Abstract
The dynamic behavior of thermodynamic system, described by one order parameter and one control parameter, in a small neighborhood of ordinary and bifurcation equilibrium values of the system parameters is studied. Using the general methods of investigating the branching (bifurcations) of solutions for nonlinear equations, we performed an exhaustive analysis of the order parameter dependences on the control parameter in a small vicinity of the equilibrium values of parameters, including the stability analysis of the equilibrium states, and the asymptotic behavior of the order parameter dependences on the control parameter (bifurcation diagrams). The peculiarities of the transition to an unstable state of the system are discussed, and the estimates of the transition time to the unstable state in the neighborhood of ordinary and bifurcation equilibrium values of parameters are given. The influence of an external field on the dynamic behavior of thermodynamic system is analyzed, and the peculiarities of the system dynamic behavior are discussed near the ordinary and bifurcation equilibrium values of parameters in the presence of external field. The dynamic process of magnetization of a ferromagnet is discussed by using the general methods of bifurcation and stability analysis presented in the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mullin, J.W.: Crystallization. Butterworth-Heinemann, Oxford (2001)
Nicolis, G., Nicolis, C.: Kinetics of phase transitions in the presence of an intermediate metastable state: a generic model. Physica A 351, 22–39 (2005)
Solé, R.V.: Phase Transitions. Princeton University Press, Princeton (2011)
Mihara, A.: Thermodynamic geometry and critical aspects of bifurcations. Phys. Rev. E 94, 012144 (2016)
Vainberg, M.M., Trenogin, V.A.: Theory of Branching of Solutions of Non-linear Equations (Monographs and textbooks on Pure and Applied Mathematics). Wolters-Noordhoff B.V., Groningen (1974)
Pontryagin, L.S.: Ordinary Differential Equations (Adiwes International Series in Mathematics). Pergamon Press, London (1962)
Paladi, F.: Effects of asymmetry and external field on phase transitions in the presence of an intermediate metastable state. Physica A 389, 1986–1992 (2010)
Barsuk, A.A., Gamurari, V., Gubceac, G., Paladi, F.: Bifurcation and stability analysis for phase transitions in the presence of an intermediate state: a general solution. Physica A 392, 1931–1945 (2013)
Barsuk, A.A., Paladi, F.: Generalized parametric model for phase transitions in the presence of an intermediate metastable state and its application. Physica A 487, 74–92 (2017)
Fikhtengol’ts, G.M.: The Fundamentals of Mathematical Analysis (International Series of Monographs on Pure and Applied Mathematics). Pergamon Press, London (1965)
Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory. Springer, New York (1989)
Nicolis, G., Nicolis, C.: Foundations of Complex Systems: Nonlinear Dynamics, Statistical Physics, Information and Prediction. World Scientific Publishing, Singapore (2007)
Haken, H.: Synergetics: An Introduction Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology. Springer, New York (1978)
Acknowledgements
Authors gratefully acknowledge support provided by the Moldova State University through the Grant No. 15.817.02.29F, and the Academy of Sciences of Moldova through the STCU Project No. 6219. Authors are also indebted to anonymous referees for constructive comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barsuk, A.A., Paladi, F. Bifurcation and Stability Analysis of the Equilibrium States in Thermodynamic Systems in a Small Vicinity of the Equilibrium Values of Parameters. J Stat Phys 171, 361–381 (2018). https://doi.org/10.1007/s10955-018-2011-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-018-2011-3