Qiu-Dong Wang

Interests

Research

Dynamical Systems, ordinary Differential Equations, Celestial Mechanics

Courses

Scholarly Contributions

Journals/Publications

• Chen, F., Oksasoglu, A., & Wang, Q. (2013). Heteroclinic tangles in time-periodic equations. Journal of Differential Equations, 254(3), 1137-1171.
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  Abstract: In this paper we prove that, when a heteroclinic loop is periodically perturbed, three types of heteroclinic tangles are created and they compete in the space of μ where μ is a parameter representing the magnitude of the perturbations. The three types are (a) transient heteroclinic tangles containing no Gibbs measures; (b) heteroclinic tangles dominated by sinks representing stable dynamical behavior; and (c) heteroclinic tangles with strange attractors admitting SRB measures representing chaos. We also prove that, as μ. →. 0, the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections. The theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical Birkhoff-Melnikov-Smale method. © 2012 Elsevier Inc.
• Kening, L. u., Wang, Q., & Young, L. (2013). Strange attractors for periodically forced parabolic equations. Memoirs of the American Mathematical Society, 224(1054), 1-97.
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  Abstract: We prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given. © 2012 American Mathematical Society.
• Wang, Q., & Young, L. (2013). Dynamical profile of a class of rank-one attractors. Ergodic Theory and Dynamical Systems, 33(4), 1221-1264.
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  Abstract: This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349-480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai-Ruelle-Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with 'strange attractors'. © 2012 Cambridge University Press.
• Punoševac, P., & Wang, Q. (2012). Regularization of simultaneous binary collisions in some gravitational systems. Rocky Mountain Journal of Mathematics, 42(1), 257-283.
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  Abstract: In this paper we construct coordinate transforms that regularize the singularities of simultaneous binary collisions in a pair of decoupled Kepler problems and in a restricted collinear four-body problem. This is the first time regularization transforms are introduced for collisions involving more than one colliding pair in the study of the Newtonian gravitational systems. Copyright © 2012 Rocky Mountain Mathematics Consortium.
• Takahasi, H., & Wang, Q. (2012). Nonuniformly expanding 1D maps with logarithmic singularities. Nonlinearity, 25(2), 533-550.
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  Abstract: For a certain parametrized family of maps on a circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behaviour. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations. © 2012 IOP Publishing Ltd & London Mathematical Society.
• Kening, L. u., & Wang, Q. (2011). Chaotic behavior in differential equations driven by a Brownian motion. Journal of Differential Equations, 251(10), 2853-2895.
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  Abstract: In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation. © 2011 Elsevier Inc.
• Wang, Q., & Oksasoglu, A. (2011). Dynamics of homoclinic tangles in periodically perturbed second-order equations. Journal of Differential Equations, 250(2), 710-751.
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  Abstract: We obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second-order ordinary differential equations with dissipation. Let Μ be the size of perturbation and ΛΜ be the entire homoclinic tangle. We prove in particular that (i) for infinitely many disjoint open sets of Μ, ΛΜ contains nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many disjoint open sets of Μ, ΛΜ contains nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure sets of Μ so that ΛΜ admits strange attractors with Sinai-Ruelle-Bowen measure. We also use the equation. d2q/dt2+(Λ-γq2)dq/dt-q+q2=Μq2sinωt to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation. © 2010 Elsevier Inc.
• Wang, Q., & Oksasoglu, A. (2011). Strange attractors and their periodic repetition. Chaos, 21(1).
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  PMID: 21456842;Abstract: In this paper, we present some important findings regarding a comprehensive characterization of dynamical behavior in the vicinity of two periodically perturbed homoclinic solutions. Using the Duffing system, we illustrate that the overall dynamical behavior of the system, including strange attractors, is organized in the form of an asymptotic invariant pattern as the magnitude of the applied periodic forcing approaches zero. Moreover, this invariant pattern repeats itself with a multiplicative period with respect to the magnitude of the forcing. This multiplicative period is an explicitly known function of the system parameters. The findings from the numerical experiments are shown to be in great agreement with the theoretical expectations. © 2011 American Institute of Physics.
• Wang, Q., & Ott, W. (2011). Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability. Communications on Pure and Applied Mathematics, 64(11), 1439-1496.
