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Is the Hamiltonian geometrical criterion for chaos always reliable?
Journal of Geometry and Physics 59 (2009) 1357–1362
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Review
Is the Hamiltonian geometrical criterion for chaos always reliable? Xin Wu Department of Physics, Nanchang University, Nanchang 330031, China
article
info
Article history: Received 19 January 2009 Received in revised form 22 June 2009 Accepted 1 July 2009 Available online 9 July 2009 MSC: 37D45 53Z05 70H14 70K55
abstract It is found that the application of a newly developed geometrical criterion, in which negative eigenvalues of the associated matrix determined by the dynamical curvature of a conformal metric for a Hamiltonian system are used to identify the onset of local instability or chaos, is somewhat problematic in some circumstances. In fact, this criterion is neither necessary nor sufficient for the prediction of instability of orbits on a same energy hypersurface because it is not in good agreement with information on unstable or chaotic behavior given by the maximal Lyapunov exponent in general. © 2009 Elsevier B.V. All rights reserved.
Keywords: Hamiltonian system Manifold Fluctuations of curvature Parametric instability Chaos
Contents 1. 2. 3.
Introduction............................................................................................................................................................................................. 1357 Imperfection of the criterion C3............................................................................................................................................................. 1359 Conclusions.............................................................................................................................................................................................. 1361 Acknowledgments .................................................................................................................................................................................. 1361 References................................................................................................................................................................................................ 1361
1. Introduction In the last decade or so, much effort has been devoted to formulate invariant chaos indicators in general relativity (for reviews and references see [1–6]). For example, Sota et al. gave a geometrical criterion for chaos based on the eigenvalues associated with the Weyl curvature tensor [5]. The geometrical criteria like this are also applied to Hamiltonian systems in classical mechanics [7,8]. For isotropic manifolds given by a Hamiltonian system 1
p2 + V (x) 2m with V as a potential function of space variables, the geodesic deviation equation takes a simple form H =
D2 J ds2
+ K J = 0,
E-mail address: [email protected]. 0393-0440/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2009.07.001
(1)
(2)
1358
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
where D denotes the covariant derivative, K is the constant sectional curvature of the manifold, and s is considered as a measure of time. When the configuration space is two-dimensional, the scalar curvature reads as K = 12 R with
(∇ V )2 ∆V + , (E − V )3 (E − V )2
R=
(3)
where ∇ , ∆ and E stand, respectively, for the Euclidean gradient, Laplacian operators and energy. By a projection along the direction normal to the geodesic, covariant derivatives become ordinary derivatives, i.e., D/ds ≡ d/ds. The geodesic flow is unstable only if K < 0. This means the onset of chaos in the case of compact manifolds. On the other hand, the geodesic flow is stable if K > 0. This instability criterion by negative curvatures is constructed on the Jacobi metric. Hereafter the criterion is labeled as C1. It is worth stressing that this criterion C1 should be carefully used when the curvature K is no longer a constant. It is still suitable for the case that the nonconstant curvature K is everywhere negative. In fact, the case corresponds to that of hyperbolicity, viewed as an important mechanism leading to the origin of the instability of the geodesics. However, C1 is somewhat questionable to treat the manifolds whose curvature is neither constant nor everywhere negative. In practice, it was found that chaos can be caused not only by negative curvatures but also by positive nonconstant curvatures [6,9–11]. In other words, a negative curvature is not necessary at all for the presence of chaos in a geodesic flow. These facts show sufficiently that besides the mechanism of the hyperbolicity another mechanism inducing chaos in geodesic flows of physical relevance, namely, parametric instability1 due to the variability of curvature along the geodesics, should be present [10–13]. This mechanism is obviously active also when the mechanical manifold is mainly positively curved. Besides the reason why a fluctuating positive nonconstant curvature along the geodesic can produce instability, these articles gave an analytic formula for the largest Lyapunov exponent depending on the evolution of the averages and fluctuations of the curvature of the configuration space with varying energy. Here are some details. In order to cope with the limitation of C1, Refs. [10,11] replaced the Jacobi equation (2) with the following expression d2 ψ
+ hkR iµ ψ + σΩ η(t )ψ = 0,
dt 2
(4)
where ψ denotes any of the components about the Jacobi field J with N dimensions, and η is a Gaussian function with zero mean and unit variance. In addition, the average Ricci curvature and its fluctuation are respectively written as
Ω0 = hkR iµ =
σΩ2 =
1 N −1
1 N −1
h∆V iµ ,
hδ 2 KR iµ =
1 N −1
(5)
[h(∆V )2 iµ − h∆V i2µ ],
(6)
where hiµ stands for static averages computed with the microcanonical measure µ on the constant energy surface of phase space. In a word, both of them are functions of the energy E. They also determine the largest Lyapunov exponent
4Ω0 λ(Ω0 , σΩ , τ ) = Λ− 2 3Λ 1
(7)
with
Λ=
2τ σΩ2 +
"
4Ω0 3
3
+ (2τ σΩ2 )2
#1/2 1/3
,
(8)
1/2
2τ =
π Ω0 . 2[Ω0 (Ω0 + σΩ )]1/2 + π σΩ
(9)
Of course, λ is still a function of the energy E. Hence the criterion marked as C2, by applying curvature fluctuations of the manifolds to find the energy dependence of the geometric instability exponent, is constructed. Some examples have displayed that it is more sensitive to detect instability or chaos than the method C1. Readers are also recommended to see a thorough discussion about similar topics which is given in the recently published book entitled ‘‘Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics’’ by Pettini [14]. Additionally, it is worth mentioning that although several examples in [10,11] described that analytic results of Eq. (7) for the largest Lyapunov exponent vs the energy density coincide basically with numeric results, the Lyapunov exponent by Eq. (7) is quite different from the usual sense of the Lyapunov exponent in the known literature (e.g. see [15]). The Lyapunov exponent from Eq. (7), as a function of the energy, is of help for telling one which energy is possible or impossible to bring chaos, but it is very difficult to provide any details about the
