Modifying some foliated dynamical systems to guide their trajectories to specified sub-manifolds

@article{Vaidya2011,
abstract = {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.},
author = {Vaidya, Prabhakar G., Majumder, Swarnali},
journal = {Mathematica Bohemica},
keywords = {manifold; foliation; duffing oscillator; manifold; foliation; Duffing oscillator},
language = {eng},
number = {4},
pages = {439-448},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Modifying some foliated dynamical systems to guide their trajectories to specified sub-manifolds},
url = {http://eudml.org/doc/196666},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Vaidya, Prabhakar G.
AU - Majumder, Swarnali
TI - Modifying some foliated dynamical systems to guide their trajectories to specified sub-manifolds
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 4
SP - 439
EP - 448
AB - 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.
LA - eng
KW - manifold; foliation; duffing oscillator; manifold; foliation; Duffing oscillator
UR - http://eudml.org/doc/196666
ER -