| Title:
|
Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator (English) |
| Author:
|
Pimenov, Alexander |
| Author:
|
Rachinskii, Dmitrii |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
285-298 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed. (English) |
| Keyword:
|
robust homoclinic |
| Keyword:
|
orbit Preisach operator |
| Keyword:
|
operator-differential equations |
| Keyword:
|
predator-prey model |
| MSC:
|
37L15 |
| MSC:
|
47J40 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 06362259 |
| idMR:
|
MR3238840 |
| DOI:
|
10.21136/MB.2014.143855 |
| . |
| Date available:
|
2014-07-14T08:31:13Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143855 |
| . |
| Reference:
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[1] Appelbe, B., Flynn, D., McNamara, H., O'Kane, P., Pimenov, A., Pokrovskii, A., skii, D. Rachin-\allowbreak, Zhezherun, A.: Rate-independent hysteresis in terrestrial hydrology.Control Systems Magazine, IEEE 29 (2009), 44-69 DOI 10.1109/MCS.2008.930923. 10.1109/MCS.2008.930923 |
| Reference:
|
[2] Appelbe, B., Rachinskii, D., Zhezherun, A.: Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis.Physica B: Condensed Matter 403 (2008), 301-304 DOI 10.1016/j.physb.2007.08.034. 10.1016/j.physb.2007.08.034 |
| Reference:
|
[3] Bertotti, G., Mayergoyz, I. D., Serpico, C.: Nonlinear magnetization dynamics. Switching and relaxation phenomena.The Science of Hysteresis II. Physical Modeling, Micromagnetics, and Magnetization Dynamics G. Bertotti, I. D. Mayergoyz Elsevier, Amsterdam 435-565 (2006). Zbl 1148.78001, MR 2307930 |
| Reference:
|
[4] Bertotti, G., Mayergoyz, I., eds.: The Science of Hysteresis.Elsevier, Amsterdam (2006). MR 2307931 |
| Reference:
|
[5] Brokate, M., Pokrovskii, A., Rachinskii, D.: Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis.J. Math. Anal. Appl. 319 (2006), 94-109. Zbl 1111.34035, MR 2217849, 10.1016/j.jmaa.2006.02.060 |
| Reference:
|
[6] Brokate, M., Pokrovskii, A., Rachinskii, D., Rasskazov, O.: Differential equations with hysteresis via a canonical example.The Science of Hysteresis I. Mathematical Modeling and Applications G. Bertotti, I. D. Mayergoyz Elsevier, Amsterdam 125-291 (2006). Zbl 1142.34026, MR 2307931 |
| Reference:
|
[7] Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions.Applied Mathematical Sciences 121 Springer, New York (1996). Zbl 0951.74002, MR 1411908, 10.1007/978-1-4612-4048-8_5 |
| Reference:
|
[8] Chiorino, G., Auger, P., Chassé, J.-L., Charles, S.: Behavioral choices based on patch selection: a model using aggregation methods.Math. Biosci. 157 (1999), 189-216. MR 1686474, 10.1016/S0025-5564(98)10082-2 |
| Reference:
|
[9] Cross, R., McNamara, H., Pokrovskii, A., Rachinskii, D.: A new paradigm for modelling hysteresis in macroeconomic flows.Physica B: Condensed Matter 403 (2008), 231-236 DOI 10.1016/j.physb.2007.08.017. 10.1016/j.physb.2007.08.017 |
| Reference:
|
[10] Davino, D., Krejčí, P., Visone, C.: Fully coupled modeling of magneto-mechanical hysteresis through `thermodynamic' compatibility.Smart Materials and Structures 22 (2013), 14 pages DOI 10.1088/0964-1726/22/9/095009. 10.1088/0964-1726/22/9/095009 |
| Reference:
|
[11] Diamond, P., Kuznetsov, N., Rachinskii, D.: On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity.J. Differ. Equations 175 (2001), 1-26. Zbl 0984.34029, MR 1849221, 10.1006/jdeq.2000.3916 |
| Reference:
|
[12] Diamond, P., Rachinskii, D., Yumagulov, M.: Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity.Nonlinear Anal., Theory Methods Appl. 42 (2000), 1017-1031. Zbl 0963.34034, MR 1780452 |
| Reference:
|
[13] Eleuteri, M., Kopfová, J., Krejčí, P.: Magnetohydrodynamic flow with hysteresis.SIAM J. Math. Anal. 41 (2009), 435-464. MR 2507458, 10.1137/080718383 |
| Reference:
|
[14] Guardia, M., Seara, T. M., Teixeira, M. A.: Generic bifurcations of low codimension of planar Filippov systems.J. Differ. Equations 250 (2011), 1967-2023. Zbl 1225.34046, MR 2763562, 10.1016/j.jde.2010.11.016 |