1 It means that parameters vary periodically or quasiperiodically in time.
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
1359
dynamics of orbits on the same energy hypersurface. On the contrary, the usual Lyapunov exponent computed by means of the standard algorithm [15], as the average over the whole orbit, depends on the orbit of phase space as well as the energy. The good agreement between the former and the latter lies mainly in that from the global point of view they give almost the same classification of two energy domains,2 but it is not generally based on the classification of orbits from the local point of view. On the other hand, Horwitz et al. [16] had recently developed a new geometrical criterion for local instability, according to the sign of eigenvalues of the matrix determined by the dynamical curvature of a conformal metric for the Hamiltonian (1). The matrix V
Vij =
3 m2 v 2
∂V ∂V 1 ∂ 2V + i j ∂x ∂x m ∂ xi ∂ xj
(10)
gives the geodesic deviation equation D2 ξ Dt 2
= −V Pξ,
(11)
where P ij = δ ij −
vi vj
(12)
v2
with v i = x˙ i , is a projection into a direction orthogonal to the velocity v i . The superscripts i = 1, 2, . . ., and j = 1, 2, . . .. For the case of a two-dimensional potential with mass m = 1, the matrix V in the physical region E − V ≥ 0 corresponds to its eigen-equation
λ2 − bλ + c = 0,
(13)
where b and c are given by b = V11 + V22
=
"
3 2(E − V )
∂V ∂x
2
∂V ∂x
2
+
∂V ∂y
2 # +
∂ 2V ∂ 2V + , ∂ x2 ∂ y2
(14)
2 c = V11 V22 − V12
=
"
3 2(E − V )
∂ 2V + ∂ y2
∂V ∂y
2
# 2 2 ∂ 2V ∂ V ∂ V ∂ 2V ∂ 2V ∂ 2V ∂ V − 2 + − . 2 2 2 ∂x ∂ x ∂ y ∂ x∂ y ∂x ∂y ∂ x∂ y
(15)
Set λ1 and λ2 to be two solutions of Eq. (13), that is, eigenvalues of the matrix V . Then the geodesic deviation equation for the component orthogonal to the motion becomes d2 (v⊥ · ξ) dt 2
= −[λ1 cos2 φ + λ2 sin2 φ](v⊥ · ξ).
(16)
Here φ represents the angle between v⊥ and the eigenvector for λ1 . In terms of Eq. (16), the authors of Ref. [16] proposed a new criterion (C3) that ‘‘instability should occur if at least one of eigenvalues of V is negative’’. For a further investigation to the new criterion, the authors of Ref. [17] thought that the criterion C3 is somewhat different from the criterion C1 in the choice of metrics, so C3 can provide a clearer signal of both local and global instability. Additionally, they claimed that ‘‘the method (C3) contains information on chaotic behavior of the same nature as the maximal Lyapunov exponent’’. Although C3 is indeed superior to C1 in most cases, it is still questionable from the theoretical point of view. There is a discussion with respect to this problem in the following. 2. Imperfection of the criterion C3 Intuitively, C3 should have its limitations like C1 because they originate from a similar mechanism, i.e., curvature tensors in the geodesic deviation equations. That is to say, Eq. (16) as well as Eq. (2) cannot be regarded as a universal means to estimate the dynamical instability of various manifolds. The application of Eq. (16) is conditional. I believe that method C3 should be valid if both λ1 and λ2 of Eq. (13) are constant or everywhere negative eigenvalues. In the case of the requirement of compactness, the method C3 seems to tell one that the orbit should be stable if both λ1 and λ2 are positive. On the other hand, local instability occurs when both λ1 and λ2 are negative. However, the case becomes complicated if the matrix of C3 has both a positive and negative eigenvalue on the configuration space (suppose that λ1 < 0 and λ2 > 0 without loss of
