KRASNYUK I.B.
ABSTRACT
The paper deals with a linear hyperbolic equation for a nonconserved order parameter in the diblock copolymer system with nonlinear differential boundary conditions which models the evolution of an ordered phase in a nonordered phase (in the melt). It is shown that for the ideal polymer systems the asymptotic periodic piecewise constant distributions of the order parameter with a finite or infinite set of points of discontinuities on a period appear in the melt (when bulk
perturbations in the melt are small and, hence, surface perturbations are dominating). For the nonideal systems there are limit quasi-periodic distributions that admits the period doubling bifurcations as the problem parameters are changing. Particularly, these distributions are the elements of the strange unchaotic attractor.
KEYWORDS
diblock copolymers, strange nonchaotic attractor, difference equation with quasiperiodic perturbations, period-doubling bifurcations on the torus.
REFERENCES
1. Krasnyuk, I.B., Taranets, R.M., Stefanovich, L.I. and Yurchenko, V.M. (2009), “Cascade process of nanostructure formation in the crystallization of the melt”, Materialovedenie, no. 7, pp. 8-15.
2. Binder, K., Frisch, H.L. and Stepanov, S. (1997), “Surface Effects on Block Copolymer Melts Above the Order-Disorder Transition: Linear Theory of Equilibrium Properties and the Kinetics of Surface Induced Ordering”, J. Phys. II France, no. 7, pp. 1353-1378.
3. Maxwell, J. C. (1879), “On Stresses in Rarified Gases Arising from Inequalities of Temperature”, Phil. Trans. R. Soc., no. 170, pp. 231-256.
4. Levanov, E.I. and Sotskiy, E.N. (1997), “Heat transfer with heat flux relaxation”, Matematicheskoe modelirovanie uravneniy matematicheskoy fiziki, pp. 155-190.
5. Krasnyuk, I.B. (1994), “Singularity theory in problems of heat transfer: solutions of relaxation type”, Inzh.-fiz. zhurn., no. 1-2, pp. 162-167.
6. Brandenburg, A., Kаpylа, P.J. and Mohammed, A. (2004), “Non-Fickian Diffusion and tau approximation From Numerical Turbulence”, Physics Fluids, Vol. 16, no. 4, pp. 1020-1027.
7. Krasnyuk, I.B. and Taranets, R.M. (2009),”The Spatiotemporal Oscillations of Order Parameter for Isothermal Model of the Surface-Directed Spinodal Decomposition ib Bounded Binary Mixtures”, Hindavi Publishing Corporation Research Letters in Physics, Volume 2009, Article ID 250203, 4 pages doi: 10.//55/2009/250203.
8. Krasnyuk, I.B. and Taranets, R.M. (2011), “The Spatial-Temporal Lamellar Structures in the Confined Ideal Polymer Blends”, J. Stat. Phys., no. 6, pp. 1485-1498.
9. Sharkovskiy, A.N., Maystrenko, Yu.L. and Romanenko, Ye.Yu. (1986), Raznostnye uravneniya i ikh prilozheniya [Difference equations and their applications], Naukova dumka, Kiev, Ukraine.
10. Krasnyuk, I.B., Taranets, R.M. and Yurchenko, V.M. (2010), “Pulse structures lamellar type in limited polymer systems”, Matematicheskoe modelirovanie, Vol. 22, no. 12, pp. 65-81.
11. Kuznetsov, S.P. (1984), “About the impact of periodic external perturbation to the system, showing the transition order - chaos via period-doubling bifurcation”, Pisma ZHETF, Vol. 39, no. 3, pp. 113-116.
12. Krasnyuk, I.B., Melnik, T.N. and Yurchenko, V.M. (2009), “Spatio-temporal structure of the lamellar type in limited polymer blends”, Uchenyye zapiski Tavricheskogo universiteta im. V.I. Vernadskogo. Seriya Fizika, Vol. 22 (61), no. 1, pp. 182-193.
13. Feudel, U., Kuznetsov, S. and Pikovsky, A. (2006), Strange Nonchaotic Attractors: Dynamics Between Order and Chaos in Quasiperiodically Forced Systems, World Scientific Publishing Co. Pte. Ltd.
14. Grebodi, C., Ott, E., Peliken, S. and Jork, J.A. (1964), “Strange Attractors that are not Chaotic”, Physica D, Vol. 13, no. 1-2, pp. 261-268.
15. Boylo, I.B., Melnik, T.N., Krasnyuk, I.B. and Yurchenko, V.M. (2011), “Stochastic nanostructures in polymer physics”, Neobratimyye protsessy v prirode i tekhnike, pp. 259-262.
16. Fredrickson, G.H. (1987), “Surface Ordering Phenomena in Block Copolymer Melts”, Macromolecules, no. 20, pp. 2535-2542.
17. De Zhen, P. (1982), Idei skeylinga v fizike polimerov [Scaling ideas in polymer physics], Mir, Moscow, Russia.
18. Sharkovskiy, A.N. (1964), “Coexistence of cycles of a continuous mapping line onto itself “, Ukr. mat. zhurn., Vol. 16, no. 1, pp. 61-71.
19. Sharkovskiy, A.N., Maystrenko, Yu.L. and Romanenko, E.Yu. (1986), Raznostnye uravneniya i ikh prilozheniya [Difference equations and their applications], Naukova Dumka, Kiev, Ukraine.
20. Sharkovskiy, A.N., Kolyada, S.F., Sivak, A.G. and Fedorenko, V.V. (1986), Dinamika odnomernykh otobrazheniy [The dynamics of one-dimensional mappings], Naukova dumka, Kiev, Ukraine.
21. Ivankov, N.Yu. (1997), “Scaling properties of the parameter space logistic mapping under external periodic force”, Izv. vuzov. Prikladnaya nelineynaya dinamika, Vol. 5, no. 2-3, pp. 118-127.