The chaotic behaviour on transition points between parabolic orbits

Abstract

The potential energy surfaces interact each other and their curvilinear coordinates have the critical information about disturbance at interaction points. Therefore, transition points between parabolic orbits that are solutions of one differential equation with variable coefficients is studied in this paper. Also we present an approach for the chaotic behaviour on transition points of the parabolic orbits

Keywords

• Chaotic Behaviour,Differential Equation,Parabolic Intersections,Quadratic Function,Schwarzian Derivative

The chaotic behaviour on transition points between parabolic orbits

Öz

Kaynakça

1. J.M.Jirstrand. Nonlinear Control System Design by Quantiier Elimination, J. Symbolic Computation, 24, 137-152, August, 1997, pp: 137-152.
2. F.Sicilia, L.Blancafort, M. J. Bearpark, and M. A. Robb. Quadratic Description of Conical Intersections: Characterization of Critical Points on the Extended Seam. J. Phys. Chem. A111, 2007, pp:2182-2192.
3. H.Riecke. Methods of Nonlinear Analysis 412 Engineering Sciences and Applied Mathematics. Northwestern University, 2008.
4. R.M. May. Simple Mathematical Models with very Complicated Dynamics. Nature, 1976, pp:261 459-67.
5. M.J. Bearpark, M. A. Robb, H. B. Schlege. A direct method for the location of the lowest energy point on a potential surface crossing. Chemical Physics Letters. 223, 1994, pp:269-274.
6. M. J. Paterson, M. J. Bearpark, and M. A. Robba. The curvature of the conical intersection seam: An approximate second-order analysis. Journal of Chemical Physics. Volume 121, Number 23. 15 December 2004.
7. D. A. Brue, X. Li, and G. A. Parker Conical intersection between the lowest spin-aligned Li3(4A)… potential-energy surfaces. Journal of Chemical Physics.123, 091101, 2005.
8. B. G. Levine, C. Ko, J. Quenneville and T. J. Martinez Conical intersections and double excitations in time-dependent density functional theory. Molecular Physics, Vol. 104, Nos. 5–7, 1039–1051, 10 March–10 April 2006.

9. M.W. Hirsch, S.Smale, R.L. Devaney. Differential Equations, Dynamical Systems and Introduction to Chaos., Elsevier Academic Press, 2004.
10. J. S. Kozlovski. Getting rid of the negative Schwarzian derivative condition, Annals of Mathematics, 152 (2000), 743-762.
11. J T.R. Scavo, J. B. Thoo. On the Geometry of Halley’s Method, The American Mathematical Montly, 1994.
12. L.-S. Yao Computed chaos or numerical errors. Nonlinear Analysis: Modelling and Control, Vol. 15, No. 1, 2010, pp: 109-126.
13. H. Kocak, K. J. Palmer Lyapunov Exponents and Sensitive Dependence. J Dyn Diff Equat, 22, 2010, pp:381-398.
14. T. Theivasanthi Bifurcations and chaos in simple dynamical systems. International Journal of Physical Sciences Vol. 4 (12), December, 2009, pp. 824-834.
15. J G. V. Weinberg and A. Alexopoulos. Examples of a Class of Chaotic Radar Signals, ISBN 0-387-94677-2.
16. J. K.T.Alligood, T.Sauer, J.A.Yorke. CHAOS An Introduction to Dynamical Systems, Elsevier Academic Press, Chapter:11, Page:447-496, ISBN 0-387-94677-2, 2000.
17. E. R. Scheinerman. Invitation to Dynamical Systems,Department of Mathematical Sciences The Johns Hopkins University, Chapter:4.2, Page:127-136, ISBN 0-13-185000-8, 2000.
18. Figure 1: The Interactions Between Surfaces of Parabolic Intersections,f x f x, for various a 1( ), 2( )