2 One refers to that chaos can occur, and the other means that chaos cannot occur at all.
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X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
generality). If λ1 cos2 φ + λ2 sin2 φ < 0 in Eq. (16), the local instability becomes rather explicit. But the system is still stable if λ1 cos2 φ + λ2 sin2 φ > 0. Thus, even if Eq. (16) can be taken as a diagnostic reference, I do not think that one negative eigenvalue of the matrix is sufficient for the prediction of instability of orbits on the configuration space. However, Eq. (16) is not always suitable for the manifolds whose eigenvalue (curvature) is neither constant nor everywhere negative. An orbit with a negative eigenvalue is either unstable or stable. Perhaps an orbit with positive eigenvalues is still unstable. This point is completely different from that of Ref. [16]. Next, I list some examples as a detailed illustration. One example I consider is the Yang–Mills potential V = 12 x2 y2 . If x2 + y2 6= 0, I can easily obtain ∆V = x2 + y2 > 0, i.e., R > 0. At once, it is shown with C1 that the system is always stable for any positive energies and orbits on the configuration space. The result is completely wrong. Now, let me use C3 to evaluate the dynamics of the system. It is easy to get
b = (x + y ) 1 + 2
2
c = −3x2 y2
1+
3x2 y2
2(E − V )
x2 y 2 E−V
≤ 0.
> 0, (17)
Thus, there are a negative eigenvalue λ1 and a positive eigenvalue λ2 in Eq. (13). Then, it can be inferred from the method C3 that the Yang–Mills system is everywhere unstable, even chaotic except at the origin. The results are almost the same as earlier claims that the system is fully ergodic [18]. This seems to display that C3 is very successful to explore the energy dependence of instability. In other words, C3 describes that the Yang–Mills system is always chaotic for any positive energy. Certainly this fact should also be supported by the method C2.3 However, C3 is not completely true to achieve the dynamical instability of orbits on the x–y plane. In fact, Dahlqvist and Russberg refuted the earlier claims by finding the presence of one family of stable periodic orbits in the system [20]. Pollner et al. [21] proved further that the regular islands cover less than 0.005% of the phase space. All these facts illustrate that negative eigenvalues of the matrix support both the existence of large chaotic regions and the existence of small regular islands in the Yang–Mills system. In other words, C3 gives the regular islands wrong information of instability or chaos. As stated above, the reason is that local instability is determined by not only the negative eigenvalue λ1 , but the combination of both λ1 and λ2 , λ1 cos2 φ + λ2 sin2 φ , when Eq. (16) becomes valid. In this sense, no single negative eigenvalue for an orbit can be described as being responsible for causing instability. In fact, the examples studied in Ref. [16] imply a similar case, too. Note that the dark area of negative eigenvalues for the matrix V along the straight line x = 0 of Fig. 1(b) in Ref. [16] is the region of ymin ≤ y < −0.5 (ymin being the minimum value of y in the case of physically allowable motion for E = 3). As is expect, the Poincaré plot in Fig. 2(b) of Ref. [16] does display that the dark area corresponds to the onset of both chaotic and regular orbits. On the other hand, orbits on compact manifolds with positive eigenvalues may be chaotic. For example, there are chaotic regions, which are distributed in the region of y > −0.5 on the Poincaré plot given by Fig. 2(b) of Ref. [16]. It should be worth emphasizing that these regions just have positive eigenvalues, as shown in Fig. 1(b) of Ref. [16]. In this case, Eq. (16) cannot be taken as a diagnostic reference of instability or chaos of orbits at all. As far as the relation between the criterion C3 and the maximal Lyapunov exponent is concerned, the above statement (‘‘the method (C3) contains information · · ·’’) itself is slightly ambiguous. As claimed in the Introduction, the maximal Lyapunov exponent with the standard algorithm [15], is computed through the average over the whole orbit. On the other hand, by looking at Eq. (16) one can find that this exponent does depend on not only −λ1 , but the expressional form of −(λ1 cos2 φ + λ2 sin2 φ). Some facts from the Yang–Mills system support this clearly. In my opinion, fluctuations of the Ricci curvature give rise to parametric instability so that the method C3 is not always reliable to distinguish the classification of orbits on a same energy bounded hypersurface. Here these words ‘‘not always reliable’’ I use mean that some results are correct. This seems to ensure that C3 works well in distinguishing the two energy domains defined by Footnote 2 in the general cases. Note that, the mechanism of C3 distinguishing the two energy domains is typically different from that of C2. The method C3 distinguishes the two energy domains by finding negative eigenvalues through scanning all the orbits on the same energy hypersurface, while C2 does by calculating Lyapunov exponents through static averages to the constant energy surface. The difference leads to their distinct performance in the study of instability. On the other hand, even if the curvatures are not fluctuating along the geodesics, C3 may have bad performance. This can be seen clearly from the following two simple unbounded systems, where C3 becomes completely useless in distinguishing both the two energy domains and the dynamical details of orbits. For a rather simple potential V = x + y, C1 cannot still get the true dynamical information on the system because ∆V = 0, i.e., R > 0. Let me recall the application of C3. There are two identical positive eigenvalues for Eq. (13) in the physical region. In the light of the criterion C3, one can get a wrong result that the system is stable. If the criterion C2 is adopted, I have Ω0 = σΩ ≡ 0. In this case, I have no way to measure the Largest Lyapunov exponent with Eq. (7). But I can use Eq. (4) to obtain the evolution of the average geodesic deviation according to ψ = At + B , where A and B are constant. Only if A 6= 0, is the system always unstable. This shows that Eq. (16) with Eq. (2) becomes useless, but Eq. (4) does not. 3 It is rather cumbersome to estimate the dependence of the maximal Lyapunov exponent on the varying energy if Eq. (7) is adopted. In this sense, it is convenient to use numeric results instead of analytic results because the agreement between them is generally good, as stated in the Introduction. Numerical computations of Lyapunov exponents in Ref. [19] look to show this clearly.
