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1.
In this paper we discuss a perturbed extension of hyperbolic twist mappings to a 3-dimensional measure-preserving mapping $$\begin{array}{*{20}c} {T:\left\{ {\begin{array}{*{20}c} {x_{n + 1} = s(x_n \cos \varphi _n - y_n \sin \varphi _n ) + A\cos z_n ,} \\ {y_{n + 1} = s^{ - 1} (x_n \sin \varphi _n + y_n \cos \varphi _n ) + B\sin z_n ,} \\ {z_{n + 1} = z_n + C\cos (x_{n + 1} + y_{n + 1} ) + D,(\bmod 2\pi )} \\ \end{array} } \right.} \\ {\varphi _n = (x_n^2 + y_n^2 )^k } \\ \end{array}$$ wheres, k are parameters andA, B, C, D are perturbation parameters. We find that the ordered regions near the fixed point of the hyperbolic twist mapping is destroyed by the perturbed extension more easily than the ones distant from it. The size of the ordered region decreases with increasing perturbation parameters and is insensitive to the parameterD for the same parametersA, B, C.  相似文献   

2.
In this paper, using two methods: LCN'S (Lyapunov characteristic numbers) method and slice cutting method, we study numerically two mappings with odd dimension: $$T_1 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + z_n ,} \\ {y_{n + 1} = y_n + x_{n + 1} , (\bmod 2\pi )} \\ {z_{n + 1} = z_n + A\sin y_{n + 1} ,} \\ \end{array} } \right. T_2 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + y_n + B \sin z_n ,} \\ {y_{n + 1} = y_n + A \sin x_{n + 1} , (\bmod 2\pi ),} \\ {z_{n + 1} = z_n + B \sin y_{n + 1} ,} \\ \end{array} } \right.$$ whereA, B are parameters. For the mappingT 1 the whole region is stochastic; however, we find two-dimensional invariant manifolds for the mappingT 2.  相似文献   

3.
Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).  相似文献   

4.
In this paper, we study the following three-dimensional mappings $$T:\left\{ \begin{gathered} x_{n + 1} = x_n + y_n + B sin z_n , \hfill \\ y_{n + 1} = y_n + A sin x_{n + 1} , \hfill \\ z_{n + 1} = z_n + C sin y_{n + 1} + D, \hfill \\ \end{gathered} \right.\left( {\bmod 2\Pi } \right)$$ where A, B, C, D are parameters. When D>BC and 2π/D is an irrational number, we find numerically-two-dimensional and one-dimensional invariant manifolds, but when DBC and 2π/D is a rational number we find numerically one-dimensional manifolds and the fixed points for some cycles.  相似文献   

5.
If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 $$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$ Herey is the momentum-vectory k withk=1,2?N, x 1 is thecritical argument, andx k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian. The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by 1 $$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$ The solution isglobal in the sense that it is valid for all values of α2 in the range 1 $$0 \leqslant \alpha ^2 \leqslant \infty ,$$ which embrances thelibration and thecirculation regimes of the co-ordinatex 1, associated with α2 < 1 and α2 > 1, respectively. The solution includes asymptotically the limit α2 → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ2, and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε1/2. The singularities that commonly arise within the libration region α2 < 1 and on the separatrix α2 = 1 of the phase-plane have been suppressed by means of aregularizing function 1 $$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$ introduced into the new Hamiltonian. The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α2 < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α2 > 1/μ, where the classical solution is valid. Thedemarcation point 1 $$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$ conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 $$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$ It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).  相似文献   

6.
The Ideal Resonance Problem, defined by the Hamiltonian $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x,\mu \ll 1,$$ has been solved in Garfinkelet al. (1971). As a perturbed simple pendulum, this solution furnishes a convenient and accurate reference orbit for the study of resonance. In order to preserve the penduloid character of the motion, the solution is subject to thenormality condition, which boundsAB" andB' away from zero indeep and inshallow resonance, respectively. For a first-order solution, the paper derives the normality condition in the form $$pi \leqslant max(|\alpha /\alpha _1 |,|\alpha /\alpha _1 |^{2i} ),i = 1,2.$$ Herep i are known functions of the constant ‘mean element’y', α is the resonance parameter defined by $$\alpha \equiv - {\rm B}'/|4AB\prime \prime |^{1/2} \mu ,$$ and $$\alpha _1 \equiv \mu ^{ - 1/2}$$ defines the conventionaldemarcation point separating the deep and the shallow resonance regions. The results are applied to the problem of the critical inclination of a satellite of an oblate planet. There the normality condition takes the form $$\Lambda _1 (\lambda ) \leqslant e \leqslant \Lambda _2 (\lambda )if|i - tan^{ - 1} 2| \leqslant \lambda e/2(1 + e)$$ withΛ 1, andΛ 2 known functions of λ, defined by $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$   相似文献   

