Doctor of Sciences (DSc), Hungarian Academy of Sciences 1997
Doctor Habil., TU Budapest, 1996
Technical Doctor, TU of Budapest, 1990
I summarize below my previous research by grouping my papers around five main topics and giving a brief overview to each.
Large deformations of thin, elastic filaments
The geometrically exact, large deformation theory of thin, elastic filaments originating in the works of Euler and Kirchhoff recently found several applications in biology: three-dimensional shapes of macromolecules as well as of botanical filaments have been identified by these methods. With Phil Holmes, Tim Healey, Andy Ruina and Barrie Fraser I have worked on these problems for many years, using both computational and theoretical approaches. In particular, I was interested in identifying symmetric shapes as well as in understanding constrained/contact configurations.
Our work with Tim Healey was among the first to establish rigorous results on spatial configurations of macromolecules. Work with Phil Holmes on contact configurations was the first result on global classification of solutions for a geometrically nonlinear contact problem and has been often cited in engineering applications about drill strings.
Discrete models in dynamicals systems: population dynamics
With Domokos Szász I worked on the modelling of noisy, discretized, ergodic maps, a classical problem, raised originally by Ulam. Based on these results, with István Scheuring we found several applications in population dynamics and could also explain some experimental data. One of our main results was to define and to identify robust cycles in discrete population dynamics: these cycles survive environmental noise and can be detected in laboratory experiment data series. Robust cycles are crucial when predicting populations with chaotic behaviour, the latter is not uncommon.
Spatial chaos
With Phil Holmes and György Károlyi I have worked on understanding spatial (time-independent) chaos. The link to dynamical chaos is provided by the classical analogy due to Kirchhoff, describing the connection between the equations of the mathematical pendulum and the flexible elastic strut. Our first paper with Holmes extended and explored this analogy for the discrete case, establishing connection between discrete elastic linkages and the standard map. Later we proved fundamental properties of spatially chaotic boundary value problems (e.g. exponential growth of solutions) and finally we identified ghost solutions in these problems. The latter solve the initial value problem, can be arbitrarily well approximated by suitable discretizations, however, they are not close to any solution of the continuous boundary value problem. In a recent study our discovery was cited as the key element to the computation of giant (“rogue”) ocean waves.
Broken symmtery in evolution and structures
With Péter Varkonyi and András Sipos recently we wrote some papers discussing the role of slightly perturbed symmetry in seeking evolutionary and structural optima. While slightly asymmetric creatures are abundant in Nature, engineers tend to make their constructions perfectly symmetrical. Using adaptive dynamics we explained the emergence of left-right asymmetry in Nature and by applying tools from group representation theory we identified exact criteria under which symmetric engineering structures may be improved by small perturbations. We also provided practical examples in structural engineering.
Static equilibria on frigid bodies
Static equilibria of rigid bodies is one of my recurrent research interests. I started working with Andy Ruina and Jim Papadopoulos on the equilibria of planar disks 15 years ago. A conversation with V.I. Arnold in 1995 led to a recent discovery, jointly with Péter Várkonyi; the construction of the first so-called mono-monostatic object, a homogeneous, convex, three-dimensional body with just two points of equilibria (dubbed „Gömböc”). The Gömböc aroused worldwide attention within and beyond academics. Recently, the New York Times Magazine listed it among the 70 most interesting inventions of 2007 and the gomboc homepage has received so far over 3 million hits from 136 countries, including visits from over 800 universities.
Domokos G., Ruina A. ,1993. A circle construction based on elastostatics and hydrodynamics. Mechanics Research Communications Vol 20 (3) 181-185.
Domokos G., Holmes P.J. 1993: On non-inflexional solutions of non-uniform elasticae. Int'l. J. of Nonlin. Mech. Vol 28 (6) pp 677-685
Domokos G. 1994. Global description of elastic bars. Zeitschrift f. Angew. Math.Mech. Vol 74 (4) pp T289-T291.
Domokos G. 1995 A group-theoretic approach to the geometry of elastic rings . J. of Nonlinear Sci. Vol.5, pp.453-478
Domokos G., Holmes P.J., Royce, B. 1997. Constrained euler buckling. J. of Nonlinear Science 7 pp.281-314.
Holmes, P.J., Domokos G., Szeberenyi, I. and Schmitt, J. 1999. Constrained euler buckling: an interplay of computation and analysis. Comp. Meth. in Appl. Mech. and Eng . 170, pp 175-207.
