An Introduction to Differentiable Manifolds
Issue:
Volume 2, Issue 5, October 2016
Pages:
32-35
Received:
7 September 2016
Accepted:
1 November 2016
Published:
23 November 2016
DOI:
10.11648/j.ml.20160205.11
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Abstract: A manifold is a Hausdorff topological space with some neighborhood of a point that looks like an open set in a Euclidean space. The concept of Euclidean space to a topological space is extended via suitable choice of coordinates. Manifolds are important objects in mathematics, physics and control theory, because they allow more complicated structures to be expressed and understood in terms of the well–understood properties of simpler Euclidean spaces. A differentiable manifold is defined either as a set of points with neighborhoods homeomorphic with Euclidean space, Rn with coordinates in overlapping neighborhoods being related by a differentiable transformation or as a subset of R, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions. This paper aims at making a step by step introduction to differential manifolds.
Abstract: A manifold is a Hausdorff topological space with some neighborhood of a point that looks like an open set in a Euclidean space. The concept of Euclidean space to a topological space is extended via suitable choice of coordinates. Manifolds are important objects in mathematics, physics and control theory, because they allow more complicated structur...
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Complete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different 4D Nonlinear Dynamical Systems
Maysoon M. Aziz,
Saad Fawzi AL-Azzawi
Issue:
Volume 2, Issue 5, October 2016
Pages:
36-41
Received:
22 September 2016
Accepted:
1 November 2016
Published:
30 November 2016
DOI:
10.11648/j.ml.20160205.12
Downloads:
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Abstract: Three important phenomena of chaos synchronization are considered in this paper, In detailed, complete synchronization, anti- synchronization and hybrid synchronization based on the nonlinear active control approach between two different (non-identical) 4D hyperchaotic systems, i. e. Modified Pan and Liu are study herein. The Modified hyperchaotic Pan system is taken as drive system and hyperchaotic Liu system as response. Stabilization of error dynamics for each phenomenon is realized by satisfying two analytical approaches; Lyapunov's second method and linear system theory. Controllers are designed by using the relevant variable of drive and response systems. Theoretical analysis and numerical simulations are shown to verify the results.
Abstract: Three important phenomena of chaos synchronization are considered in this paper, In detailed, complete synchronization, anti- synchronization and hybrid synchronization based on the nonlinear active control approach between two different (non-identical) 4D hyperchaotic systems, i. e. Modified Pan and Liu are study herein. The Modified hyperchaotic ...
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