A numerical study of coupled maps representing energy exchange processes between twoenvironmental interfaces regarded as biophysical complex systems

ABSTRACT

The field of environmental sciences is abundant with various interfaces and is the right place for the application of new fundamental approaches leading towards a better understanding of environmental phenomena. Following the definition of environmental interface by Mihailovic and Bala? [1], such interface can be, for example, placed between: human or animal bodies and surrounding air, aquatic species and water and air around them, and natural or artificially built surfaces (vegetation, ice, snow, barren soil, water, urban communities) and the atmosphere, cells and surrounding environment, etc. Complex environmental interface systems are (i) open and hierarchically organised (ii) interactions between their constituent parts are nonlinear, and (iii) their interaction with the surrounding environment is noisy. These systems are therefore very sensitive to initial conditions, deterministic external perturbations and random fluctuations always present in nature. The study of noisy non-equilibrium processes is fundamental for modelling the dynamics of environmental interface regarded as biophysical complex system and for understanding the mechanisms of spatio-temporal pattern formation in contemporary environmental sciences. In this paper we will investigate an aspect of dynamics of energy flow based on the energy balance equation. The energy exchange between interacting environmen- tal interfaces regarded as biophysical complex systems can be represented by coupled maps. Therefore, we will numerically investigate coupled maps representing that exchange. In ana- lysis of behaviour of these maps we applied Lyapunov exponent and cross sample entropy.

The field of environmental sciences is abundant with various interfaces and is the right place for the application of new fundamental approaches leading towards a better understanding of environmental phenomena. Following the definition of environmental interface by Mihailovic and Bala? [1], such interface can be, for example, placed between: human or animal bodies and surrounding air, aquatic species and water and air around them, and natural or artificially built surfaces (vegetation, ice, snow, barren soil, water, urban communities) and the atmosphere, cells and surrounding environment, etc. Complex environmental interface systems are (i) open and hierarchically organised (ii) interactions between their constituent parts are nonlinear, and (iii) their interaction with the surrounding environment is noisy. These systems are therefore very sensitive to initial conditions, deterministic external perturbations and random fluctuations always present in nature. The study of noisy non-equilibrium processes is fundamental for modelling the dynamics of environmental interface regarded as biophysical complex system and for understanding the mechanisms of spatio-temporal pattern formation in contemporary environmental sciences. In this paper we will investigate an aspect of dynamics of energy flow based on the energy balance equation. The energy exchange between interacting environmen- tal interfaces regarded as biophysical complex systems can be represented by coupled maps. Therefore, we will numerically investigate coupled maps representing that exchange. In ana- lysis of behaviour of these maps we applied Lyapunov exponent and cross sample entropy.

KEYWORDS

Environmental Interface; Nonlinearity; Chaos; Logistic Equation; Energy Balance Equation; Coupled Maps, Hierarchy, Biophysical Complex Systems

Environmental Interface; Nonlinearity; Chaos; Logistic Equation; Energy Balance Equation; Coupled Maps, Hierarchy, Biophysical Complex Systems

Cite this paper

Mihailović, D. , Budinčević, M. , Kapor, D. , Balaž, I. and Perišić, D. (2011) A numerical study of coupled maps representing energy exchange processes between twoenvironmental interfaces regarded as biophysical complex systems.*Natural Science*, **3**, 75-84. doi: 10.4236/ns.2011.31011.

Mihailović, D. , Budinčević, M. , Kapor, D. , Balaž, I. and Perišić, D. (2011) A numerical study of coupled maps representing energy exchange processes between twoenvironmental interfaces regarded as biophysical complex systems.

References

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[2] Rosen, R. (1991) Life itself: A comprehensive inquiry into the nature, origin, and fabrication of life. Columbia University Press, New York.

[3] Selvam, A.M. and Fadnavis, S. (1998) Signatures of a universal spectrum for atmospheric interannual variability in some disparate climatic regimes. Meteorology and Atmospheric Physics, 66, 87-112. doi:10.1007/BF01030450

[4] Sivertsen, T.H. (2005) Discussing the scientific method and a documentation system of meteorological and biological parameters. Physics and Chemistry of the Earth, 30, 35-43.