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  Abstract: We prove that when subjected to periodic forcing of the form $$p-{\mu ,\rho ,\omega } (t) = \mu (\rho h(x,y) + \sin (\omega t)),$$ certain two-dimensional vector fields with dissipative homoclinic loops generate strange attractors with Sinai-Ruelle-Bowen measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov and applies the recent theory of rank 1 maps developed by Wang and Young. We prove a general theorem and then apply this theorem to an explicit model: a forced Duffing equation of the form $${{d^2 q} \over {dt^2 }} + (\lambda - \gamma q^2){{dq} \over {dt}} - q + q^3 = \mu \sin (\omega t).$$ © 2011 Wiley Periodicals, Inc.
• Kening, L. u., & Wang, Q. (2010). Chaos in differential equations driven by a nonautonomous force. Nonlinearity, 23(11), 2935-2975.
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  Abstract: Nonautonomous forces appear in many applications. They could be periodic, quasiperiodic and almost periodic in time; or they could take the form of a sample path of a random forcing driven by a stochastic process, which is without any periodicity in time. In this paper, we study the chaotic behaviour of differential equations driven by a general nonautonomous forcing without assuming any periodicity in time, aiming at applications to systems driven by a bounded random force. As a direct application, we prove that, for the Duffing equation driven by a bounded stationary stochastic process induced by a Brownian motion, chaotic dynamics exist almost surely. We also obtain various chaotic behaviour that are exclusively associated with equations driven by nonautonomous forcing without any periodicity in time. It has turned out that, unlike the systems driven by a periodic or almost periodic forcing, the transversal intersections of the stable and unstable manifolds are neither necessary nor sufficient for chaotic dynamics to exist. Finally, we apply all our results to the Duffing equation. © 2010 IOP Publishing Ltd & London Mathematical Society.
• Oksasoglu, A., & Wang, Q. (2010). Rank one chaos in a switch-controlled Chua's circuit. Journal of the Franklin Institute, 347(9), 1598-1622.
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  Abstract: In this paper, we study the existence of strange attractors in a switch-controlled Chua's circuit. This circuit is obtained from the original Chua's circuit by adding externally controlled switches to it in such a way to modulate the system state variables. This investigation is conducted from the perspective of a recent chaos theory of rank one maps. The externally controlled switches are used for the purposes of realizing the general settings of the theory. Both synchronous and asynchronous switch control schemes providing periodic kicks in various directions are investigated, and their effects on the resulting chaotic attractors are discussed. The results of the numerical simulations presented are in close agreement with the expectations of the theory. © 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
• Ott, W., & Wang, Q. (2010). Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps. Discrete and Continuous Dynamical Systems, 26(3), 1035-1054.
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  Abstract: We analyze parametrized families of multimodal 1D maps that arise as singular limits of parametrized families of rank one maps. For a generic 1-parameter family of such maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure exhibits nonuniformly expanding dynamics characterized by the existence of a positive Lyapunov exponent and an absolutely continuous invariant measure. Under a mild combinatoric assumption, we prove that each such parameter is an accumulation point of the set of parameters admitting superstable periodic sinks.
• Wang, Q. D., & Oksasoglu, A. (2010). Periodic occurrence of chaotic behavior of homoclinic tangles. Physica D: Nonlinear Phenomena, 239(7), 387-395.
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  Abstract: In this article, we illustrate, through numerical simulations, some important aspects of the dynamics of the periodically perturbed homoclinic solutions for a dissipative saddle. More explicitly, we demonstrate that, when homoclinic tangles are created, three different dynamical phenomena, namely, horseshoes, periodic sinks, and attractors with Sinai-Ruelle-Bowen measures, manifest themselves periodically with respect to the magnitude of the forcing function. In addition, when the stable and the unstable manifolds are pulled apart so as not to intersect, first, rank 1 attractors, then quasi-periodic attractors are added to the dynamical scene. © 2009 Elsevier B.V. All rights reserved.
• Oksasoglu, A., Ozoguz, S., Demirkol, A. S., Akgul, T., & Wang, Q. (2009). Experimental verification of rank 1 chaos in switch-controlled Chua circuit. Chaos, 19(1).
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  PMID: 19334980;Abstract: In this paper, we provide the first experimental proof for the existence of rank 1 chaos in the switch-controlled Chua circuit by following a step-by-step procedure given by the theory of rank 1 maps. At the center of this procedure is a periodically kicked limit cycle obtained from the unforced system. Then, this limit cycle is subjected to periodic kicks by adding externally controlled switches to the original circuit. Both the smooth nonlinearity and the piecewise linear cases are considered in this experimental investigation. Experimental results are found to be in concordance with the conclusions of the theory. © 2009 American Institute of Physics.