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
1361
As another example, I take the potential V = − 12 (x + y)2 . First, I consider the use of C1. Noting ∆V = −2, I have 1
R=
(E − V )3
[(x + y)2 − 2E ]. √
(18)
√
Obviously, R > 0 when |x + y| > 2E with E > 0, but R < 0 if |x + y| < 2E. It is shown with C1 that some regions in the system are stable, while others are unstable. For E < 0, R is everywhere positive in the physically accessible region, that is to say, the system is always stable. Second, I use C3 to study this system. It is easy to get c = 0 and b=
2 E−V
[(x + y)2 − E ].
(19)
√
Then I have two √ eigenvalues of Eq. (13), λ1 = 0 and λ2 = b. There is a demonstration like the above if I replace R and 2E with λ2 and E, respectively. The results obtained from C1 and C3 are wrong. In fact, the system must be unbounded and unstable for any energies or orbits. This can be seen clearly from the analytical solution x=
A2 √2t A1 −√2t e e − + B1 t + B2 , 4 4A2
y=
A2 √2t A1 −√2t e − e − B1 t − B2 , 4 4A2
(20)
√
where A1 , A2 , B1 and B2 are constants. Obviously, the maximal Lyapunov exponent is 2.4 It can be concluded that the eigenvalues given by Eq. (13) do not depend on the largest Lyapunov exponents again, and negative eigenvalues in the method C3 are not a necessary condition for local instability. Finally, see the validity of C2. I can easily attain Ω0 = −2 and σ √Ω ≡ 0. Although I fail to compute the largest Lyapunov exponent with Eq. (7), for any energies I can work out its value, 2, in the light of Eq. (4). This value is just consistent with the analytical result. This shows again that Eq. (16) with Eq. (2) becomes useless, but Eq. (4) is valid. 3. Conclusions In brief, parametric instability, as the source of chaos on manifolds of positive nonconstant curvature, makes the instability criterion C3 have more limited applications. In some circumstances, only one negative eigenvalue of the associated matrix is neither necessary nor sufficient criteria for instability of orbits on the configuration space. The refined geometrical criteria C3 by the conformal metric are not always in good agreement with the maximal Lyapunov exponents. Seen from the theoretical point of view, the negative eigenvalues seem to bear no relation to the maximal Lyapunov exponents. Because both the refined criteria C3 and the original geometrical criteria C1 by the Jacobi metric basically originate from a similar mechanism, the limitations of the refined criteria similar to those of the original criteria cannot be avoided to some extent. On the other hand, the best performance of the method C2 in measuring instability further illustrates that there are more problems in the application of the method C3. Therefore, the new criterion C3 can be used merely as a preliminary indicator of instability or chaos in dynamical systems. To get more rigorous and perfect indicators, one should develop Hamiltonian geometrical criteria for instability in other ways. Acknowledgments I would like to thank the referee for honest comments and significant suggestions. This research is supported by the Natural Science Foundation of China under Contract No. 10873007. It is also supported by the Science Foundation of Jiangxi Education Bureau (GJJ09072), and the Program for Innovative Research Team of Nanchang University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
X. Wu, T.Y. Huang, Phys. Lett. A 313 (2003) 77. X. Wu, T.Y. Huang, H. Zhang, Phys. Rev. D 74 (2006) 083001. X. Wu, Y. Xie, Phys. Rev. D 76 (2007) 124004. X. Wu, Y. Xie, Phys. Rev. D 77 (2008) 103012. Y. Sota, S. Zuzuki, K. Maeda, Classical Quantum Gravity 13 (1996) 1241. W.M. Vieira, P.S. Letelier, Classical Quantum Gravity 13 (1996) 3115. V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1989. J. Zsczesny, ¸ T. Dobrowolski, Ann. Phys. 277 (1999) 161. A. Saa, Ann. Phys. 314 (2004) 508. L. Casetti, C. Clementi, M. Pettini, Phys. Rev. E 54 (1996) 5969. L. Casetti, M. Pettini, E.G.D. Cohen, Phys. Rep. 337 (2000) 237.
4 The dynamics is not chaotic when an integrable system has a positive Lyapunov exponent on the non-compact admissible region.
