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Introduction: Theory of hybrid dynamical systems and its applications to biological and medical systems. (English) Zbl 1211.37099

Summary: In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on-off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.

MSC:

37N25 Dynamical systems in biology
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[1] IBM J RES DEV 47 pp 5– (2003)
[2] PROC. IEEE 90 pp 919– (2002)
[3] COMMUN. ACM 38 pp 103– (1995)
[4] PHYS. LETT. A 144 pp 333– (1990)
[5] 71 pp 2782– (1993)
[6] McTiernan, Nature reviews. Cancer 8 (3) pp 227– (2008) · doi:10.1038/nrc2329
[7] PHYS TODAY 39 pp 38– (1986)
[8] Battogtokh, Physical Review Letters 96 (14) pp 148102– (2006) · doi:10.1103/PhysRevLett.96.148102
[9] SIAM J COMPUT 26 pp 1411– (1997)
[10] ERGOD THEORY DYN SYST 15 pp 821– (1995) · Zbl 0836.58026 · doi:10.1017/S0143385700009652
[11] THEOR COMPUT SCI 168 pp 417– (1996)
[12] THEOR COMPUT SCI 138 pp 67– (1995)
[13] IEEE TRANS AUTOM CONTROL 43 pp 31– (1998)
[14] Bruchovsky, Cancer Research 50 (8) pp 2275– (1990)
[15] Journal of Theoretical Biology 2 pp 204– (1961)
[16] PHIL TRANS R SOC A 368 pp 5071– (2010) · Zbl 1211.37104 · doi:10.1098/rsta.2010.0171
[17] New Scientist (1971) 2082 pp 30– (1997)
[18] IEEE TRANS CIRCUITS SYST I FUNDAM THEORY APPL 48 pp 308– (2001)
[19] PHYS REV E 70 pp 011909– (2004)
[20] IEEE TRANS CIRCUITS SYST 33 pp 1072– (1986)
[21] Meccanica 41 pp 241– (2006)
[22] PROC R SOC LOND A 400 pp 97– (1985)
[23] PHIL TRANS R SOC A 368 pp 4915– (2010) · Zbl 1211.37063 · doi:10.1098/rsta.2010.0198
[24] 19 pp 25– (1978)
[25] Tsukada, Neural networks : the official journal of the International Neural Network Society 9 (8) pp 1303– (1996) · Zbl 0899.92008 · doi:10.1016/S0893-6080(96)00054-8
[26] IEEE CONTROL SYST MAG 29 pp 28– (2009)
[27] ILLINOIS J MATH 44 pp 465– (2000)
[28] IEEE TRANS COMMUN 35 pp 481– (1987)
[29] 18 pp 3789– (2008)
[30] J MATH KYOTO UNIV 22 pp 155– (1982)
[31] PHIL TRANS R SOC A 368 pp 4937– (2010) · Zbl 1211.93074 · doi:10.1098/rsta.2010.0187
[32] Hirata, Journal of Theoretical Biology 264 (2) pp 517– (2010) · doi:10.1016/j.jtbi.2010.02.027
[33] Hodgkin, The Journal of Physiology 117 (4) pp 500– (1952) · doi:10.1113/jphysiol.1952.sp004764
[34] Physica. D 237 pp 1215– (2008)
[35] J NONLINEAR SCI 18 pp 593– (2008)
[36] SYST CONTROL INFORM 51 pp 230– (2007)
[37] PHIL TRANS R SOC A 368 pp 4977– (2010) · Zbl 1211.37107 · doi:10.1098/rsta.2010.0176
[38] IRE TRANS SPACE ELECTRON TELEMETR 8 pp 204– (1962)
[39] PHIL TRANS R SOC A 368 pp 5061– (2010) · Zbl 1211.37108 · doi:10.1098/rsta.2010.0130
[40] Physical Review Letters 50 pp 1637– (1983)
[41] PROC SYMP ON NONLINEAR CIRCUIT ANALYSIS VI NEW YORK NY APRIL VI pp 273– (1956)
[42] KOLATA, Science 196 (4287) pp 287– (1977) · doi:10.1126/science.196.4287.287
[43] 82 pp 985– (1975)
[44] PHYS. LETT. A 123 pp 162– (1987)
[45] May, Nature; Physical Science (London) 261 (5560) pp 459– (1976) · Zbl 1369.37088 · doi:10.1038/261459a0
[46] BULL MATH BIOPHY 5 pp 115– (1943)
[47] PHYS. LETT. A 169 pp 41– (1992)
[48] Moore, Physical Review Letters 64 (20) pp 2354– (1990) · Zbl 1050.37510 · doi:10.1103/PhysRevLett.64.2354
[49] 4 pp 199– (1991)
[50] THEOR COMPUT SCI 162 pp 23– (1996)
[51] Nagumo, Kybernetik 10 (3) pp 155– (1972) · Zbl 0235.92001 · doi:10.1007/BF00290514
[52] PHIL TRANS R SOC A 368 pp 5013– (2010) · Zbl 1211.37113 · doi:10.1098/rsta.2010.0173
[53] Ott, Physical Review Letters 64 (11) pp 1196– (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[54] PHIL TRANS R SOC A 368 pp 4961– (2010) · Zbl 1211.37115 · doi:10.1098/rsta.2010.0139
[55] PHIL TRANS R SOC A 368 pp 5087– (2010) · Zbl 1211.93099 · doi:10.1098/rsta.2010.0134
[56] Sato, Neural Computation 19 (12) pp 3335– (2007) · Zbl 1135.62089 · doi:10.1162/neco.2007.19.12.3335
[57] 65 pp 579– (1991)
[58] Scheffer, Nature; Physical Science (London) 461 (7260) pp 53– (2009) · doi:10.1038/nature08227
[59] Shimada, Mathematical biosciences 214 (1-2) pp 134– (2008) · Zbl 1143.92023 · doi:10.1016/j.mbs.2008.03.001
[60] PHIL TRANS R SOC A 368 pp 4995– (2010) · Zbl 1211.37117 · doi:10.1098/rsta.2010.0211
[61] J NONLINEAR SCI 9 pp 255– (1999)
[62] MATH MAG 81 pp 291– (2008)
[63] Europhysics Letters 66 pp 28– (2004)
[64] DISCRETE CONTIN DYN SYST 13 pp 515– (2005)
[65] PHIL TRANS R SOC A 368 pp 5045– (2010) · Zbl 1211.93069 · doi:10.1098/rsta.2010.0220
[66] DYNAMICAL SYSTEMS AND BIFURCATIONS 1125 pp 90– (1984)
[67] Physica. D 237 pp 2616– (2008)
[68] PHIL TRANS R SOC A 368 pp 5029– (2010) · Zbl 1211.37118 · doi:10.1098/rsta.2010.0221
[69] MATH MODELS METHODS APPL SCI 19 pp 2177– (2009)
[70] PHYS LETT 85A pp 4– (1981)
[71] IEEE TRANS AUTOM CONTROL 11 pp 161– (1966)
[72] IEEE TRANS AUTOM CONTROL 43 pp 461– (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.