Forced Lorenz system, important in modeling of monsoonlike phenomena, is analyzed for the existence of heteroclinic orbit. This is done in the light of the suggested new mechanism for the onset of chaos by Magnitskii and Sidorov (2006, “Finding Homoclinic and Heteroclinic Contours of Singular Points of Nonlinear Systems of Ordinary Differential Equations,” Diff. Eq., 39, pp. 1593–1602), where heteroclinic orbits plays important and dominant roles. The analysis is performed based on the theory laid down by Shilnikov. An analytic expression in the form of uniformly convergent series is obtained. The same orbit is also obtained numerically by a technique enunciated by Magnitskii and Sidorov, reproducing the necessary important features.
Issue Section:
Research Papers
Keywords:
chaos,
nonlinear differential equations,
nonlinear dynamical systems,
Shilnikov chaos,
heteroclinic orbit,
forced Lorenz system
Topics:
Chaos
1.
Guckenheimer
, J.
, and Holmes
, P.
, 1990, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
, Springer-Verlag
, New York
.2.
Sparrow
, C.
, 1982, The Lorenz Equation: Bifurcation, Chaos and Strange Attractors
, Springer-Verlag
, New York
.3.
Zhou
, T.
, Chen
, G.
, and Celikovsky
, S.
, 2005, “Shilnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems
,” Nonlinear Dyn.
0924-090X, 39
, pp. 319
–334
.4.
Medrano-T
, R. O.
, Baptista
, M. S.
, and Caldas
, I. L.
, 2005, “Basic Structures of the Shilnikov Homoclinic Bifurcation Scenario
,” Chaos
1054-1500, 15
, p. 033112
.5.
Robinson
, C.
, 1989, “Homoclinic Bifurcation to a Transitive Attractor of Lorenz Type
,” Nonlinearity
0951-7715, 2
, pp. 495
–518
.6.
Mancho
, A. M.
, Small
, D.
, Wiggins
, S.
, and Ide
, K.
, 2003, “Computation of Stable and Unstable Manifolds of Hyperbolic Trajectories in Two-Dimensional, Aperiodically Time-Dependent Vector Fields
,” Physica D
0167-2789, 182
, pp. 188
–222
.7.
Wiggins
, S.
, 1988, Global Bifurcation and Chaos
, Springer-Verlag
, New York
.8.
Mittal
, A. K.
, Dwivedi
, S.
, and Pandey
, A. C.
, 2005, “Bifurcation Analysis of a Paradigmatic Model of Monsoon Prediction
,” Nonlinear Processes Geophys.
1023-5809, 12
, pp. 707
–715
.9.
Magnitskii
, N. A.
, and Sidorov
, S. V.
, 2003, “Finding Homoclinic and Heteroclinic Contours of Singular Points of Nonlinear Systems of Ordinary Differential Equations
,” Diff. Eq.
0012-2661, 39
, pp. 1593
–1602
.10.
Silva
, C. P.
, 1993, “Shilnikov Theorem—A Tutorial
,” IEEE Trans. Circuits Syst., I: Regul. Pap.
1549-8328, 40
, pp. 675
–682
.11.
Awrejcewicz
, J.
, and Krysko
, V. A.
, 2006, Introduction to Asymptotic Methods
, Chapman and Hall
, London
/CRC
, Boca Raton, FL
.12.
Awrejcewicz
, J.
, and Holicke
, M. M.
, 2007, Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods
, Vol. 60
(World Scientific Series on Nonlinear Science-Series A
), World Scientific
, Singapore
.13.
Lorenz
, E. N.
, 1963, “Deterministic Nonperiodic Flow
,” J. Atmos. Sci.
0022-4928, 20
, pp. 130
–141
.Copyright © 2010
by American Society of Mechanical Engineers
You do not currently have access to this content.
