International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 6, No. 1, 8 Feb. 2020
Cover page and Table of Contents: PDF (size: 490KB)
Application of the Flow Curvature Method in Lorenz-Haken Model
Full Text (PDF, 490KB), PP.33-48
Views: 0 Downloads: 0
Author(s)
Index Terms
Flow Curvature Method, Slow Manifold, Dynamical System, Differential Geometry.
Abstract
We consider a recently developed new approach so-called the flow curvature method based on the differential geometry to analyze the Lorenz-Haken model. According to this method, the trajectory curve or flow of any dynamical system of dimension considers as a curve in Euclidean space of dimension . Then the flow curvature or the curvature of the trajectory curve may be computed analytically. The set of points where the flow curvature is null or empty defines the flow curvature manifold. This manifold connected with the dynamical system of any dimension directly describes the analytical equation of the slow invariant manifold incorporated with the same dynamical system. In this article, we apply the flow curvature method for the first time on the three-dimensional Lorenz-Haken model to compute the analytical equation of the slow invariant manifold where we use the Darboux theorem to prove the invariance property of the slow manifold. After that, we determine the osculating plane of the dynamical system and find the relation between flow curvature manifold and osculating plane. Finally, we find the nature of the fixed point stability using flow curvature manifold.
Cite This Paper
A. K. M. Nazimuddin, Md. Showkat Ali," Application of the Flow Curvature Method in Lorenz-Haken Model ", International Journal of Mathematical Sciences and Computing (IJMSC), Vol.6, No.1, pp.33-48, 2020. DOI: 10.5815/ijmsc.2020.01.04.
Reference
[1]Andronov, A.A., Chaikin, S.E (1937). Plane Theory of Oscillators, I, Moscow.
[2]Tikhonov, A.N. (1948). On the dependence of solutions of differential equations on a small parameter, Mat. Sbornik N. S., 31:575–586.
[3]Levinson, N., (1949). A second-order differential equation with singular solutions, Ann. Math, 50:127–153.
[4]Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J, 21:193–225.
[5]Fenichel, N. (1974). Asymptotic stability with rate conditions, Indiana Univ. Math. J, 23:1109–1137.
[6]Fenichel, N. (1977). Asymptotic stability with rate conditions II, Indiana Univ. Math. J, 26:81–93.
[7]Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31: 53–98.
[8]Wasow, W.R. (1965). Asymptotic Expansions for Ordinary Differential Equations, Wiley-Interscience, New York.
[9]Cole, J.D. (1968). Perturbation Methods in Applied Mathematics, Blaisdell, Waltham.
[10]O’Malley, R.E. (1974). Introduction to Singular Perturbations, Academic Press, New York.
[11]O’Malley, R.E. (1991). Singular Perturbations Methods for Ordinary Differential Equations, Springer, New York.
[12]Ginoux, J.M. and Rossetto, B. (2006). Differential geometry and mechanics applications to chaotic dynamical systems, Int. J. Bifurc. Chaos, 4(16): 887–910.
[13]Ginoux, J.M., Rossetto, B. and Chua, L.O. (2008). Slow invariant manifolds as curvature of the flow of dynamical systems, Int. J. Bifurc. Chaos, 11(18): 3409–3430.
[14]Ginoux, J.M. (2009). Differential geometry applied to dynamical systems, In: World Scientific Series on Nonlinear Science, Series A, 66, World Scientific, Singapore.
[15]Ginoux, J.M.and Llibre, J. (2011). The flow curvature method applied to canard explosion, J. Phys. A Math. Theor., 44: 465203.
[16]Ginoux, J.M., Llibre, J. and Chua, L.O. (2013). Canards from Chua’s circuit, Int. J. Bifurc. Chaos, 23(4): 1330010.
[17]Ginoux, J. M., & Rossetto, B. (2014). Slow invariant manifold of heartbeat model, arXiv preprint arXiv:1408.4988.
[18]Ginoux, J. M. (2014). The slow invariant manifold of the Lorenz–Krishnamurthy model, Qualitative theory of dynamical systems, 13(1): 19–37.
[19]Rossetto, B., Lenzini, T., Ramdani, S. & Suchey, G. (1998). Slow–fast autonomous dynamical systems, Int. J. Bifurcation and Chaos, 8(11): 2135–2145.
[20]Ramdani, S. (2000). Slow manifolds of some chaotic systems with applications to laser systems, Int. J of bifurcation and Chaos, 10 (12): 2729–2744.
[21]Cai, G., Tian, L., & Huang, J. (2006). Slow manifolds of Lorenz-Haken system and its application, International Journal of Nonlinear Science, 1(2): 93–104.
[22]Haken, H. (1975). Analogy between higher instabilities in fluids and lasers, Phys. Lett. A, 53(1):77–78.
[23]Schlomiuk D. (1999). Elementary first integrals of differential equations and invariant algebraic curves , Expositiones Mathematicae, 11: 433–454.
[24]Llibre J . & Medrado J. C. (2007). On the invariant hyperplanes for d-dimensional polynomial vector fields , J. Phys. A. Math. Theor., 40 : 8385–8391.