθcosθ+1,dτdθ=ε1−ρ2sin2θετsinθcosθ+1.It is not difficult to find that equation (17) is reversible under the transformation ρ,τ⟶ρ,−τ since g1ρ,−τ,−θ=−g1ρ,τ,θ and g2ρ,−τ,−θ=g2ρ,τ,θ where g1 and g2 denote the functions at the right hands of equation (12), respectively.Note that the transformations above appeared in [10] either. By the averaging method, Messias and Reinol proved that equation (17) has periodic solutions tending to 2,0 as ε tends to 0. Different from the method and result in [10], we focus on the Poincaré mapping to obtain invariant torus. Based on this, the subsequent transformation is carried out after a translation:(18)ρ¯=ρ−2,τ=τ.Correspondingly, equation (17) is transformed into(19)dρ¯dθ=ερ¯+2τsin2θετsinθcosθ+1,dτdθ=ε1−ρ¯+22sin2θετsinθcosθ+1.This system is reversible under the transformationρ¯,τ⟶ρ¯,−τ.Lastly, writing system (19) in polar coordinates(20)ρ¯=Icosφ,τ=Isinφ,it becomes(21)dIdθ=−2εsinφsin2θIcosφ+2+εsinφεIsinφsinθcosθ+1,dφdθ=εcosφ−εIcosφ+2I+2cosφsin2θIεIsinφsinθcosθ+1.We also note that system (21) is reversible under the involution I,φ⟶I,−φ since g3I,−φ,−θ=−g3I,φ,θ and g4I,−φ,−θ=g4I,φ,θ, where(22)g3=−2εsinφsin2θIcosφ+2+εsinφεIsinφsinθcosθ+1,g4=εcosφ−εIcosφ+2I+2cosφsin2θIεIsinφsinθcosθ+1.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.


## 2.1. Some Facts on Reversible Systems

Before performing transformations on system (3), we first give some definitions and facts related to reversible systems ([12, 16]).Definition 1.
LetΩ∈ℝn be open and Z=Zz,t:Ω×ℝ⟶ℝn be continuous and 2π-periodic in the time variable t. Moreover, suppose G:ℝn⟶ℝn is an involution (i.e., G is a C1-diffeomorphism such that G2=idℝn) satisfying GΩ=Ω. The system(5)z′=Zz,t,is called reversible with respect to G, if(6)DGGzZGz,−t=−Zz,t,z∈Ω,t∈ℝ,with DG denoting the Jacobian of G.
Usually, for the two-dimensional system, we are interested in the special involutionGx,y=−x,y and G¯x,y=x,−y for z=x,y∈ℝ2, under which condition (6) is transformed into an obvious form for G:(7)Z1−x,y,−t=Z1x,y,t,Z2−x,y,−t=−Z2x,y,t,and for G¯:(8)Z1x,−y,−t=−Z1x,y,t,Z2x,−y,−t=Z2x,y,t,with Z=Z1,Z2.Definition 2.
LetT⋅,t be an invertible transformation of Ω for every fixed t, and suppose that G is an involution with GΩ=Ω. Then, T is called G-invariant if(9)GTz,t=TGz,−t,z∈Ω,t∈ℝ.
Returning to the special exampleG¯x,y=x,−y from above, condition (9) requires(10)T1x,−y,−t=T1x,y,t,T2x,−y,−t=−T2x,y,t,where T=T1,T2.
Next lemma is useful to show that a transformed version of a reversible system is again reversible.Lemma 1.
Assume that (5) is reversible with respect to G. If a transformation T⋅,t:Ω⟶ℝn is G-invariant, then the transformed system, i.e., the system satisfied by z¯t=Tzt,t, is also reversible with respect to G.
Finally, reversible systems lead to reversible Poincaré maps, in the following sense.Definition 3.
Letf:Ω⟶ℝn be a homeomorphism onto its image, and let G be an involution. Then, f is called reversible with respect to G on a set D⊂Ω∩fΩ with GD=D if(11)f−1=G∘f∘G,inD.Lemma 2.
Suppose the Cauchy problems associated with (5) are uniquely solvable and let the time 2π-map P and its inverse P−1 be defined. Then, P is reversible with respect to G, provided that (5) is reversible with respect to G.