1362 [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362 M. Cerruti-Sola, M. Pettini, Phys. Rev. E 53 (1996) 179. M. Pettini, R. Valdettaro, Chaos 5 (1995) 646. M. Pettini, Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics, in: IAM Series, vol. 33, Springer, New York, 2007. G. Tancredi, A. Sánchez, F. Roig, Astron. J. 121 (2001) 1171. L. Horwitz, Y.B. Zion, M. Lewkowicz, M. Schiffer, J. Levitan, Phys. Rev. Lett. 98 (2007) 234301. Y.B. Zion, L. Horwitz, Phys. Rev. E 76 (2007) 046220. O.K. Saviddy, Phys. Lett. B 130 (1983) 303. I.Y. Aref’eva, P.B. Medvedev, O.A. Rytchkov, I.V. Volovich, Chaos, Solitons Fractals 10 (1999) 213. P. Dahlqvist, G. Russberg, Phys. Rev. Lett. 65 (1990) 2837. P. Pollner, G. Vattay, Phys. Rev. Lett. 76 (1996) 4155.
Contents lists available at ScienceDirect
Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Review
Is the Hamiltonian geometrical criterion for chaos always reliable? Xin Wu Department of Physics, Nanchang University, Nanchang 330031, China
article
info
Article history: Received 19 January 2009 Received in revised form 22 June 2009 Accepted 1 July 2009 Available online 9 July 2009 MSC: 37D45 53Z05 70H14 70K55
abstract It is found that the application of a newly developed geometrical criterion, in which negative eigenvalues of the associated matrix determined by the dynamical curvature of a conformal metric for a Hamiltonian system are used to identify the onset of local instability or chaos, is somewhat problematic in some circumstances. In fact, this criterion is neither necessary nor sufficient for the prediction of instability of orbits on a same energy hypersurface because it is not in good agreement with information on unstable or chaotic behavior given by the maximal Lyapunov exponent in general. © 2009 Elsevier B.V. All rights reserved.
Keywords: Hamiltonian system Manifold Fluctuations of curvature Parametric instability Chaos
Contents 1. 2. 3.
Introduction............................................................................................................................................................................................. 1357 Imperfection of the criterion C3............................................................................................................................................................. 1359 Conclusions.............................................................................................................................................................................................. 1361 Acknowledgments .................................................................................................................................................................................. 1361 References................................................................................................................................................................................................ 1361
1. Introduction In the last decade or so, much effort has been devoted to formulate invariant chaos indicators in general relativity (for reviews and references see [1–6]). For example, Sota et al. gave a geometrical criterion for chaos based on the eigenvalues associated with the Weyl curvature tensor [5]. The geometrical criteria like this are also applied to Hamiltonian systems in classical mechanics [7,8]. For isotropic manifolds given by a Hamiltonian system 1
p2 + V (x) 2m with V as a potential function of space variables, the geodesic deviation equation takes a simple form H =
D2 J ds2
+ K J = 0,
E-mail address: [email protected]. 0393-0440/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2009.07.001
(1)
(2)
1358
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
where D denotes the covariant derivative, K is the constant sectional curvature of the manifold, and s is considered as a measure of time. When the configuration space is two-dimensional, the scalar curvature reads as K = 12 R with
(∇ V )2 ∆V + , (E − V )3 (E − V )2
R=
(3)
where ∇ , ∆ and E stand, respectively, for the Euclidean gradient, Laplacian operators and energy. By a projection along the direction normal to the geodesic, covariant derivatives become ordinary derivatives, i.e., D/ds ≡ d/ds. The geodesic flow is unstable only if K < 0. This means the onset of chaos in the case of compact manifolds. On the other hand, the geodesic flow is stable if K > 0. This instability criterion by negative curvatures is constructed on the Jacobi metric. Hereafter the criterion is labeled as C1. It is worth stressing that this criterion C1 should be carefully used when the curvature K is no longer a constant. It is still suitable for the case that the nonconstant curvature K is everywhere negative. In fact, the case corresponds to that of hyperbolicity, viewed as an important mechanism leading to the origin of the instability of the