• Wang, Q., & Oksasoglu, A. (2008). Rank one chaos: Theory and applications. International Journal of Bifurcation and Chaos, 18(5), 1261-1319.
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  Abstract: The main purpose of this tutorial is to introduce to a more application-oriented audience a new chaos theory that is applicable to certain systems of differential equations. This new chaos theory, namely the theory of rank one maps, claims a comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain systems of differential equations, the existence of the indicated phenomenon of chaos can be verified through a well-defined computational process. Applications to the well-known Chua's and MLC circuits employing controlled switches are also presented to demonstrate the usefulness of the theory. We try to introduce this new chaos theory by using a balanced combination of examples, numerical simulations and theoretical discussions. We also try to create a standard reference for this theory that will hopefully be accessible to a more application-oriented audience. © 2008 World Scientific Publishing Company.
• Wang, Q., & Young, L. (2008). Toward a theory of rank one attractors. Annals of Mathematics, 167(2), 349-480.
• Zhu, X., Schülzgen, A., Li, H., Li, L., Wang, Q., Suzuki, S., Temyanko, V. L., Moloney, J. V., & Peyghambarian, N. (2008). Single-transverse-mode output from a fiber laser based on multimode interference. Optics Letters, 33(9), 908-910.
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  PMID: 18451935;Abstract: An alternative original approach to achieve single-transverse-mode laser emissions from multimode (MM) active fibers is demonstrated. The fiber cavity is constructed by simply splicing a conventional passive single-mode fiber (SMF-28) onto a few centimeters-long active MM fiber section whose length is precisely controlled. Owing to the self-imaging property of multimode interference (MMI) in the MM fiber, diffraction-limited laser output is obtained from the end of the SMF-28, and the MMI fiber laser is nearly as efficient as the corresponding MM fiber laser. Moreover, because of the spectral filtering effect during in-phase MMI, the bandwidth of the MMI fiber laser is below 0.5 nm. © 2008 Optical Society of America.
• Wang, Q., & Oksasoglu, A. (2007). Rank one chaos in switch-controlled piecewise linear Chua's circuit. Journal of Circuits, Systems and Computers, 16(5), 769-789.
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  Abstract: In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations. © 2007 World Scientific Publishing Company.
• Oksasoglu, A., & Wang, Q. (2006). A new class of chaotic attractors in Murali-lakshmanan-chua circuit. International Journal of Bifurcation and Chaos, 16(9), 2659-2670.
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  Abstract: In this paper, we study the existence of a new class of chaotic attractors, namely the rank-one attractors, in the MLC (Murali-Lakshmanan-Chua) circuit [Murali et al., 1994] by numerical simulations based on a theory of rank-one maps developed in [Wang & Young, 2005]. With the guidance of the theory in [Wang & Young, 2005], weakly stable limit cycles, found through Hopf bifurcations and other numerical means, are subjected to periodic pulses with long relaxation periods to produce rank-one attractors. The periodic pulses are applied directly as an input. Periodic pulses have been used before in various schemes of chaos. However, for this scheme of creating rank-one attractors to work, the applied periodic pulses must have short pulse widths and long relaxation periods. This is one of the key components in creating this new class of chaotic attractors. © World Scientific Publishing Company.
• Oksasoglu, A., Dongsheng, M. a., & Wang, Q. (2006). Rank one chaos in switch-controlled Murali-lakshmanan-chua circuit. International Journal of Bifurcation and Chaos, 16(11), 3207-3224.
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  Abstract: In this paper, we investigate the creation of strange attractors in a switch-controlled MLC (Murali-Lakshmanan-Chua) circuit. The design and use of this circuit is motivated by a recent mathematical theory of rank one attractors developed by Wang and Young. Strange attractors are created by periodically kicking a weakly stable limit cycle emerging from the center of a supercritical Hopf bifurcation, and are found in numerical simulations by following a recipe-like algorithm. Rigorous conditions for chaos are derived and various switch control schemes, such as synchronous, asynchronous, single-, and multi-pulse, are investigated in numerical simulations. © World Scientific Publishing Company.
• Wang, Q., & Young, L. (2006). Nonuniformly expanding 1D maps. Communications in Mathematical Physics, 264(1), 255-282.
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  Abstract: This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through ''Misiurewicz points'' have invariant densities for positive measure sets of parameters.
• Li, L., Schülzgen, A., Temyanko, V. L., Qiu, T., Morrell, M. M., Wang, Q., Mafi, A., Moloney, J. V., & Peyghambarian, N. (2005). Short-length microstructured phosphate glass fiber lasers with large mode areas. Optics Letters, 30(10), 1141-1143.