[5] Glazier, J. and Graner, F. (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Physical Review E, 47, 2128-2154. doi:10.1103/PhysRevE.47.2128

[6] Martins, M.L., Ceotto, G., Alves, G., Bufon, C.C.B., Silva, J.M. and Laranjeira, F.F. (2000) Cellular automata model for citrus variegated chlorosis. Physical Review E, 62, 7024-7030. doi:10.1103/PhysRevE.62.7024

[7] Nikolov, N., Massman, W. and Schoettle, A. (1995) Coupling biochemical and biophysical processes at leaf level: an equilibrium photosynthesis model for leaves of C3 plants. Ecological Modelling, 80, 205-235. doi:10.1016/0304-3800(94)00072-P

[8] Niyogi, D.S. and Raman, S. (2001) Numerical modelling of gas deposition and bidirectional surface–atmosphere exchanges in mesoscale air pollution systems. In: Boybeyi, Z. Ed., Advances in Air Pollution, WIT Publications, Southampton, 1-51.

[9] Cushman-Roisin, B., Gualtieri, C. and Mihailovic, D.T. (2008) Environmental fluid mechanics: Current issues and future outlook. In: Gualtieri, C. and Mihailovic, D.T. Eds., Fluid Mechanics of Environmental Interfaces, Taylor & Francis, Leiden, 1-16.

[10] Collier, J.D. (2003) Fundamental properties of self-organization. In: Arshinov, V. and Fuchs, C. Eds., Causality, Emergence, Self-Organisation, NIA-Priroda, Moscow, 150-166.

[11] Arshinov, V. and Fuchs, C. (2003) Preface. In: Arshinov, V. and Fuchs, C. Eds., Causality, Emergence, Self-Organisation, NIA-Priroda, Moscow, 1-18.

[12] Edmonds, B. (1999) What is Complexity?: The philosophy of complexity per se with application to some examples in evolution. In: Heylighen F., Bollen, J. and Riegler, A. Eds., The Evolution of Complexity, Kluwer, Dordrecht, 1-17.

[13] Kauffman, S. (1993) The origins of order: Self-organization and selection in evolution. Oxford University Press, Oxford.

[14] Heylighen, F. (1999) The growth of structural and functional complexity during evolution. In: Heylighen F., Bollen, J. and Riegler, A. Eds., The Evolution of Complexity, Kluwer, Dordrecht, 17-47.

[15] Kaneko, K. (1983) Transition from torus to chaos accompanied by the frequency locking with symmetry breaking - In connection with the coupled logistic map. Progress of Theoretical Physics, 69, 1427-1442. doi:10.1143/PTP.69.1427

[16] Midorikawa, S., Takayuki, K. and Taksu, C. (1995) Folded bifurcation in coupled asymmetric logistic maps. Progress of Theoretical Physics, 94, 571-575. doi:10.1143/PTP.94.571

[17] Mihailovic, D.T. (2008) Two interacting environmental Interfaces: Folded bifurcation in coupled asymmetric logistic maps. 4th Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008), 134-140.

[18] Balaz, I. and Mihailovic, D.T. (2008) Evolvable biological interfaces: Outline of the new computing system. 4th Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008), Barcelona, 7-10 July 2008, 104-113.

[19] Van der Vaart, H.R. (1973) A comparative investigation of certain difference equations and related differential equations: implications for model building. Bulletin of Mathematical Biology, 35, 195-211.

[20] Bhumralkar, C.M. (1975) Numerical experiments on the computation of ground surface temperature in an atmospheric general circulation model. Journal of Applied Meteorology, 14, 1246-1258. doi:10.1175/1520-0450(1975)014%3c1246:NEOTCO%3e2.0.CO;2

[21] Holtslag A.A. and Van Ulden, A.P. (1983) A simple scheme for daytime estimates of the surface fluxes from routine weather data. Journal of Applied Meteorology, 22, 517-529. doi:10.1175/1520-0450(1983)022%3c0517:ASSFDE%3e2.0.CO;2

[22] Kennel, M.B., Brown, R. and Abarbanel, H.D.I. (1992) Determining embedding dimension for phase-space reconstruction using a geometrical construction. Physical Review A, 45, 3403-3411. doi:10.1103/PhysRevA.45.3403

[23] Richman, J.S. and Moorman, J.R. (2000) Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology: Heart & Circulatory Physiology, 278, H2039-H2049.