## 2.2. Transformed System

Obviously, the equations of motion (3) are invariant under the transformation of coordinates x,y,z⟶−x,−y,z. After a rescaling of time t, equation (3) is firstly written into(12)x′=−y,y′=x+yz,z′=a−y2.Passing to cylindrical coordinates(13)x=rcosθ,y=rsinθ,z=z,via a short calculation (12) becomes(14)r′=rzsin2θ,θ′=zsinθcosθ+1,z′=a−r2sin2θ.Now, we introduce a small quantity by the change in variables(15)r=ερ,z=ετ,with ε=a>0. Then, system (14) is rewritten into(16)ρ′=ερτsin2θ,θ′=ετsinθcosθ+1,τ′=ε1−ρ2sin2θ.Noticingετsinθcosθ+1≠0 for ε small enough and taking θ as the independent variable, we get(17)dρdθ=ερτsin2θετsinθcosθ+1,dτdθ=ε1−ρ2sin2θετsinθcosθ+1.It is not difficult to find that equation (17) is reversible under the transformation ρ,τ⟶ρ,−τ since g1ρ,−τ,−θ=−g1ρ,τ,θ and g2ρ,−τ,−θ=g2ρ,τ,θ where g1 and g2 denote the functions at the right hands of equation (12), respectively.Note that the transformations above appeared in [10] either. By the averaging method, Messias and Reinol proved that equation (17) has periodic solutions tending to 2,0 as ε tends to 0. Different from the method and result in [10], we focus on the Poincaré mapping to obtain invariant torus. Based on this, the subsequent transformation is carried out after a translation:(18)ρ¯=ρ−2,τ=τ.Correspondingly, equation (17) is transformed into(19)dρ¯dθ=ερ¯+2τsin2θετsinθcosθ+1,dτdθ=ε1−ρ¯+22sin2θετsinθcosθ+1.This system is reversible under the transformationρ¯,τ⟶ρ¯,−τ.Lastly, writing system (19) in polar coordinates(20)ρ¯=Icosφ,τ=Isinφ,it becomes(21)dIdθ=−2εsinφsin2θIcosφ+2+εsinφεIsinφsinθcosθ+1,dφdθ=εcosφ−εIcosφ+2I+2cosφsin2θIεIsinφsinθcosθ+1.We also note that system (21) is reversible under the involution I,φ⟶I,−φ since g3I,−φ,−θ=−g3I,φ,θ and g4I,−φ,−θ=g4I,φ,θ, where(22)g3=−2εsinφsin2θIcosφ+2+εsinφεIsinφsinθcosθ+1,g4=εcosφ−εIcosφ+2I+2cosφsin2θIεIsinφsinθcosθ+1.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.