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  PMID: 15943293;Abstract: We report fabrication and testing of the first phosphate glass microstructured fiber lasers with large Er-Yb-codoped cores. For an 11-cm-long cladding-pumped fiber laser, more than 3 W of continuous wave output power is demonstrated, and near single-mode beam quality is obtained for an active core area larger than 400 μm2. © 2005 Optical Society of America.
• Wang, Q., & Oksasoglu, A. (2005). Strange attractors in periodically kicked Chua's circuit. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 15(1), 83-98.
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  Abstract: In this paper, we discuss a new mechanism for chaos in light of some new developments in the theory of dynamical systems. It was shown in [Wang & Young, 2002b] that strange attractors occur when an autonomous system undergoing a generic Hopf bifurcation is subjected to a weak external forcing that is periodically turned on and off. For illustration purposes, we apply these results to the Chua's system. Derivation of conditions for chaos along with the results of numerical simulations are presented. © World Scientific Publishing Company.
• Wang, Q., & Young, L. (2003). Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations. Communications in Mathematical Physics, 240(3), 509-529.
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  Abstract: We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations.
• Wang, Q., & Young, L. (2002). From invariant curves to strange attractors. Communications in Mathematical Physics, 225(2), 275-304.
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  Abstract: We prove that simple mechanical systems, when subjected to external periodic forcing, can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied. In particular, the existence of global strange attractors with fully stochastic properties is proved for a class of second order ODEs. © Springer-Verlag 2002.
• Wang, Q. D. (2001). Power series solutions and integral manifold of the n-body problem. Regular and Chaotic Dynamics, 6(4), 433-442.
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  Abstract: In this article we discuss our solutions for two of the questions asked in the history of the n-body problem: the construction of the global power series solution of the n-body problem and the bifurcation of the integral manifold of the three-body problem.
• Wang, Q., & Young, L. (2001). Strange attractors with one direction of instability. Communications in Mathematical Physics, 218(1), 1-97.
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  Abstract: We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d = 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy.
• Wang, Q. (2000). The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete and Continuous Dynamical Systems, 6(2), 255-274.
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  Abstract: We improve Mather's proof on the existence of the connecting orbit around rotation number zero (Proposition 8.1 in [7]) in this paper. Our new proof not only assures the existence of the connecting orbit, but also gives a quantitative estimation on the diffusion time.
• Meyer, K., & Wang, Q. D. (1995). The Collinear Three-Body Problem with Negative Energy. Journal of Differential Equations, 119(2), 284-309.
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  Abstract: The geometry of the global phase space of the collinear three-body problem with negative energy is presented in this paper. A set of transformations is introduced to create fictitious boundaries to make the phase space compact. At first, the binary collisions are not regularized. Then one of the binary collisions (the collision between m2 and m3) is regularized and we analyze the phase structure of this "half regularized" system. Finally, the second binary collision (the collision between m1 and m2) is regularized and we analyze how the phase structure is transformed by this regularization. The whole analysis provides a vivid picture of the phase flow of the collinear three-body problem. © 1995 Academic Press. All rights reserved.
• Qiu-Dong, W. (1990). The global solution of the N-body problem. CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY, 50(1), 73-88.
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  Abstract: The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum. A new 'blowing up' transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum. © 1991 Kluwer Academic Publishers.
• Wang, Q. (1986). The existence of global solution of the n-body problem. Chinese Astronomy and Astrophysics, 10(2), 135-142.
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  Abstract: Based on McGehee's transformation f = I-1q; g = I 1 2p;, dt dt′ = I 3 2, I introduce the transformtion X = (gTM-1g) 1 2, h ≥ 0X = (gTM-1g) - 2Ih) 1 2, h≤0, u = IX-2; F = X2f; G = X-1g; dt′ dt = X-3. I prove that these variables may be continued to every point of the new time axis t for any initial value, and the whole axis corresponds to the "time interval of existence of the global solution". Also, F, G, u are O(eBτ). I then obtain a region H on the complex plane τ, |Im(τ)| < A exp (B Re τ2), over which F and G are analytic. Here, A, B, C are constants related only to the masses and the initial value. Lastly, a conformal mapping is established which maps a subregion of H, H, onto the unit circle of the new complex variable, thus obtaining a global solution of the n-body problem. The convergence of my power series is admittedly unsatisfactory and so the present result is of limited value for practical calculation. © 1986.