| Reference:
|
[15] Harrison, G. W.: Multiple stable equilibria in a predator-prey system.Bull. Math. Biol. 48 (1986), 137-148. Zbl 0585.92023, MR 0845634, 10.1007/BF02460019 |
| Reference:
|
[16] Hodgkin, A. L., Huxley, A. F.: A quantitative description of membrane current and its application to conduction and excitation in nerve.J. Physiol. 117 (1952), 500-544 DOI 10.1007/BF02459568. 10.1113/jphysiol.1952.sp004764 |
| Reference:
|
[17] Krasnosel'skii, A., Rachinskii, D.: On a bifurcation governed by hysteresis nonlinearity.NoDEA, Nonlinear Differ. Equ. Appl. 9 (2002), 93-115. Zbl 1013.34036, MR 1891697, 10.1007/s00030-002-8120-2 |
| Reference:
|
[18] Krasnosel'skij, A., Rachinskij, D. I.: On the continua of cycles in systems with hysteresis.Dokl. Math. 63 (2001), 339-344. Zbl 1052.34052 |
| Reference:
|
[19] Krasnosel'skij, M. A., Pokrovskij, A. V.: Systems with Hysteresis. Translated from the Russian.Springer, Berlin (1989). |
| Reference:
|
[20] Krejčí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations.GAKUTO International Series. Mathematical Sciences and Applications 8 Gakkotosho, Tokyo (1996). MR 2466538 |
| Reference:
|
[21] Krejčí, P.: On Maxwell equations with the Preisach hysteresis operator: The one-dimensional time-periodic case.Apl. Mat. 34 (1989), 364-374. Zbl 0701.35098, MR 1014077 |
| Reference:
|
[22] Krejčí, P.: Resonance in Preisach systems.Appl. Math. 45 (2000), 439-468. Zbl 1010.34038, MR 1800964, 10.1023/A:1022333500777 |
| Reference:
|
[23] Krejčí, P., O'Kane, J. P., Pokrovskii, A., Rachinskii, D.: Stability results for a soil model with singular hysteretic hydrology.Journal of Physics: Conference Series 268 (2011), 19 pages DOI 10.1088/1742-6596/268/1/012016. 10.1088/1742-6596/268/1/012016 |
| Reference:
|
[24] Krejčí, P., O'Kane, J. P., Pokrovskii, A., Rachinskii, D.: Properties of solutions to a class of differential models incorporating Preisach hysteresis operator.Physica D. Nonlinear Phenomena 241 (2012), 2010-2028. MR 2994340, 10.1016/j.physd.2011.05.005 |
| Reference:
|
[25] Kuhnen, K., Krejčí, P.: Compensation of complex hysteresis and creep effects in piezoelectrically actuated systems---a new Preisach modeling approach.IEEE Trans. Automat. Control 54 (2009), 537-550. MR 2191546, 10.1109/TAC.2009.2012984 |
| Reference:
|
[26] Kuznetsov, Yu. A.: Elements of Applied Bifurcation Theory.Applied Mathematical Sciences 112 Springer, New York (2004). Zbl 1082.37002, MR 2071006, 10.1007/978-1-4757-3978-7 |
| Reference:
|
[27] Kuznetsov, Yu. A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems.Int. J. Bifurcation Chaos Appl. Sci. Eng. 13 (2003), 2157-2188. Zbl 1079.34029, MR 2012652, 10.1142/S0218127403007874 |
| Reference:
|
[28] Mayergoyz, I. D.: Mathematical Models of Hysteresis and Their Applications.Elsevier, Amsterdam (2003). MR 1083150 |
| Reference:
|
[29] McCarthy, S., Rachinskii, D.: Dynamics of systems with Preisach memory near equilibria.Math. Bohem. 139 (2014), 39-73. MR 3231429 |
| Reference:
|
[30] Pimenov, A., Kelly, T. C., Korobeinikov, A., O'Callaghan, M. J., Pokrovskii, A. V., Rachinskii, D.: Memory effects in population dynamics: spread of infectious disease as a case study.Math. Model. Nat. Phenom. 7 (2012), 204-226. MR 2928740, 10.1051/mmnp/20127313 |
| Reference:
|
[31] Pimenov, A., Rachinskii, D.: Linear stability analysis of systems with Preisach memory.Discrete Contin. Dyn. Syst., Ser. B 11 (2009), 997-1018. Zbl 1181.47075, MR 2505656, 10.3934/dcdsb.2009.11.997 |
| Reference:
|
[32] Visintin, A.: Differential Models of Hysteresis.Applied Mathematical Sciences 111 Springer, Berlin (1994). Zbl 0820.35004, MR 1329094, 10.1007/978-3-662-11557-2 |
| Reference:
|
[33] Visone, C.: Hysteresis modelling and compensation for smart sensors and actuators.Journal of Physics: Conference Series 138 (2008), 23 pages DOI 10.1088/1742-6596/138/\allowbreak1/012028. 10.1088/1742-6596/138/\allowbreak1/012028 |
| . |
