Abstract
The study of nervous system and neurological diseases attracted the attention of scientists working in medical sciences and quick progress in present times than that of previous, only because of simulated models, computational power, and advances in experimental methods. There is a considerable increase in the neurological disorders (Alzheimer’s disease, Parkinson's disease, headaches, stroke, amyotrophic lateral sclerosis epilepsy, and seizures, etc.) cases over the past quarter century. Neurological disorders (NDs) are the leading cause of death and disability in the world today. In general combination of experimental and theoretical (mathematical) approaches are used to investigate the complex and dynamic interactions within a biological system or process in systems biology. In systems biology, a mathematical model is used to study the biological, chemical, and physical processes within living organisms by integrating genetics, signal transduction, biochemistry, and cell biology.
These mathematical models are of immense use in understanding the mechanisms behind the gene regulation and also in the identification of pathways that reveal the root causes for diseases. It is always easy to solve a linear dynamic model by using high-performance computing (HPC) cluster but it is still hard to construct solutions of nonlinear dynamic model. Most of the time it is much more difficult to find an analytical solution than a numerical solution of a nonlinear dynamic model even by utilizing the high-performance computing and some state-of-the-art symbolic computation software such as MATLAB, Mathematica, Maple, Axiom, Scilab, FriCAS, and so on. This chapter deals with analytic, qualitative, qualitative, and numerical treatment of nonlinear dynamical systems. This chapter present linearization, various notions qualitative properties, analytical methods, and numerical methods to handle nonlinear dynamical systems. The presented results and methods are illustrated and explained by numerical examples.
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References
Achdou Y, Franchi B, Marcello N, Tesi MC (2013) A qualitative model for aggregation and diffusion of beta-amyloid in Alzheimer’s disease. J Math Biol 67:1369–1392. https://doi.org/10.1007/s00285-012-0591-0
Anand PVS (2009) Controllability and observability of the matrix Lyapunov systems. In: Proceedings of the international conference on Recent Advances in Mathematical Science and Applications (RAMSA) held at Vizag, pp 117–131
Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New York
Bakshi S, Chelliah V, Chen C, van der Graaf PH (2019) Mathematical biology models of Parkinson’s disease. CPT Pharmacometrics Syst Pharmacol 8:77–86
Banwarth-Kuhn M, Sindi S (2020) How and why to build a mathematical model: a case study using prion aggregation. J Biol Chem 295(15):5022–5035. https://www.jbc.org/cgi/doi/10.1074/jbc.REV119.009851
Boyce WE, DiPrima RC (1997) Elementary differential equations and boundary value problems. Wiley, New York
Brown GC (1984) Stability in an insect pathogen model incorporating age-dependent immunity and seasonal host reproduction. Bull Math Biol 46:139–153
Cartwright ML (1956) On the stability of solution of certain differential equations of the fourth order. Quart J Mech Appl Math 9:185–194
Chin PSM (1988) Stability of nonlinear systems via the intrinsic method. Int J Control 48(4):1561–1567
Coddington EE (1989) An introduction to ordinary differential equations. Dover Publications, New York, p c1961
Coddington EE, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill/Tata McGraw-Hill, New York/New Delhi
Faghidian SA, Moghimi Zand M, Farjami Y, Farrahi GH (2011) Application of homotopy-pade technique to the Volterra’s prey and predator problem. Appl Comput Math 10(2):262–270
Fernt’andez FM (2009) On some approximate methods for nonlinear models. arXiv:0904.4044v1 [math-ph] 26 Apr 2009
Gibson JE, Schultz DG (1962) The variable gradient method of generating Liapunov functions with application to automatic control systems. Doctoral dissertation. Purdue University
Guckenheimer J, Holmes P (2002) Nonlinear oscillations, dynamical systems and bifurcations of vector fields, 6th edn. Springer, New York. 55(4): 273–289
He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178(3U′′ 4):257–262
He JH (2003) Homotopy perturbation technique: a new nonlinear analytical technique. Appl Math Comput 135:73–79
He JH (2004) Asymptotology by homotopy perturbation method. Appl Math Comput 156:591–596