Holmes, P.J., Domokos G., and Hek, G 2000. Euler buckling in potential field. J. Nonlin. Sci, Vol 10.pp 477-505
Domokos G., Healey, T. 2001: Hidden symmetry of global solutions in twisted elastic rings. J. Nonlin.Sci, Vol. 11, pp 47-67.
Domokos G., Fraser W.B. Symmetry-breaking bifurcations of the uplifted elastic strip. Physica D Vol 185/2 pp 67-77 (2003)
Domokos G., Healey, T. 2005. Multiple helical perversions of finite, intristically curved rodsInt.J. Bifurcation and Chaos Vol 15., No.3 , pp 871-890
Domokos G., Scheuring I. 2002: Random Perturbations and Lattice Effects in Chaotic Population Dynamics. Science Vol 297, No.5590,p2163
Domokos G., Szász, D. 2003: Ulam's scheme revisited: digital modeling of chaotic attractors via micro-perturbations.Discrete and Continuous Dynamical Systems, Ser. A. Vol 9. No.4. Pp 859-876 (2003).
Domokos G., Scheuring I.. 2003: Discrete and continuous state population models in a noisy worldJ. Theoretical Biology, Vol 227, pp 535-545.
Domokos G. 2005. Coarse-grained observation of discretized maps Int.J. Bifurcation and Chaos Vol 15., No.3 , pp 871-890
Scheuring I. , Domokos G. 2005. Sturdy cycles in the chaotic Tribolium castaneum data seriesJ. Theor. Pop. Dyn. Vol 67 (2005) pp 127-139.
Scheuring I. , Domokos G. 2007. Only noise can induce chaos in discrete populationsOikos Vol 166, pp361-366
Domokos G., Holmes, P.J. , 1993 . Euler's problem, Euler's method and the standard map or the discrete charm of buckling . J. Nonlinear Science Vol 3 pp 109-151
Domokos G. 1997. Static solitary waves as limits of discretization: a plausible argument.Phil. Trans. of the Royal Society London, A vol 355 pp 2099-2116
Domokos G., Karolyi G. 1999: Symbolic dynamics of infinite depth: finding global invariants for BVPs . Physica D 134, pp 316-336.
Holmes, P.J. Doelman, A., Hek, G. and Domokos, G. 2001: Homoclinic orbits and chaos in three- and four-dimensional flows. Proc. Royal Soc. London (A), 359, pp 1429-1438
Domokos G., Holmes P.J. 2003: On nonlinear BVPs: ghosts, parasites and discretizations. Proc. Roy. Soc. London A Vol 459, 1535-1561.
Kapsza E., Károlyi G., Kovacs, S. and Domokos G. 2003: Regular and random patterns in complex bifurcation diagramsDiscrete and Continuous Dynamical Systems, Ser. B Vol 3 No.4 pp 519-540
Várkonyi, P. Domokos, G. 2006. Symmetry and bifurcation of optima in structural design. Nonlinear Dynamics , Vol 43, pp 47-58.
Várkonyi, P., Domokos, G. 2006. Emergence of asymmetry in evolution. Theor. Pop. Dyn. Vol 70: pp 63-75.
Sipos A.Á,, Domokos G. 2007.Slightly asymmetric beams: examples of a new class of structural optima. Int'l. J. of Nonlin. Mech Vol 42 (3) pp 504-514
Várkonyi P., Domokos, G. 2007. Imperfect symmetry: a new approach to structural optima via group representation theory .Int'l. J. of Solids and Structures
Vol 44, pp4723-4741
Domokos, G., Papadopoulos, J., Ruina, A., 1994. Static equilibria of planar, rigid bodies: is there anything new?J. Elasticity 36, 59-66.
Domokos, G., 2006 My lunch with Arnold . Math. Intelligencer 28, 31-33.
Várkonyi, P. L., Domokos, G., 2006 Static equilibria of rigid bodies: dice, pebbles and the Poincaré-Hopf Theorem J. Nonlinear Sci. 16, 255-281.
Várkonyi, P.L., Domokos, G., 2006 Mono-monostatic bodies: the answer to Arnold ’s question . Math. Intelligencer 28, 34-38.
Domokos G., Várkonyi P.L.: Geometry and self-righting oh turtles. Proc. Roy. Soc. B., Volume 275, Number 1630 / January 07, 2008 pp 11-17
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