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manifold; foliation; duffing oscillator
We show that dynamical systems in inverse problems are sometimes foliated if the embedding dimension is greater than the dimension of the manifold on which the system resides. Under this condition, we end up reaching different leaves of the foliation if we start from different initial conditions. For some of these cases we have found a method by which we can asymptotically guide the system to a specific leaf even if we start from an initial condition which corresponds to some other leaf. We demonstrate the method by two examples. In the chosen cases of the harmonic oscillator and Duffing's oscillator we find an alternative set of equations which represent a collapsed foliation, such that no matter what initial conditions we choose, the system would asymptotically reach the same desired sub-manifold of the original system. This process can lead to cases for which a system begins in a chaotic region, but is guided to a periodic region and vice versa. It may also happen that we could move from an orbit of one period to an orbit of another period.
[1] Anosov, D. V.: Foliation. Encyclopedia of Mathematics. Michiel Hazewinkel Kluwer Academic Publishers (2001). MR 1935796
[2] Moerdijk, I., Mrcun, J.: Introduction to foliations and Lie groupoids. Cambridge Studies in Advanced Mathematics 91 (2003), 318-320. MR 2012261 | Zbl 1029.58012
[3] Packard, N. H., Crutchfield, J. P., Farmer, J. D., Shaw, R. S.: Geometry from a Time Series. Physical Review Letter 45 (1980), 712-715. DOI 10.1103/PhysRevLett.45.712
[4] Takens, F.: Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics 898 D. A. Rand and L. S. Young (1981), 366-381. DOI 10.1007/BFb0091924 | MR 0654900 | Zbl 0513.58032
[5] Vaidya, P. G., Majumder, S.: Embedding in higher dimension causes ambiguity for the problem of determining equation from data. European Physics Journal, special topic 165 (2008), 15-24. DOI 10.1140/epjst/e2008-00845-1
[6] Vaidya, P. G., Angadi, S.: A Computational Procedure to Generate a Difference Equations from Differential Equation. New Progress in Difference Equations Proceedings of the 6th ICDEA in Augsburg (2003), 539-548.
[7] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983). MR 0709768 | Zbl 0515.34001
[8] Sepulchre, J. A., MacKay, R. S.: Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators. Nonlinearity 10 (1997), 679-713. DOI 10.1088/0951-7715/10/3/006 | MR 1448582 | Zbl 0905.39004
[9] Pokorny, P.: Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum. Math. Probl. Eng. 2009, Article ID 104547, p. 15, doi: 10.1155/2009/104547. MR 2530049
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