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Research Papers

Heteroclinic Orbit, Forced Lorenz System, and Chaos

[+] Author and Article Information
Dibakar Ghosh

Department of Mathematics, Dinabandhu Andrews College, Garia, Calcutta 700 084, India; Department of Physics, High Energy Physics Division, Jadavpur University, Calcutta 700032, Indiadrghosh_chaos@yahoo.com

Anirban Ray

Department of Physics, High Energy Physics Division, Jadavpur University, Calcutta 700032, Indiaanirban.chaos@gmail.com

A. Roy Chowdhury

Department of Physics, High Energy Physics Division, Jadavpur University, Calcutta 700032, Indiaasesh_r@yahoo.com

J. Comput. Nonlinear Dynam 5(1), 011008 (Nov 18, 2009) (7 pages) doi:10.1115/1.4000318 History: Received October 21, 2008; Revised June 03, 2009; Published November 18, 2009; Online November 18, 2009

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.

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Figures

Grahic Jump Location
Figure 1

Fc as a function of r for a=10 and b=8/3

Grahic Jump Location
Figure 2

(a) Parameter region of r for heteroclinic orbit where Δ>0 and Δ1<0. (b) Two parameter region of r, F for heteroclinic orbit.

Grahic Jump Location
Figure 3

Chaotic attractors of forced Lorenz system for (a) F=0.0, (b) F=−1.0, (c) F=−2.0, and (d) F=−2.5, where r=28.0, b=8/3, and a=10.0

Grahic Jump Location
Figure 4

Variation in largest Lyapunov exponent with respect to r for different values of forcing F: (a) F=0, (b) F=−1.0, (c) F=−2.0, and (d) F=−2.5

Grahic Jump Location
Figure 5

(a) (x1,x2) projection and (b) (x1,x3) projection of heteroclinic contour joining saddle node with saddle focus in the forced Lorenz system for ϵ=10−7

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