Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 20, No. 11 (2010) 3391–3441 c  World Scientific Publishing Company DOI: 10.1142/S0218127410027799 A REVIEW STUDY OF THE 3-PARTICLE TODA LATTICE AND HIGHER ORDER TRUNCATIONS: THE EVEN-ORDER CASES (PART II) LOUKAS ZACHILAS Lecturer in Applied Mathematics, Department of Economics, University of Thessaly, 43 Korai str., GR-38333, Volos, Greece zachilas@uth.gr Received February 13, 2010 We complete the study of the numerical behavior of the truncated 3-particle Toda lattice (3pTL) with even truncations at orders n =2k, k =2,..., 10. We use (as in Part I): (a) the method of Poincar´ e surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits. We derived some similarities and quite many differences between the odd and even order expansions. Keywords : Toda lattice; Hamiltonian dynamics; Poincar´ e surface of section; periodic orbits. 1. Introduction The special case of the periodic 3-particle Toda lattice (3pTL) was studied by Contopoulos and Polymilis (henceforth CP) [1987]: H = 1 2 (˙ q 2 1 +˙ q 2 2 +˙ q 2 3 ) + exp(q 1 - q 2 ) + exp(q 2 - q 3 ) + exp(q 3 - q 1 ) - 3 (1) where q i , ˙ q i (i = 1, 2, 3) are the positions and momenta of the three particles, respectively. In the present paper (Part II), we study the transformed and truncated Toda lattice systems of order n =2k, k =2,..., 10. The choice of truncat- ing the approximation at the 20th order is due to the complexity and the heavy computer consump- tion of the 117 terms of H 20 . The general form of the truncated Hamiltoni- ans is: H i = 1 2 (˙ x 2 +˙ y 2 )+Φ i (x, y)= E (2) where Φ i (x, y) is the potential function of each approximation (i =2k, k =2,..., 10) and E is the energy integral. Similarly to Part I [Zachilas, 2010], we have chosen the plane x = 0 as surface of section. We integrate numerically the equations of motion for- ward in time and find the successive intersections of the orbit by the surface (Poincar´ e consequents). Equation (2) defines the curve of zero velocity (i.e. “limiting curve”), if we consider x =˙ x = 0: E = 1 2 ˙ y 2 +Φ i (0,y) (3) for given value of the energy E. In Sec. 2 we present the even-order truncations. Section 2.1 provides some general remarks about the nature of those potentials, and Sec. 2.2 describes the first even-order potential, the fourth order one (H 4 ). As it is a quite well studied potential [Udry & Martinet, 1990], we just underline the main characteristic features and we present the crucial differences with respect to the third order 3391