[24] Lake, D.E., Richman, J.S., Grif?n, M.P. and Moorman, J.R. (2002) Sample entropy analysis of neonatal heart rate variability. American Journal of Physiology: Regulatory Integrative and Comparative Physiology, 283, R789-R797.

[25] Pincus, S.M. (1991) Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences of the United States, 88, 2297-2301. doi:10.1073/pnas.88.6.2297

[26] Bandt, C. and Pompe, B. (2002) Permutation entropy: A natural complexity measure for time series. Physical Review Letter, 88, 174102. doi:10.1103/PhysRevLett.88.174102

[27] Parker, T.S. and Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York.

[28] Hogg, T. and Huberman, B.A. (1984) Generic behavior of coupled oscillators. Physical Review A, 29, 275-281. doi:10.1103/PhysRevA.29.275

[29] Feit, D. (1978) Characteristic exponents and strange attractors. Communications in Mathematical Physics, 61, 249. doi:10.1007/BF01940767

[30] Pincus, S. and Singer B.H. (1995) Randomness and degrees of irregularity. Proceedings of the National Academy of Sciences, 93, 2083-2088. doi:10.1073/pnas.93.5.2083

[31] Pincus, S.M., Mulligan, T., Iranmanesh, A., Gheorghiu, S., Godschalk, M. and Veldhuis J.D. (1996) Older males secrete luteinizing hormone and testosterone more irregularly, and jointly more asynchronously, than younger males. Proceedings of the National Academy of Sciences, 93, 14100-14105. doi:10.1073/pnas.93.24.14100

[1] Mihailovic, D.T. and Balaz, I. (2007) An essay about modeling problems of complex systems in environmental fluid mechanics. Idojaras, 111, 209-220.

[2] Rosen, R. (1991) Life itself: A comprehensive inquiry into the nature, origin, and fabrication of life. Columbia University Press, New York.

[3] Selvam, A.M. and Fadnavis, S. (1998) Signatures of a universal spectrum for atmospheric interannual variability in some disparate climatic regimes. Meteorology and Atmospheric Physics, 66, 87-112. doi:10.1007/BF01030450

[4] Sivertsen, T.H. (2005) Discussing the scientific method and a documentation system of meteorological and biological parameters. Physics and Chemistry of the Earth, 30, 35-43.

[5] Glazier, J. and Graner, F. (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Physical Review E, 47, 2128-2154. doi:10.1103/PhysRevE.47.2128

[6] Martins, M.L., Ceotto, G., Alves, G., Bufon, C.C.B., Silva, J.M. and Laranjeira, F.F. (2000) Cellular automata model for citrus variegated chlorosis. Physical Review E, 62, 7024-7030. doi:10.1103/PhysRevE.62.7024

[7] Nikolov, N., Massman, W. and Schoettle, A. (1995) Coupling biochemical and biophysical processes at leaf level: an equilibrium photosynthesis model for leaves of C3 plants. Ecological Modelling, 80, 205-235. doi:10.1016/0304-3800(94)00072-P

[8] Niyogi, D.S. and Raman, S. (2001) Numerical modelling of gas deposition and bidirectional surface–atmosphere exchanges in mesoscale air pollution systems. In: Boybeyi, Z. Ed., Advances in Air Pollution, WIT Publications, Southampton, 1-51.

[9] Cushman-Roisin, B., Gualtieri, C. and Mihailovic, D.T. (2008) Environmental fluid mechanics: Current issues and future outlook. In: Gualtieri, C. and Mihailovic, D.T. Eds., Fluid Mechanics of Environmental Interfaces, Taylor & Francis, Leiden, 1-16.

[10] Collier, J.D. (2003) Fundamental properties of self-organization. In: Arshinov, V. and Fuchs, C. Eds., Causality, Emergence, Self-Organisation, NIA-Priroda, Moscow, 150-166.

[11] Arshinov, V. and Fuchs, C. (2003) Preface. In: Arshinov, V. and Fuchs, C. Eds., Causality, Emergence, Self-Organisation, NIA-Priroda, Moscow, 1-18.

