Victor Roytburd

Bio: Victor Roytburd is an academic researcher from Rensselaer Polytechnic Institute. The author has contributed to research in topics: Free boundary problem & Attractor. The author has an hindex of 10, co-authored 21 publications receiving 165 citations.

Finite-dimensional dynamical system modeling thermal instabilities

Abstract: We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary problem modeling solid combustion and rapid solidification, and is capable of generating its major dynamical patterns. These patterns include a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. A computer-assisted bifurcation analysis uncovers some novel mechanisms of stability exchange. The most striking of them is an infinite period bifurcation which resembles the classical Shilnikov bifurcation, but instead of a funnel-shaped spiral along which the period is continually increasing, the continuation produced a series of isolas. Each isola is a closed branch of solutions of roughly the same period, and with the same number of oscillations. The isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other, and appear to accumulate on a saddle-focus homoclinic connection of Shilnikov type. ©2000 Elsevier Science B.V. All rights reserved.

A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics☆

Abstract: A reaction-diffusion equation related to some mathematical models of gasless combustion of solid fuel is studied. A formal bifurcation analysis by B. J. Matkowsky and G. I. Sivashinsky (SIAM J. Appl. Math. 35 (1978), 465–478) shows that solutions demonstrate behavior typical for the Hopf bifurcation. A rigorous treatment of this phenomenon is developed. The problem is considered as an evolution equation in a Banach space. To circumvent difficulties involving a possible resonance with the continuous spectrum, appropriate weighted norms are introduced. A suitable version of the Hopf bifurcation theorem is developed and the existence of time periodic solutions is proved for values of the parameter near some critical value.

Dynamical portrait of a model of thermal instability: cascades, chaos, reversed cascades and infinite period bifurcations

Abstract: The paper presents results of numerical simulations on a model free boundary problem which is qualitatively equivalent to the free interface problems describing solid combustion and exothermic phase transitions. The model problem has been recently shown to exhibit transition to chaotic oscillations via a sequence of period doubling, assuming an Arrhenius type boundary kinetics. In the present paper we demonstrate that for a slightly different class of kinetics the behavior pattern, while retaining the above scenario, may undergo a drastic change. This behavior is characterized by slowly expanding oscillations followed by a powerful burst, after which the system returns to near equilibrium and the scenario is repeated periodically. As the bifurcation parameter approaches the stability threshold, the total period tends to infinity due to an increasingly prolonged “accumulation phase.” Additional scenarios corresponding to increasing supercriticality of the bifurcation parameter include finite period doublin...

Stability and attractors for the quasi-steady equation of cellular flames

Abstract: We continue to study a simple integro-differential equation: the Quasi-Steady equation (QS) of flame front dynamics. This second order quasi-linear parabolic equation with a non-local term is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. In [FGS03], where it was introduced, its well-posedness and proximity for finite time intervals to the KS equation in Sobolev spaces of periodic functions were established. Here we demonstrate that QS possesses a universal absorbing set, and a compact attractor. Furthermore we show that the attractor is of a finite Hausdorff dimension.We discuss relationship with the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations.

A free boundary problem modeling thermal instabilities: well-posedness

Abstract: In this paper, the authors analyze a simple free boundary model associated with solid combustion and some phase transition processes. There is strong evidence that this “one-phase” model captures many salient features of dynamical behavior of more realistic (and complicated) combustion and phase transition models. The main result is a global existence and uniqueness theorem whose proof is based on a uniform a priori estimate on the growth of solutions. The techniques employed are quite elementary and involve some maximum principle type estimates as well as parabolic potential estimates for the equivalent integral equation.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Traveling wave solutions of parabolic systems

Abstract: Part I. Stationary waves: Scalar equation Leray-Schauder degree Existence of waves Structure of the spectrum Stability and approach to a wave Part II. Bifurcation of waves: Bifurcation of nonstationary modes of wave propagation Mathematical proofs Part III. Waves in chemical kinetics and combustion: Waves in chemical kinetics Combustion waves with complex kinetics Estimates and asymptotics of the speed of combustion waves Asymptotic and approximate analytical methods in combustion problems (supplement) Bibliography.

Theoretical and numerical structure for unstable one-dimensional detonations

Abstract: The spatio-temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods. A new high resolution numerical method for computing unstable detonations is developed. This method combines the piecewise parabolic method (PPM) with conservative shock tracking and adaptive mesh refinement. A new nonlinear asymptotic theory for the spatio-temporal growth of instabilities is also developed. This asymptotic theory involves a nonclassical “Hopf bifurcation”, because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free-boundary problem for a nonlinear hyperbolic equation. An interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio-temporal mechanisms of nonlinear stability near the transition boundary; in particular, a quantitative-qualitative explanation is developed for the experimentally observed instabilities...

Theoretical and numerical structure of unstable detonations

Abstract: In this review, we emphasize the recent progress achieved in understanding the behaviour of unstable detonations through the interaction of theoretical, asymp­ totic, and numerical ideas. Theoretical predictions and numerical simulations for unstable one-dimensional detonations are described in detail as an important testing ground for the more complex ideas and phenomena that occur in several space dimensions. The linear and nonlinear theories for unstable detonations are generalized to several space dimensions. A new dedicated numerical method leads to better insight into the physical phenomena of unstable detonations, such as the nature of the turbulence generated in the wake of the front. Simplified models derived through asymptotics and comparisons between theoretical and numerical predictions are stressed throughout this paper.