geodesics. However, C1 is somewhat questionable to treat the manifolds whose curvature is neither constant nor everywhere negative. In practice, it was found that chaos can be caused not only by negative curvatures but also by positive nonconstant curvatures [6,9–11]. In other words, a negative curvature is not necessary at all for the presence of chaos in a geodesic flow. These facts show sufficiently that besides the mechanism of the hyperbolicity another mechanism inducing chaos in geodesic flows of physical relevance, namely, parametric instability1 due to the variability of curvature along the geodesics, should be present [10–13]. This mechanism is obviously active also when the mechanical manifold is mainly positively curved. Besides the reason why a fluctuating positive nonconstant curvature along the geodesic can produce instability, these articles gave an analytic formula for the largest Lyapunov exponent depending on the evolution of the averages and fluctuations of the curvature of the configuration space with varying energy. Here are some details. In order to cope with the limitation of C1, Refs. [10,11] replaced the Jacobi equation (2) with the following expression d2 ψ
+ hkR iµ ψ + σΩ η(t )ψ = 0,
dt 2
(4)
where ψ denotes any of the components about the Jacobi field J with N dimensions, and η is a Gaussian function with zero mean and unit variance. In addition, the average Ricci curvature and its fluctuation are respectively written as
Ω0 = hkR iµ =
σΩ2 =
1 N −1
1 N −1
h∆V iµ ,
hδ 2 KR iµ =
1 N −1
(5)
[h(∆V )2 iµ − h∆V i2µ ],
(6)
where hiµ stands for static averages computed with the microcanonical measure µ on the constant energy surface of phase space. In a word, both of them are functions of the energy E. They also determine the largest Lyapunov exponent
4Ω0 λ(Ω0 , σΩ , τ ) = Λ− 2 3Λ 1
(7)
with
Λ=
2τ σΩ2 +
"
4Ω0 3
3
+ (2τ σΩ2 )2
#1/2 1/3
,
(8)
1/2
2τ =
π Ω0 . 2[Ω0 (Ω0 + σΩ )]1/2 + π σΩ
(9)
Of course, λ is still a function of the energy E. Hence the criterion marked as C2, by applying curvature fluctuations of the manifolds to find the energy dependence of the geometric instability exponent, is constructed. Some examples have displayed that it is more sensitive to detect instability or chaos than the method C1. Readers are also recommended to see a thorough discussion about similar topics which is given in the recently published book entitled ‘‘Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics’’ by Pettini [14]. Additionally, it is worth mentioning that although several examples in [10,11] described that analytic results of Eq. (7) for the largest Lyapunov exponent vs the energy density coincide basically with numeric results, the Lyapunov exponent by Eq. (7) is quite different from the usual sense of the Lyapunov exponent in the known literature (e.g. see [15]). The Lyapunov exponent from Eq. (7), as a function of the energy, is of help for telling one which energy is possible or impossible to bring chaos, but it is very difficult to provide any details about the
1 It means that parameters vary periodically or quasiperiodically in time.
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
1359
dynamics of orbits on the same energy hypersurface. On the contrary, the usual Lyapunov exponent computed by means of the standard algorithm [15], as the average over the whole orbit, depends on the orbit of phase space as well as the energy. The good agreement between the former and the latter lies mainly in that from the global point of view they give almost the same classification of two energy domains,2 but it is not generally based on the classification of orbits from the local point of view. On the other hand, Horwitz et al. [16] had recently developed a new geometrical criterion for local instability, according to the sign of eigenvalues of the matrix determined by the dynamical curvature of a conformal metric for the Hamiltonian (1). The matrix V
Vij =
3 m2 v 2
∂V ∂V 1 ∂ 2V + i j ∂x ∂x m ∂ xi ∂ xj
(10)
gives the geodesic deviation equation D2 ξ Dt 2
= −V Pξ,
(11)
where P ij = δ ij −
vi vj
(12)
v2
with v i = x˙ i , is a projection into a direction orthogonal to the velocity v i . The superscripts i = 1, 2, . . ., and j = 1, 2, . . .. For the case of a two-dimensional potential with mass m = 1, the matrix V in the physical region E − V ≥ 0 corresponds to its eigen-equation
λ2 − bλ + c = 0,
(13)
where b and c are given by b = V11 + V22
=
"
3 2(E − V )
∂V ∂x
2
∂V ∂x
2
+
∂V ∂y
2 # +
∂ 2V ∂ 2V + , ∂ x2 ∂ y2
(14)
2 c = V11 V22 − V12
=
"
3 2(E − V )
∂ 2V + ∂ y2
∂V ∂y
2
# 2 2 ∂ 2V ∂ V ∂ V ∂ 2V ∂ 2V ∂ 2V ∂ V − 2 + − . 2 2 2 ∂x ∂ x ∂ y ∂ x∂ y ∂x ∂y ∂ x∂ y
(15)
Set λ1 and λ2 to be two solutions of Eq. (13), that is, eigenvalues of the matrix V . Then the geodesic deviation equation for the component orthogonal to the motion becomes d2 (v⊥ · ξ) dt 2
= −[λ1 cos2 φ + λ2 sin2 φ](v⊥ · ξ).