[12] Edmonds, B. (1999) What is Complexity?: The philosophy of complexity per se with application to some examples in evolution. In: Heylighen F., Bollen, J. and Riegler, A. Eds., The Evolution of Complexity, Kluwer, Dordrecht, 1-17.

[13] Kauffman, S. (1993) The origins of order: Self-organization and selection in evolution. Oxford University Press, Oxford.

[14] Heylighen, F. (1999) The growth of structural and functional complexity during evolution. In: Heylighen F., Bollen, J. and Riegler, A. Eds., The Evolution of Complexity, Kluwer, Dordrecht, 17-47.

[15] Kaneko, K. (1983) Transition from torus to chaos accompanied by the frequency locking with symmetry breaking - In connection with the coupled logistic map. Progress of Theoretical Physics, 69, 1427-1442. doi:10.1143/PTP.69.1427

[16] Midorikawa, S., Takayuki, K. and Taksu, C. (1995) Folded bifurcation in coupled asymmetric logistic maps. Progress of Theoretical Physics, 94, 571-575. doi:10.1143/PTP.94.571

[17] Mihailovic, D.T. (2008) Two interacting environmental Interfaces: Folded bifurcation in coupled asymmetric logistic maps. 4th Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008), 134-140.

[18] Balaz, I. and Mihailovic, D.T. (2008) Evolvable biological interfaces: Outline of the new computing system. 4th Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008), Barcelona, 7-10 July 2008, 104-113.

[19] Van der Vaart, H.R. (1973) A comparative investigation of certain difference equations and related differential equations: implications for model building. Bulletin of Mathematical Biology, 35, 195-211.

[20] Bhumralkar, C.M. (1975) Numerical experiments on the computation of ground surface temperature in an atmospheric general circulation model. Journal of Applied Meteorology, 14, 1246-1258. doi:10.1175/1520-0450(1975)014%3c1246:NEOTCO%3e2.0.CO;2

[21] Holtslag A.A. and Van Ulden, A.P. (1983) A simple scheme for daytime estimates of the surface fluxes from routine weather data. Journal of Applied Meteorology, 22, 517-529. doi:10.1175/1520-0450(1983)022%3c0517:ASSFDE%3e2.0.CO;2

[22] Kennel, M.B., Brown, R. and Abarbanel, H.D.I. (1992) Determining embedding dimension for phase-space reconstruction using a geometrical construction. Physical Review A, 45, 3403-3411. doi:10.1103/PhysRevA.45.3403

[23] Richman, J.S. and Moorman, J.R. (2000) Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology: Heart & Circulatory Physiology, 278, H2039-H2049.

[24] Lake, D.E., Richman, J.S., Grif?n, M.P. and Moorman, J.R. (2002) Sample entropy analysis of neonatal heart rate variability. American Journal of Physiology: Regulatory Integrative and Comparative Physiology, 283, R789-R797.

[25] Pincus, S.M. (1991) Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences of the United States, 88, 2297-2301. doi:10.1073/pnas.88.6.2297

[26] Bandt, C. and Pompe, B. (2002) Permutation entropy: A natural complexity measure for time series. Physical Review Letter, 88, 174102. doi:10.1103/PhysRevLett.88.174102

[27] Parker, T.S. and Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York.

[28] Hogg, T. and Huberman, B.A. (1984) Generic behavior of coupled oscillators. Physical Review A, 29, 275-281. doi:10.1103/PhysRevA.29.275

[29] Feit, D. (1978) Characteristic exponents and strange attractors. Communications in Mathematical Physics, 61, 249. doi:10.1007/BF01940767

[30] Pincus, S. and Singer B.H. (1995) Randomness and degrees of irregularity. Proceedings of the National Academy of Sciences, 93, 2083-2088. doi:10.1073/pnas.93.5.2083

[31] Pincus, S.M., Mulligan, T., Iranmanesh, A., Gheorghiu, S., Godschalk, M. and Veldhuis J.D. (1996) Older males secrete luteinizing hormone and testosterone more irregularly, and jointly more asynchronously, than younger males. Proceedings of the National Academy of Sciences, 93, 14100-14105. doi:10.1073/pnas.93.24.14100