7.
From the values of period changes for 6 close binary stars the mass transfer rate was calculated. Comparing these values Mt with the values of shell masses Msh, the expression $$lg \dot M_t = \begin{array}{*{20}c} {4.24} \\ { \pm 24} \\ \end{array} + \begin{array}{*{20}c} {0.63} \\ { \pm 6} \\ \end{array} lg M_{sh} $$ Was derived. The analysis of this expression points out the initial character of the outflow of matter, and one may determine the time interval of the substitution of the shell matter. So one may conclude that for a certain mass transfer rate, a certain amount of matter accumulates in the nearby regions of the system. The study of orbital period changes of close binary stellar systems led to the idea that these secular and irregular changes are due to the mass loss and to the redistribution of masses in a close binary. Secular changes of orbital periods are known for approximately 400 eclipsing binary stars. For many stars, including cataclysmic binaries, irregular period changes are known. Thus, the mass loss and the matter redistribution in close binaries are often observed phenomena.  相似文献   

8.
The development of the post-nova light curve of V1500 Cyg inUBV andHβ, for 15 nights in September and October 1975 are presented. We confirm previous reports that superimposed on the steady decline of the light curve are small amplitude cyclic variations. The times of maxima and minima are determined. These together with other published values yield the following ephemerides from JD 2 442 661 to JD 2 442 674: $$\begin{gathered} {\text{From}} 17 {\text{points:}} {\text{JD}}_{ \odot \min } = 2 442 661.4881 + 0_{^. }^{\text{d}} 140 91{\text{n}} \hfill \\ \pm 0.0027 \pm 0.000 05 \hfill \\ {\text{From}} 15 {\text{points:}} {\text{JD}}_{ \odot \max } = 2 442 661.5480 + 0_{^. }^{\text{d}} 140 89{\text{n}} \hfill \\ \pm 0.0046 \pm 0.0001 \hfill \\ \end{gathered} $$ with standard errors of the fits of ±0 . d 0052 for the minima and ±0 . d 0091 for the maxima. Assuming V1500 Cyg is similar to novae in M31, we foundr=750 pc and a pre-nova absolute photographic magnitude greater than 9.68.  相似文献   

9.
A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$\begin{gathered} R_{ik} = - \mu \varphi _i \varphi _k ,\varphi _i = \frac{{\partial \varphi }}{{\partial x^i }}, \varphi ^i = g^{ik} \varphi _k , \hfill \\ \hfill \\ \end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$\phi _{;i}^i = ( - g)^{ - 1/2} \frac{\partial }{{\partial x^i }}\left[ {( - g)^{1/2} g^{ik} \frac{\partial }{{\partial x^k }}} \right] = - 4\pi ( -g )^{ - 1/2} \rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.  相似文献   

10.
Techniques are developed to facilitate the transformation of a perturbed Keplerian system into Deláunay normal form at first order. The implicit dependence of the Hamiltonian on 1, the mean anomaly, through the explicit variable f, the true anomaly, or E, the eccentric anomaly, is removed through first order for terms of the form:
  相似文献   

11.
The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric functionp F q of one variable, Appell and Kampé de Fériet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.  相似文献   

12.
From new observational material we made a curve of growth analysis of the penumbra of a large, stable sunspot. The analysis was done relative to the undisturbed photosphere and gave the following results (⊙ denotes photosphere, * denotes penumbra): $$\begin{gathered} (\theta ^ * - \theta ^ \odot )_{exe} = 0.051 \pm 0.007 \hfill \\ {{\xi _t ^ * } \mathord{\left/ {\vphantom {{\xi _t ^ * } {\xi _t }}} \right. \kern-\nulldelimiterspace} {\xi _t }}^ \odot = 1.3 \pm 0.1 \hfill \\ {{P_e ^ * } \mathord{\left/ {\vphantom {{P_e ^ * } {P_e ^ \odot = 0.6 \pm 0.1}}} \right. \kern-\nulldelimiterspace} {P_e ^ \odot = 0.6 \pm 0.1}} \hfill \\ {{P_g ^ * } \mathord{\left/ {\vphantom {{P_g ^ * } {P_g }}} \right. \kern-\nulldelimiterspace} {P_g }}^ \odot = 1.0 \pm 0.2 \hfill \\ \end{gathered} $$ The results of the analysis are in satisfactory agreement with the penumbral model as published by Kjeldseth Moe and Maltby (1969). Additionally we tested this model by computing the equivalent widths of 28 well selected lines and comparing them with our observations.  相似文献   

13.
Non-linear stability of the libration point L 4 of the restricted three-body problem is studied when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion, Moser's conditions are utilised in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff's normal form with the help of double D'Alembert's series. It is found that L 4 is stable for all mass ratios in the range of linear stability except for the three mass ratios: $$\begin{gathered} \mu _{c1} = 0.0242{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.1790{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c2} = 0.0135{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0993{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c3} = 0.0109{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0294{\text{ }}...{\text{ }}A_1 . \hfill \\ \end{gathered} $$   相似文献   