(16)
Here φ represents the angle between v⊥ and the eigenvector for λ1 . In terms of Eq. (16), the authors of Ref. [16] proposed a new criterion (C3) that ‘‘instability should occur if at least one of eigenvalues of V is negative’’. For a further investigation to the new criterion, the authors of Ref. [17] thought that the criterion C3 is somewhat different from the criterion C1 in the choice of metrics, so C3 can provide a clearer signal of both local and global instability. Additionally, they claimed that ‘‘the method (C3) contains information on chaotic behavior of the same nature as the maximal Lyapunov exponent’’. Although C3 is indeed superior to C1 in most cases, it is still questionable from the theoretical point of view. There is a discussion with respect to this problem in the following. 2. Imperfection of the criterion C3 Intuitively, C3 should have its limitations like C1 because they originate from a similar mechanism, i.e., curvature tensors in the geodesic deviation equations. That is to say, Eq. (16) as well as Eq. (2) cannot be regarded as a universal means to estimate the dynamical instability of various manifolds. The application of Eq. (16) is conditional. I believe that method C3 should be valid if both λ1 and λ2 of Eq. (13) are constant or everywhere negative eigenvalues. In the case of the requirement of compactness, the method C3 seems to tell one that the orbit should be stable if both λ1 and λ2 are positive. On the other hand, local instability occurs when both λ1 and λ2 are negative. However, the case becomes complicated if the matrix of C3 has both a positive and negative eigenvalue on the configuration space (suppose that λ1 < 0 and λ2 > 0 without loss of
2 One refers to that chaos can occur, and the other means that chaos cannot occur at all.
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generality). If λ1 cos2 φ + λ2 sin2 φ < 0 in Eq. (16), the local instability becomes rather explicit. But the system is still stable if λ1 cos2 φ + λ2 sin2 φ > 0. Thus, even if Eq. (16) can be taken as a diagnostic reference, I do not think that one negative eigenvalue of the matrix is sufficient for the prediction of instability of orbits on the configuration space. However, Eq. (16) is not always suitable for the manifolds whose eigenvalue (curvature) is neither constant nor everywhere negative. An orbit with a negative eigenvalue is either unstable or stable. Perhaps an orbit with positive eigenvalues is still unstable. This point is completely different from that of Ref. [16]. Next, I list some examples as a detailed illustration. One example I consider is the Yang–Mills potential V = 12 x2 y2 . If x2 + y2 6= 0, I can easily obtain ∆V = x2 + y2 > 0, i.e., R > 0. At once, it is shown with C1 that the system is always stable for any positive energies and orbits on the configuration space. The result is completely wrong. Now, let me use C3 to evaluate the dynamics of the system. It is easy to get
b = (x + y ) 1 + 2
2
c = −3x2 y2
1+
3x2 y2
2(E − V )
x2 y 2 E−V
≤ 0.
> 0, (17)
Thus, there are a negative eigenvalue λ1 and a positive eigenvalue λ2 in Eq. (13). Then, it can be inferred from the method C3 that the Yang–Mills system is everywhere unstable, even chaotic except at the origin. The results are almost the same as earlier claims that the system is fully ergodic [18]. This seems to display that C3 is very successful to explore the energy dependence of instability. In other words, C3 describes that the Yang–Mills system is always chaotic for any positive energy. Certainly this fact should also be supported by the method C2.3 However, C3 is not completely true to achieve the dynamical instability of orbits on the x–y plane. In fact, Dahlqvist and Russberg refuted the earlier claims by finding the presence of one family of stable periodic orbits in the system [20]. Pollner et al. [21] proved further that the regular islands cover less than 0.005% of the phase space. All these facts illustrate that negative eigenvalues of the matrix support both the existence of large chaotic regions and the existence of small regular islands in the Yang–Mills system. In other words, C3 gives the regular islands wrong information of instability or chaos. As stated above, the reason is that local instability is determined by not only the negative eigenvalue λ1 , but the combination of both λ1 and λ2 , λ1 cos2 φ + λ2 sin2 φ , when Eq. (16) becomes valid. In this sense, no single negative eigenvalue for an orbit can be described as being responsible for causing instability. In fact, the examples studied in Ref. [16] imply a similar case, too. Note that the dark area of negative eigenvalues for the matrix V along the straight line x = 0 of Fig. 1(b) in Ref. [16] is the region of ymin ≤ y < −0.5 (ymin being the minimum value of y in the case of physically allowable motion for E = 3). As is expect, the Poincaré plot in Fig. 2(b) of Ref. [16] does display that the dark area corresponds to the onset of both chaotic and regular orbits. On the other hand, orbits on compact manifolds with positive eigenvalues may be chaotic. For example, there are chaotic regions, which are distributed in the region of y > −0.5 on the Poincaré plot given by Fig. 2(b) of Ref. [16]. It should be worth emphasizing that these regions just have positive eigenvalues, as shown in Fig. 1(b) of Ref. [16]. In this case, Eq. (16) cannot be taken as a diagnostic reference of instability or chaos of orbits at all. As far as the relation between the criterion C3 and the maximal Lyapunov exponent is concerned, the above statement (‘‘the method (C3) contains information · · ·’’) itself is slightly ambiguous. As claimed in the Introduction, the maximal Lyapunov exponent with the standard algorithm [15], is computed through the average over the whole orbit. On the other hand, by looking at Eq. (16) one can find that this exponent does depend on