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Fuzzy dynamics as an alternative to statistical mechanics. (English) Zbl 0940.70015

Summary: We propose a description of continuous evolution of systems with uncertainty based on min or product \(t\)-norms of fuzzy logic. We show that the dynamics has a Hamiltonian form in the extended state space composed of “physical” and “information” components, if the min \(t\)-norm is used. This description is useful when available information on a system behavior is mainly ordinal and not numerical. It is shown, also, that if the product \(t\)-norm is used, the dynamics on a continuous universe is similar to the stochastic processes dynamics, but a probability distribution is replaced by a possibility density. We show that such replacement leads to a principal extension of the stochastic process theory, and the corresponding differential equation of evolution for the possibility density covers most basic evolution equations in physics.

MSC:

70H99 Hamiltonian and Lagrangian mechanics
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
03B52 Fuzzy logic; logic of vagueness

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[1] Arnold, V. I., Mathematical Methods of Classical Mechanics, (Graduate Texts in Mathematics (1978), Springer: Springer New York) · Zbl 0386.70001
[2] Aubin, J.-P., Fuzzy differential inclusions, Prob. Control Inform. Theory, 19, 1, 55-67 (1990) · Zbl 0718.93039
[3] T. Biglic, I.B. Turksen, Measurement of membership function on the theoretical and empirical work, Handbook of Fuzzy Systems, vol. 1, Foundation, NY, to appear.; T. Biglic, I.B. Turksen, Measurement of membership function on the theoretical and empirical work, Handbook of Fuzzy Systems, vol. 1, Foundation, NY, to appear.
[4] Butnariu, D., On Triangular Norm-Based Propositional Fuzzy Logic (1994), preprint
[5] Courant, R.; Hilbert, D., Methods of Mathematical Physics (1953), Interscience Publ: Interscience Publ New York · Zbl 0729.35001
[6] Forrester, J. W., Industrial Dynamics (1961), MIT Press: MIT Press Cambridge, MA
[7] Fibiger, H. C., Mesolimbic dopmine: an analysis of its role in motivated behavior, Sen. Neurosci., 5, 321-327 (1993)
[8] Friedman, Y.; Sandler, U., A new approach to fuzzy dynamics, (Proc. of the 12th IAPR Internat. Conf. Jerusalem. Proc. of the 12th IAPR Internat. Conf. Jerusalem, Israel (1994)), 615-618
[9] Friedman, Y.; Sandler, U., Evolution of systems under fuzzy dynamics laws, Fuzzy Sets and Systems, 84, 61-74 (1996) · Zbl 0911.93040
[10] Y. Friedman, U. Sandler, Dynamics of fuzzy systems, Chaos Theory Appl., to appear.; Y. Friedman, U. Sandler, Dynamics of fuzzy systems, Chaos Theory Appl., to appear. · Zbl 0911.93040
[11] Y. Friedman, U. Sandler, to be published.; Y. Friedman, U. Sandler, to be published.
[12] Grabisch, M.; Nguyen, H. T.; Walker, E. A., Fundamentals of Uncertainty Caculi with Applications to Fuzzy Inference (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[13] Hebb, D. O., Essay on Mind (1980), Lawrence-Erlbaum Assc: Lawrence-Erlbaum Assc Hillsdale, NJ
[14] Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35, 389-396 (1990) · Zbl 0696.34005
[15] Kotlar, B. I., The Mechanism of Conditioning (1977), Moscow university
[16] Kloeden, P. E., Fuzzy dynamical systems, Fuzzy Sets and Systems, 7, 275-296 (1982) · Zbl 0509.54040
[17] Levary, R., Systems dynamics with fuzzy logic, Internat. J. System Sci., 21, 8, 1701-1707 (1990)
[18] Nazaroff, G. J., Fuzzy topological polysystems, J. Math. Anal. Appl., 41, 478-485 (1973) · Zbl 0332.94001
[19] Pavlov, I. P., (Lectures on Conditioned Reflexes (1928), Intern. Press: Intern. Press New York)
[20] Sandler, U., A new fuzzy dynamics, (Lecture on 10th Israeli Conf. on Artificial Intelligence Computer Vision and Neural Networks. Lecture on 10th Israeli Conf. on Artificial Intelligence Computer Vision and Neural Networks, 10-13 October, Tel-Aviv, Israel (1993)) · Zbl 1017.91088
[21] Sandler, U., A neural networks with multi-neuron, Neurocomput., 14, 41-62 (1997)
[22] Sandler, Yu. M.; Sergeev, V. M., Uncertainty principle in non-quantum spherical modification of the Hopfield model interacting with classical physical reality, Phys. Lett. A, 133, 4-6 (1988)
[23] Srinivasan, S. K.; Mehata, K. M., Stochastic Processes (1978), McGraw-Hill Publishing: McGraw-Hill Publishing New York · Zbl 0431.60001
[24] Teodorescu, H. N.I.; Brezulian, A., Chaos and clustering in fuzzy systems networks, Chaotic Theory Appl., 2, 17-44 (1997)
[25] Tsitolovsky, L. E., A model of motivation with chaotic neuronal dynamics, BioSystems, 5, 2, 205-232 (1997)
[26] Zadeh, L.; Yager, R. R., (An Introduction to Fuzzy Logical Applications in Intelligent Systems, vol. 1 (1992), Kluwer: Kluwer Boston) · Zbl 0755.68018
[27] Zaslavsky, G. M.; Sagdeev, R. Z., Introduction to Nonlinear Physics (1988), Nauka: Nauka Moscow · Zbl 0709.58003
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