## 3. Proof of Theorem1

The 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 thatg3andg4 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. Then, there are(23)dIdθ=−2εIcosφ+2sinφsin2θ+εsinφ+2ε22IIcosφ+2sin2φsin2θsin2θ−ε22Isin2φsin2θ−2ε34I2Icosφ+2sin3φsin2θsin22θ+ε34I2sin3φsin22θ+Oε4,(24)dφdθ=εI−1cosφ−εI−1Icosφ+2I+2cosφsin2θ−ε24sin2φsin2θ+ε22Icosφ+2I+2cosφsinφsin2θsin2θ−ε34IIcosφ+2I+2cosφsin2φsin2θsin22θ+ε34Icosφsin2φsin22θ+Oε4.
To derive an expression for the corresponding Poincaré map, we set(25)Iθ;I0,φ0=I0+εF1θ;I0,φ0,ε,φθ;I0,φ0=φ0+εF2θ;I0,φ0,ε,for the solution Iθ,φθ of (14) with I0,φ0=I0,φ0. Integrating (23) and comparing with (25), we see that(26)F1θ;I0,φ0,ε=−2∫0θsinφ0+εF2⋅A1sin2tdt+∫0θsinφ0+εF2dt+2ε2∫0θI0+εF1sin2φ0+εF2⋅A1sin2t⋅sin2tdt−ε2∫0θI0+εF1sin2φ0+εF2⋅sin2tdt−2ε24∫0θI0+εF12A1sin3φ0+εF2⋅sin2t⋅sin22tdt+ε24∫0θI0+εF12sin3φ0+εF2⋅sin22tdt+Oε3,and(27)F2θ;I0,φ0,ε=∫0θI0+εF1−1cosφ0+εF2dt−∫0θI0+εF1−1A1A2×sin2tdt−ε4∫0θsin2φ0+εF2⋅sin2tdt+ε2∫0θA1A2⋅sinφ0+εF2⋅sin2t⋅sin2tdt−ε24∫0θI0+εF1⋅A1A2⋅sin2φ0+εF2⋅sin2t⋅sin22tdt+ε24∫0θI0+εF1cosφ0+εF2⋅sin2φ0+εF2⋅sin22tdt+Oε3,where(28)A1≜I0+εF1cosφ0+εF2+2,A2≜I0+εF1+2cosφ0+εF2.
We then representF1andF2 according to the order of ε and simplify the coefficients of each item in the next lemma.Lemma 3.
We have(29)F1π=−2π4I0sin2φ0+εL1I0,φ0+ε2L2I0,φ0+Oε3,F2π=−π2I0cosφ0+2cos2φ0+2−εG1I0,φ0−ε2G2I0,φ0+Oε3,where LiI0,φ0,GiI0,φ0,i=1,2, are analytic about I0,φ0 and 2π-periodic about φ0.
In fact,LiI0,φ0,GiI0,φ0,i=1,2, have specific expressions, and we omit here.Step 2.
The Poincaré time-π-map of (21)(30)P:I0,φ0⟶Iπ;I0,φ0,φπ;I0,φ0:=I1,φ1,is well defined for ε small enough, and by Lemma 3, it has the following form:(31)I1=I0−2π4I0sin2φ0⋅ε+ε2L1I0,φ0+ε3L2I0,φ0+Oε4,φ1=φ0−π22+I0cosφ0+2cos2φ0ε−ε2G1I0,φ0−ε3G2I0,φ0+Oε4.
In order to complete the proof of Theorem1, 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. Therefore, 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.
LetA=a,b×S1S1=ℝ/2πℤ denote a cylinder with universal cover a,b×ℝ. It will be assumed that a map f:A⟶ℝ×S1 has a lift, which can be expressed in the following form:(32)θ1=θ+ω+εl1θ,r+εϕ1θ,r;ε,r1=r+εl2θ,r+εϕ2θ,r;ε.
Moreover,f is reversible with respect to the involution θ,r⟶−θ,r.Theorem 2.
Letω=2nπ with an integer n and the functions l1,l2,ϕ1, and ϕ2 satisfy(33)l1∈C6A,l1>0,∂l1∂r>0,∀θ,r∈A,l2⋅,⋅,ϕ1⋅,⋅;ε,ϕ2⋅,⋅;ε∈C5A.
In addition, we assume that there is a functionI:A⟶ℝ satisfying(34)I∈C6A,∂I∂r>0,∀θ,r∈A,l1θ,r∂I∂θθ,r+l2θ,r∂I∂rθ,r=0.
Moreover, suppose that there are two numbersa˜ and b˜ such that(35)a<a˜<b˜<b,I¯a<I¯a˜<I¯a˜<I¯b˜<I¯b˜<I¯b,I¯r:=minθ∈S1Iθ,r,I¯:=maxθ∈S1Iθ,r.
Then, there existδ>0 and Δ>0 such that if ε<Δ and(36)ϕ1C5A+ϕ2C5A<δ.
The mapf has an invariant curve in A. The constants Δ and δ are dependent on a,b,a˜,b˜,l1,l2 and I. In particular, δ is independent of ε.Remark 1.
If the signs ofl1 and ∂l1/∂r are inverse, the theorem is still valid.
Then, we turn to verify the conditions in Theorem2 for the Poincaré map P.Step 3.
Let chooseA=0,ε×S1, where ε as before is small. For I0,φ0∈A, the term π/2εI0cosφ0 in (31) can be seen as Oε2. Therefore, we rewrite the Poincaré map (31) in the form of(37)I1=I0−2π4I0sin2φ0⋅ε+Oε2,φ1=φ0−2π21+cos2φ0ε+Oε2.
Comparing the Poincaré map (37) with (32), we regard(38)l1=−2π21+cos2φ0,l2=−2π4I0sin2φ0,ω=0.
The high-order terms ofε are put into Oε2.
It is not difficult to show that, forI0,φ0∈A, l1<0,∂l1/∂I0=0. Based on the expressions of l1 and l2, we find a function J=1/2ln1+cos2φ0+lnI0, which is the one satisfying Theorem 2. In fact, there are ∂J/∂I0=1/I0>0 for I0,φ0∈A and(39)l1∂J∂φ0+l2∂J∂I0=2π21+cos2φ0⋅sinφ0cosφ01+cos2φ0−2π4I0sin2φ0⋅1I0=0.
Moreover, sinceJ¯=−ln2/2+lnI0, J¯=lnI0 and fI0≜lnI0 monotonically increases for I0>0, there must be a˜,b˜∈0,ε such that J¯0<J¯a˜<J¯a˜<J¯b˜<J¯b˜<J¯ε.
Consequently, by Theorem2 and Remark 1, the Poincaré map P has an invariant closed curve for small enough ε. In other words, there is an embedding ψ:Tπℝ/πℤ⟶A=0,ε×S1 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∗:(40)P∘ψs=ψs+ω∗,withsmodπ,where ω∗ is not the ω in (32), and it satisfies Diophantine condition:(41)ω∗−pq≥cq−2−β,for all integers p and q≠0 where β>0 and c>0 are two constants. Then, the solutions of (21) starting at time t=0 on this invariant curve determine a π-periodic cylinder in the space I,φ,θ∈A×ℝ. Denote the vector field of (21) by X. Since X is time π-periodic, the phase space is A×Tπ. Let Φt with Φ0=id be the flow of the time-independent vector field X,1 on A×Tπ and define ψs,τ:Tπ×Tπ⟶A×Tπ by setting(42)ψs,τ=Φτψs−τω∗,0=ϕτ∘ψs−τω∗,τ,where ϕt is the flow of the vector field of X and, therefore, P=ϕπ. Then,(43)ψs,τ+π=ϕτ+π∘ψs−τ+πω∗,τ+π=ϕτ∘ϕπ∘ψs−τω∗−πω∗,τ+π=ϕτ∘ψs−τω∗,τ=ψs,τ,ψs+π,τ=Φτψs+π−τω∗,0=ψs,τ.
Moreover,(44)Φt∘ψs,τ=ψs+ω∗t,τ+t.Therefore, the torus ψTπ×Tπ is quasiperiodic with the frequencies ω∗,1.
We note that the existence of invariant curves forP corresponds to the existence of invariant tori for the three-dimensional flow of (16). By the transformations performed on (3) in Section 2, it is not difficult to find that frequencies of the quasiperiodic solutions of (3) are also ω∗,1 for ε or a sufficiently small. Then, we finish the proof of Theorem 1.