not only −λ1 , but the expressional form of −(λ1 cos2 φ + λ2 sin2 φ). Some facts from the Yang–Mills system support this clearly. In my opinion, fluctuations of the Ricci curvature give rise to parametric instability so that the method C3 is not always reliable to distinguish the classification of orbits on a same energy bounded hypersurface. Here these words ‘‘not always reliable’’ I use mean that some results are correct. This seems to ensure that C3 works well in distinguishing the two energy domains defined by Footnote 2 in the general cases. Note that, the mechanism of C3 distinguishing the two energy domains is typically different from that of C2. The method C3 distinguishes the two energy domains by finding negative eigenvalues through scanning all the orbits on the same energy hypersurface, while C2 does by calculating Lyapunov exponents through static averages to the constant energy surface. The difference leads to their distinct performance in the study of instability. On the other hand, even if the curvatures are not fluctuating along the geodesics, C3 may have bad performance. This can be seen clearly from the following two simple unbounded systems, where C3 becomes completely useless in distinguishing both the two energy domains and the dynamical details of orbits. For a rather simple potential V = x + y, C1 cannot still get the true dynamical information on the system because ∆V = 0, i.e., R > 0. Let me recall the application of C3. There are two identical positive eigenvalues for Eq. (13) in the physical region. In the light of the criterion C3, one can get a wrong result that the system is stable. If the criterion C2 is adopted, I have Ω0 = σΩ ≡ 0. In this case, I have no way to measure the Largest Lyapunov exponent with Eq. (7). But I can use Eq. (4) to obtain the evolution of the average geodesic deviation according to ψ = At + B , where A and B are constant. Only if A 6= 0, is the system always unstable. This shows that Eq. (16) with Eq. (2) becomes useless, but Eq. (4) does not. 3 It is rather cumbersome to estimate the dependence of the maximal Lyapunov exponent on the varying energy if Eq. (7) is adopted. In this sense, it is convenient to use numeric results instead of analytic results because the agreement between them is generally good, as stated in the Introduction. Numerical computations of Lyapunov exponents in Ref. [19] look to show this clearly.
X. Wu / Journal of Geometry and Physics 59 (2009) 1357–1362
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As another example, I take the potential V = − 12 (x + y)2 . First, I consider the use of C1. Noting ∆V = −2, I have 1
R=
(E − V )3
[(x + y)2 − 2E ]. √
(18)
√
Obviously, R > 0 when |x + y| > 2E with E > 0, but R < 0 if |x + y| < 2E. It is shown with C1 that some regions in the system are stable, while others are unstable. For E < 0, R is everywhere positive in the physically accessible region, that is to say, the system is always stable. Second, I use C3 to study this system. It is easy to get c = 0 and b=
2 E−V
[(x + y)2 − E ].
(19)
√
Then I have two √ eigenvalues of Eq. (13), λ1 = 0 and λ2 = b. There is a demonstration like the above if I replace R and 2E with λ2 and E, respectively. The results obtained from C1 and C3 are wrong. In fact, the system must be unbounded and unstable for any energies or orbits. This can be seen clearly from the analytical solution x=
A2 √2t A1 −√2t e e − + B1 t + B2 , 4 4A2
y=
A2 √2t A1 −√2t e − e − B1 t − B2 , 4 4A2
(20)
√
where A1 , A2 , B1 and B2 are constants. Obviously, the maximal Lyapunov exponent is 2.4 It can be concluded that the eigenvalues given by Eq. (13) do not depend on the largest Lyapunov exponents again, and negative eigenvalues in the method C3 are not a necessary condition for local instability. Finally, see the validity of C2. I can easily attain Ω0 = −2 and σ √Ω ≡ 0. Although I fail to compute the largest Lyapunov exponent with Eq. (7), for any energies I can work out its value, 2, in the light of Eq. (4). This value is just consistent with the analytical result. This shows again that Eq. (16) with Eq. (2) becomes useless, but Eq. (4) is valid. 3. Conclusions In brief, parametric instability, as the source of chaos on manifolds of positive nonconstant curvature, makes the instability criterion C3 have more limited applications. In some circumstances, only one negative eigenvalue of the associated matrix is neither necessary nor sufficient criteria for instability of orbits on the configuration space. The refined geometrical criteria C3 by the conformal metric are not always in good agreement with the maximal Lyapunov exponents. Seen from the theoretical point of view, the negative eigenvalues seem to bear no relation to the maximal Lyapunov exponents. Because both the refined criteria C3 and the original geometrical criteria C1 by the Jacobi metric basically originate from a similar mechanism, the limitations of the refined criteria similar to those of the original criteria cannot be avoided to some extent. On the other hand, the best performance of the method C2 in measuring instability further illustrates that there are more problems in the application of the method C3. Therefore, the new criterion C3 can be used merely as a preliminary indicator of instability or chaos in dynamical systems. To get more rigorous and perfect indicators, one should develop Hamiltonian geometrical criteria for instability in other ways. Acknowledgments I would like to thank the referee for honest comments and significant suggestions. This research is supported by the Natural Science Foundation of China under Contract No. 10873007. It is also supported by the Science Foundation of Jiangxi Education Bureau (GJJ09072), and the Program for Innovative Research Team of Nanchang University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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