Local – Global Connection

(Statement of Purpose)

 

From the mathematical point of view self – assembly or pattern formation is called to some kinds of spatiotemporal nonlinearity leading to pattern formation. Of course, pattern formation is just one feature of the brooder landscape behaviors of nonlinearity including bifurcation, chaotic behavior and etc. The phenomenon occurs in nonlinear systems comprising of many elements each of which is governed by some evolution rules and some spatial interactions relate each element temporal dynamics to the others. It is therefore obviously that the concept comprises almost every real system such as a beam under buckling [1], a vibrating shell [2], a fluid in turbulent regime [3], the cellular automata and neural nets [4], and even alive systems and societies [5].

The behavior of such coupled systems is quite complex. However some works has been done toward the establishment of a fundamental approach for this behavior. This approach starts by the same questions as those of temporal dynamics. Questions like "what initial patterns are unchanged under the rules of evolution?" or "what set of patterns have periodic behavior for the evolution?" and etc. So the concepts introduced through studying the temporal behavior of dynamical systems like fixed points, limit cycle, attractor, and so on, get some extended application to describe the spatiotemporal behavior.

In order to approach the real physical phenomenon, discrete models have been widely simulated and studied. The so called coupled map lattices are discrete architecture of nonlinear maps connected by usually local interactions. Assume for example a logistic map: as a fundamental block or subsystem. When iterated, the map shows for some range of parameter , an irregular or chaotic behavior while a system comprising several interconnected logistic maps may show periodic patterns for the same range of [6][7]. Changing the individual dynamics or interconnection design in discrete systems though slightly may lead to thoroughly different patterns. For example some added noise to the individual dynamics may cause coherent patterns in a locally interconnected coupled media while introduction of shortcut links between non local neighbors destroys the spatial patterns and instead enhances the temporal order [8].

From this point of view complex systems reveal a different mode of design based on planning local dynamics which leads to robust global forms. This is actually an inherent design. Also besides manipulation in global via local design, a backward flow of information from global to local is possible in complex systems. In other words by varying the macroscopic parameters of the whole system the pattern formation is influenced and so does the subsystems behavior. This is what we see in a Rayleigh – Benard convection for instance [9].

Based on these features, a so called bottom – up approach has incited some efforts in technological fields such as nano manufacturing in recent years. To clarify the terminology it should be noted that currently accomplishment of a project in nano scale is estimated through two general ways i.e. top – down and bottom – up approaches. Top – down approach is assembly by manipulating components with external (much larger) devices which is more readily achievable using the current technology [10]. By bottom – up approach on the other hand, we mean to use the self – assembly abilities of synthetic and/or biological molecules [11]. There are however some combinations of these two general ways such as the templating technique. A template can be used to create a more complex initial pattern for subsequent self – assembly; alternatively, the original structure can be used as a template capable of being modified by the chemical or physical means to stabilize, or tailor the properties for a specific purpose [11]. Fabrication of 3D cantilever arrays based on single – walled carbon nanotube multilayer using layer – by – layer (LbL) nano self – assembly is one of the latest cases [12].

As an elasticity approach, pattern formation through external loading on substrates has studied in [13]. The structure control of thin soft matter films by influence of a spatiotemporal periodic pressure on the thin film evolution is presented in [14]. Some attempts have been made on diffusion models such as Cahn – Hilliard equation [14][15]. Also effects of energy and entropy as some global parameters on pattern selection and control are investigated in [2]. 

Besides afore mentioned technical applications of pattern formation such as control, analysis and design, there are some theoretical interests as to express universalities of this phenomenon shown in different physical systems. In fact it seems the same geometry (in global view) formed from different dynamics is responsible for such universalities. Certain common features for example have been reported in patterns resulting from different mechanisms [14]. The geometry of patterns also has been useful in selection of proper modes for assumed mode summation [2]. Some proper geometry has been designed as a spatial support for pattern formation processes [16].

The aim of this thesis in brief is to construct a dynamical scheme based on geometrical local – global aspects so as to describe the complex systems behavior in local as well as global view. There is hope for covering pattern formations related to different mechanical systems (solid and fluid) by this approach.

This thesis has been started with a generalization of Takens reconstruction (Embedding) theorems to spatiotemporal features of the attractors. It is notable that Takens embedding method has been widely used for reconstruction of the attractor using some time series for example see my paper: "Reconstruction and prediction of the pressure attractor ..." Nonlinear Dynamics, 48 Number 4, 2007, 437 - 447. However I think the spatiotemporal view of Takens method is more interesting.

 

 

Please contact me for discussion about details of the idea.

 

Ph.D. Courses

 

Reference:

  1. Damil, N., Potier – Ferry, M., “A generalized continuum approach to describe instability pattern formation by a multiple scale analysis”, C. R. Mecanique, 334, 674 – 678, (2006)

  2. Guan, S., Lai, C. – H., Wei, G. W., “Geometry and boundary control of pattern formation and competition”, Physica D, 176, 19 – 43, (2003)

  3. Mazzi, B., Okkels, F., Vassilicos, J. C., “A shell – model approach to fractal – induced turbulence”, The European Physical Journal B, 28, 243 – 251, (2002)

  4. Laing, C. R., Longtin, A., “Noise – induced stabilization of bumps in systems with long – range spatial coupling”, Physica D, 160, 149 – 172, (2001)

  5. Sowerby, S. J., Holm, N. G., Petersen, G. B., "Origins of life: a route to nanotechnology", BioSystems, 61, 69-78, (2001)

  6. Hilborn, R. C., Chaos and Nonlinear Dynamics, An Introduction for Scientists and Engineers, 2nd edition, Oxford University Press, New York, 2000.

  7. Bohr, T., Jensen, M. H., Paladin, G., Vulpiani, A., Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998.

  8. Perc, M., “Effects of small – world connectivity on noise – induced temporal and spatial order in neural media”, Chaos, Solitons and Fractals, 31, 280 – 291, (2007)

  9. Haken, H., Synergetics, An Introduction, 2nd Edition, Springer Verlag, Berlin, 1978

  10. Mansoori, G. A., "Advances in Atomic and Molecular Nanotechnology", UN-APCTT Tech Monitor, 53-59, (2002)

  11. Leval, J. M., Mazeran, P. E., Thomas, D., "Nanobiotechnology and its role in the development of new analytical devices", Analyst, 125, 29-33, (2000)

  12. Xue, W., Cui, T., “Carbon nanotube micropatterns and cantilever arrays fabricated with layer – by – layer nano self – assembly”, Sensors and Actuators A, 136, 510 -517, (2007)

  13. Lu, W., Kim, D., “Engineering nanophase self – assembly with elastic field”, Acta Materialia, 53, 3689 – 3694, (2005)

  14. Pototsky, A., Bestehorn, M., Theile, U., “Control of the structuring of thin soft matter films by means of different types of external disturbance”, Physica D, 199, 138 – 148, (2004)

  15. Walgraef, D., “Nanostructure initiation during the early stages of thin film growth”, Physica E, 15, 33 – 40, (2002)

  16. Tucci, K., Cosenza, M. G., “Spectral properties and pattern selection in fractal growth networks”, Physica D, 199, 91 – 104, (2004)

 

 

 

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