14.
Until very recently, there has been no identification of the significant gravitational constraints on the many common artificial earth satellite orbits in shallow resonance. Without them it is difficult to compare results derived for different sets of harmonics from different orbits. With them it is possible to extend these results to any degree without reintegration of the orbits. All such constraints are shown to be harmonic in the argument of perigee with constants determinable from tracking data: $$(C*,S*) = (C_0 ,S_0 ) + \sum\limits_{i = 1}^\infty {(C_{Ci} ,S_{Ci} )\cos i\omega + (C_{Si} ,S_{Si} )\sin i\omega .} $$ The constants are simple linear combinations of geopotential harmonics. Five such constants (lumped harmonics) have been derived for the GEOS-2 orbit (order 13, to 30th degree) whose principal resonant period is 6 days. These five lumped harmonics are shown to account for almost all (>98%) of the resonant information in the tracking. They agree well with recent gravitational models which include substantial amounts of GEOS-2 data.  相似文献   

15.
The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form $$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$ The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ 2 is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ?B/?x≡B (1) atx=x 0 characterizes the resonant nature of the problem. With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded.  相似文献   

16.
The ratio between the Earth's perihelion advance (Δθ) E and the solar gravitational red shift (GRS) (Δø s e)a 0/c 2 has been rewritten using the assumption that the Newtonian constant of gravitationG varies seasonally and is given by the relationship, first found by Gasanalizade (1992b) for an aphelion-perihelion difference of (ΔG)a?p . It is concluded that $$\begin{gathered} (\Delta \theta )_E = \frac{{3\pi }}{e}\frac{{(\Delta \phi _{sE} )_{A_0 } }}{{c^2 }}\frac{{(\Delta G)_{a - p} }}{{G_0 }} = 0.038388 \sec {\text{onds}} {\text{of}} {\text{arc}} {\text{per}} {\text{revolution,}} \hfill \\ \frac{{(\Delta G)_{a - p} }}{{G_0 }} = \frac{e}{{3\pi }}\frac{{(\Delta \theta )_E }}{{(\Delta \phi _{sE} )_{A_0 } /c^2 }} = 1.56116 \times 10^{ - 4} . \hfill \\ \end{gathered} $$ The results obtained here can be readily understood by using the Parametrized Post-Newtonian (PPN) formalism, which predicts an anisotropy in the “locally measured” value ofG, and without conflicting with the general relativity.  相似文献   

17.
Stars are gravitationally stabilized fusion reactors changing their chemical composition while transforming light atomic nuclei into heavy ones. The atomic nuclei are supposed to be in thermal equilibrium with the ambient plasma. The majority of reactions among nuclei leading to a nuclear transformation are inhibited by the necessity for the charged participants to tunnel through their mutual Coulomb barrier. As theoretical knowledge and experimental verification of nuclear cross sections increases it becomes possible to refine analytic representations for nuclear reaction rates. Over the years various approaches have been made to derive closed-form representations of thermonuclear reaction rates (Critchfield, 1972; Haubold and John, 1978; Haubold, Mathai and Anderson, 1987). They show that the reaction rate contains the astrophysical cross section factor and its derivatives which has to be determined experimentally, and an integral part of the thermonuclear reaction rate independent from experimental results which can be treated by closed-form representation techniques in terms of generalized hypergeometric functions. In this paper mathematical/statistical techniques for deriving closed-form representations of thermonuclear functions, particularly the four integrals $$\begin{gathered} I_1 (z,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - zy^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ I_2 (z,d,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - zy^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ I_3 (z,t,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - z(y + 1)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ I_4 (z,\delta ,b,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - by^\delta } e^{ - zy^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ \end{gathered} $$ will be summarized and numerical results for them will be given. The separation of thermonuclear functions from thermonuclear reaction rates is our preferred result. The purpose of the paper is also to compare numerical results for approximate and closed-form representations of thermonuclear functions. This paper completes the work of Haubold, Mathai, and Anderson (1987).  相似文献   

18.
The spheroidal harmonics expressions $$\left[ {P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) - P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ and $$\left[ {\eta ^2 P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) + \xi ^2 P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ , have ξ2+η2 as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ2+η2 in the form of a linear combination of terms of the formP 2m 2s (iξ)P 2n 2s (η)e i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.  相似文献   

19.
It is shown that the fractional increase in binding energy of a galaxy in a fast collision with another galaxy of the same size can be well represented by the formula $$\xi _2 = 3({G \mathord{\left/ {\vphantom {G {M_2 \bar R}}} \right. \kern-\nulldelimiterspace} {M_2 \bar R}}) ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {V_p }}} \right. \kern-\nulldelimiterspace} {V_p }})^2 e^{ - p/\bar R} = \xi _1 ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {M_2 }}} \right. \kern-\nulldelimiterspace} {M_2 }})^3 ,$$ whereM 1,M 2 are the masses of the perturber and the perturbed galaxy, respectively,V p is the relative velocity of the perturber at minimum separationp, and \(\bar R\) is the dynamical radius of either galaxy.  相似文献   

20.
If \(T = \sum\nolimits_{i = 1}^\infty {\varepsilon ^i } T_i\) and \(W = \sum\nolimits_{n = 1}^\infty {n\varepsilon ^{n - 1} } W^{\left( n \right)}\) are respectively the generators of Giorgilli-Galgani's and Deprit's transformations, we show that the change of variables generated byT is the inverse of the one generated byW, ifT i =W (i) for anyi. The method used is to show that the recurrence which defines the first algorithm can also be obtained with the second one.  相似文献   

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