zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Dynamical models of happiness with fractional order. (English) Zbl 1221.93234
Summary: This present study focuses on a dynamical model of happiness described through fractional-order differential equations. By categorizing people of different personality and different impact factor of memory (IFM) with different set of model parameters, it is demonstrated via numerical simulations that such fractional-order models could exhibit various behaviors with and without external circumstance. Moreover, control and synchronization problems of this model are discussed, which correspond to the control of emotion as well as emotion synchronization in real life. This study is an endeavor to combine the psychological knowledge with control problems and system theories, and some implications for psychotherapy as well as hints of a personal approach to life are both proposed.

93D15Stabilization of systems by feedback
91E10Cognitive psychology
34A08Fractional differential equations
Full Text: DOI
[1] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[2] Hilfer, R.: Applications of fractional calculus in physics, (2001) · Zbl 0998.26002
[3] Jenson, V. G.; Jeffreys, G. V.: Mathematical methods in chemical engineering, (1977) · Zbl 0413.00002
[4] Sun, H. H.; Abdelwahad, A. A.; Onaral, B.: Linear approximation of transfer function with a pole of fractional order, IEEE trans auto control 29, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[5] Ichise, M.; Nagayanagi, T.; Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode process, J electroanal chem 33, 253-265 (1971)
[6] Heaviside, O.: Electromagnetic theory, (1971) · Zbl 30.0801.03
[7] Bagley, R. L.; Calico, R. A.: Fractional order state equations for the control of visco-elastically damped structures, J guid control dyn 14, 304-311 (1991)
[8] Kusnezov, D.; Bulgac, A.; Dang, G. D.: Quantum Lévy processes and fractional kinetics, Phys rev lett 82, 1136-1139 (1999)
[9] Laskin, N.: Fractional market dynamics, Physica A 278, 482-492 (2000)
[10] Ei-Sayed, A. M. A.: Fractional order diffusion wave equation, Internat J theoret phys 35, 311-322 (1996) · Zbl 0846.35001 · doi:10.1007/BF02083817
[11] Torvik, P. J.; Bagley, R. L.: On the appearance of the fractional derivative in the behavior of real materials, Trans ASME 51, 294-298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615
[12] Tarasov, V. E.; Zaslavsky, G. M.: Fractional dynamics of systems with long-range interaction, Commun nonlinear sci numer simul 11, No. 8, 885-898 (2006) · Zbl 1106.35103 · doi:10.1016/j.cnsns.2006.03.005
[13] Korabel, N.; Zaslavsky, G. M.; Tarasov, V. E.: Coupled oscillators with power-law interaction and their fractional dynamics analogues, Commun nonlinear sci numer simul 12, No. 8, 1405-1417 (2007) · Zbl 1118.35345 · doi:10.1016/j.cnsns.2006.03.015
[14] Da Graça Marcos, Maria; Fernando, B. M. Duarte; Machado, J. A. Tenreiro: Fractional dynamics in the trajectory control of redundant manipulators, Commun nonlinear sci numer simul 13, No. 9, 1836-1844 (2008)
[15] Isabel, S.; Jesus, I. S.; Machado, J. A. Tenreiro: Implementation of fractional-order electromagnetic potential through a genetic algorithm, Commun nonlinear sci numer simul 14, No. 5, 1838-1843 (2009)
[16] Hang, X.: Analytical approximations for a population growth model with fractional order, Commun nonlinear sci numer simul 14, No. 5, 1978-1983 (2009) · Zbl 1221.65210 · doi:10.1016/j.cnsns.2008.07.006
[17] Sprott, J. C.: Dynamical models of love, Nonlinear dyn psychol life sci 8, No. 3, 303-313 (2004)
[18] Sprott, J. C.: Dynamical models of happiness, Nonlinear dyn psychol life sci 9, No. 1, 23-36 (2005)
[19] Ahmad, W. M.; Ei-Khazali, R.: Fractional-order dynamical models of love, Chaos soliton fract 33, 1367-1375 (2007) · Zbl 1133.91539 · doi:10.1016/j.chaos.2006.01.098
[20] Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor -- corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[21] Diethelm, K.; Ford, N. J.; Freed, A. D.: Detailed error analysis for a fractional Adams method, Numer algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[22] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B.: Fractal system as represented by singularity function, IEEE trans automat contr 37, 1465-1470 (1992) · Zbl 0825.58027 · doi:10.1109/9.159595
[23] Tavazoei, M. S.; Haeri, M.: Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear anal 69, 1299-1320 (2008) · Zbl 1148.65094 · doi:10.1016/j.na.2007.06.030
[24] Suh, E.; Diener, E.; Fujita, F.: Events and subjective well-being: only recent events matter, J pers soc psychol 70, 1091-1102 (1996)
[25] Matignon D. Stability results for fractional differential equations with applications to control processing. Computational engineering in systems applications, vol. 2. Lille (France): IMACS-SMC; 1996. p. 963 -- 8.
[26] Pecora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Phys rev lett 64, 821-824 (1991) · Zbl 0938.37019
[27] Curran, P. F.; Suykens, J. A. K.; Chua, L. O.: Absolute stability theory and master -- slave synchronization, Int J bifurcat chaos 7, 2891-2896 (1997) · Zbl 0911.93044 · doi:10.1142/S0218127497001977