PARAMETRIC REPRESENTATION AND BIFURCATION ANALYSIS OF THE CUBIC EQUATION SOLUTIONS WITH APPLICATION TO THE PHASE TRANSITIONS

Alexandr A. BARSUK, Florentin PALADI

Abstract


Real solutions representation for the cubic equation with real coefficients in a parametric form is given. The dependence of the solutions on the equation coefficients and the bifurcation conditions for these solutions are studied. In the parametric space regions with one and three real solutions are considered. Using parametric representations of the cubic equation solutions, the stability and bifurcation conditions of the equilibrium states for the thermodynamic systems described by the Landau-type kinetic potential are analyzed.

 

REPREZENTAREA PARAMETRICĂ ȘI ANALIZA BIFURCAȚIONALĂ A

SOLUȚIILOR ECUAȚIEI CUBICE CU APLICARE LA TRANZIȚIILE DE FAZĂ

Este expusă reprezentarea soluțiilor reale pentru ecuația cubică cu coeficienți reali într-o formă parametrică. Sunt studiate dependența soluțiilor de coeficienții ecuației și condițiile de bifurcație pentru aceste soluții. În spațiul para­metric sunt considerate regiuni cu una și trei soluții reale. Folosind reprezentările parametrice ale soluțiilor ecuației cubice, sunt analizate condițiile de stabilitate și de bifurcație ale stărilor de echilibru pentru sistemele termodinamice descrise de potențialul cinetic de tip Landau.


Keywords


phase transitions, metastable state, bifurcation analysis

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References


BANICHUK, N., BARSUK, A., JERONEN, J., NEITTAANMÄKI, P., TUOVINEN, T. On some bifurcation analysis techniques for continuous systems. In: Rakenteiden Mekaniikka (Journal of Structural Mechanics), 2016, vol.49, no.2, p.52-68. ISSN: 0783-6104

FIKHTENGOL`TS, G.M. The fundamentals of mathematical analysis. Oxford: Pergamon Press, 1965. 520 p. ISBN: 978-0080134734

BARSUK, A., PALADI, F. Generalized parametric model for phase transitions in the presence of an intermediate metastable state and its application. In: Physica A: Statistical Mechanics and its Applications, 2017, vol. 487, p.74-92. ISSN: 0378-4371

NICOLIS, G., NICOLIS, C. Kinetics of phase transitions in the presence of an intermediate metastable state: a generic model. In: Physica A: Statistical Mechanics and its Applications, 2005, vol.351, issue 1, p.22-39. ISSN: 0378-4371

PALADI, F., OGUNI, M., Anomalous generation and extinction of crystal nuclei in non-equilibrium supercooled liquid o-benzylphenol. In: Physical Review B., 2002, vol.65, no.14, 144202. 6 p. ISSN: 2469-9950

PALADI, F., OGUNI, M., Generation and extinction of crystal nuclei in an extremely non-equilibrium glassy state of salol. In: Journal of Physics: Condensed Matter., 2003, vol.15, no.23, p.3909-3917. ISSN: 0953-8984

TOMITAKA, S., MIZUKAMI, M., PALADI, F., OGUNI, M. Thermal and dielectric studies of 2,2'-dihydroxyben-zophenone: progress of crystal nucleation and growth below the glass transition temperature. In: Journal of Thermal Analysis and Calorimetry, 2005, vol.81, no.3, p.637-643. ISSN: 1388-6150

PALADI, F. Effects of asymmetry and external field on phase transitions in the presence of an intermediate metastable state. In: Physica A: Statistical Mechanics and its Applications. 2010. vol.389, p.1986-1992. ISSN: 0378-4371

PALADI, F. On the probabilistic approach to heterogeneous structure interactions in agent-based computational models. In: Applied Mathematics and Computation., 2013, vol.219, no.24, p.11430-11437. ISSN: 0096-3003

BARSUK, A., GAMURARI, V., GUBCEAC, G., PALADI, F. Bifurcation and stability analysis for phase transitions in the presence of an intermediate state: a general solution. In: Physica A: Statistical Mechanics and its Applications, 2013, vol.392, p.1931-1945. ISSN: 0378-4371

BARSUK, A., PALADI, F. Bifurcation and stability analysis of the equilibrium states in thermodynamic systems in a small vicinity of the equilibrium values of parameters. In: Journal of Statistical Physics, 2018, vol.171, p.361-381. ISSN: 0022-4715

BARSUK, A., PALADI, F. Equilibrium states of thermodynamic systems in a small vicinity of the equilibrium values of parameters: Bifurcation, stability and sensitivity analyses. In: Physica A: Statistical Mechanics and its Applications, 2019, vol.527. Article 121303. 20 p. ISSN: 0378-4371


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