## 4. Conclusion

In this paper, we prove that, fora>0 small, there exist infinitely many invariant tori. It eventually turns out that the solutions starting at t=0 on the invariant tori are quasiperiodic (see Figure 2 with different a).Figure 2
Quasiperiodic orbits of system (3) near the origin and their xyz-coordinates in function of the time t for a=10−2, a=10−3, and a=10−4, respectively.The results obtained here confirm what is stated in [18]: around a linearly stable periodic orbit, there are tori on which the orbits move quasiperiodically. The periodic solution of (3) in Figure 3 is verified in [10] in the form of(45)xt=2acost+Oa,yt=−2asint+Oa,zt=Oa.Figure 3
A periodic orbit of system (3) near the origin for a=10−4 and its xyz-coordinates in function of time t.Together with the invariant tori found by the twist theorem, we numerically obtain the periodic orbit and the nested invariant tori around it in Figure1. The orbit is dense and moves quasiperiodically on the invariant tori.By the twist Theorem2, there exist invariant curves in the neighbourhood of 0,0. On the one hand, for each I0∈0,ε, i.e.,(46)x02+y02ε−22+z0ε21/2∈0,ε,where ε=a, we obtain an invariant torus. On the other hand, (46) implies that, for sufficiently small a, once there is a initial point x0,y0,z0 satisfying (46), it corresponds to an invariant torus. Furthermore, other initial points closer to the origin than x0,y0,z0 correspond to invariant tori either. From this point, as ε⟶0, such invariant tori are infinite (see Figure 4 with different initial values for a=10−4).Figure 4
Two invariant tori with different initial values fora=10−4.



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