Hirsch MW, Smale S, Devaney RL (2004) Differential equations, dynamical systems and an introduction to chaos. Academic Press/Elsevier, San Diego, CA
Hossain MB, Hossain MJ, Miah MM, Alam MS (2017) A comparative study on fourth order and butcher’s fifth order Runge-Kutta methods with third order initial value problem (IVP). Appl Comput Math 6(6):243–253
Krasovskii NN (1963) Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay (Translated by J. L. Brenner). Stanford University Press, Stanford, CA
Lakshmikantham V, Leela S, Martynyuk AA (1989) Stability analysis of nonlinear systems. Monographs and textbooks in pure and applied mathematics, 125. Marcel Dekker, Inc., New York
Liao SJ (1992) The proposed homotopy analysis techniques for the solution of nonlinear problems. Ph.D. dissertation. Shanghai Jiao Tong University, Shanghai
Liao SJ (2003) Beyond perturbation. Introduction to the homotopy analysis method. CRC Press/Chapman and Hall, Boca Raton
Liao S (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 169:1186–1194
Liu C (2011) The essence of the homotopy analysis method. arXiv:1105.6183v1[nlin.SI] 31 May 2011
Lyapunov AM (1992) The general problem of the stability of motion. Int J Control 55(3):531–534
Moler C, Van Loan C (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev 45:1–46
Murty KN, Srinivas MAS, Prasad KR (1987) Certain mathematical models for biological systems and their approximate analytical solutions. In: Proc. Int. Conf. on Non-linear analysis and applications to Bio-mathematics. Andhra Univ, Visakhapatnam
Murty KN, Srinivas MAS, Prasad KR (1990) Approximate analytical solutions to the three-species ecological system. J Math Anal Appl 145:89–99
Murty KN, Anand PVS, Prasannam VL (1997) First order difference system existence and uniqueness. Proc Am Math Soc 125(12):3533–3539
Murty KN, Andreou S, Viswanadh KVK (2009) Qualitative properties of general first order matrix difference systems. J Nonlinear Stud 16:359–369
Murty KN, Viswanadh KVK, Ramesh P, Wu Y (2013) Qualitative properties of a system of differential equations involving Kronecker product of matrices. J Nonlinear Stud 20(3):459–467
Park JY, Evans DJ, Murugesan K, Sekar S, Murugesh V (2004) Optimal control of singular systems using the RK–Butcher algorithm. Int J Comput Math 81(2):239–249
Putcha VS (1995) Non-linear differential and difference equations existence, uniqueness, stability, observability and controllability. Ph.D. dissertation. Andhra University, Visakhapatnam
Putcha VS (2011) Two species and three species ecological modeling—homotopy analysis, diversity of ecosystems. InTech Publications, pp 221–250
Putcha VS (2014) Discrete linear Sylvester repetitive process. Nonlinear Stud 21(2):205–218
Putcha VS, Malladi R (2010) A mathematical model on detritus in mangrove estuarine ecosystem. Int J Pure Appl Math 63(2):169–181
Putcha VS, Rompicharla CN, Deekshitulu GVSR (2012) A note on fuzzy discrete dynamical systems. Int J Contemp Math Sci 7(39):1931–1939
Rafei M, Daniali H, Ganji DD, Pashaei H (2007) Solution of the prey and predator problem by homotopy perturbation method. Appl Math Comput 188:1419–1425
Rompicharla CLN, Putcha VS, Deekshithulu GVSR (2019) Controllability and observability of fuzzy matrix discrete dynamical systems. J Nonlinear Sci Appl 12:816–828
Rompicharla CLN, Putcha SV, Deekshitulu GVSR (2020) Existence of (Φ⊗Ψ) bounded solutions for linear first order Kronecker product systems. Int J Recent Sci Res 11(06):39047–39053
Sami Bataineh A, Noorani MSM, Hashim I (2008) Solving systems of ODEs by homotopy analysis method. Commun Nonlinear Sci Numer Simul 13:2060–2070
Srinivas MAS (1989) Contributions to certain nonlinear biomathematical models. Ph.D thesis. Andhra University, Visakhapatnam
Stoer J, Bulirsch R (2002) Introduction to numerical analysis, 3rd edn. Springer, New York
Stuart AM, Humphries AR (1996) Dynamical systems and numerical analysis. Cambridge University Press, Cambridge
Tewari SG, Gottipati MK, Parpura V (2016) Mathematical modeling in neuroscience: neuronal activity and its modulation by astrocytes. Front Integr Neurosci 10:3
Wilson H (1999) Spikes, decisions and actions: the dynamical foundations of neuroscience. University Press, Oxford
Zoua L, Zonga Z, Dongb GH (2008) Generalizing homotopy analysis method to solve Lotka-Volterra equation. Comput Math Appl 56:2289–2293
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Putcha, V.S., Katakol, S. (2022). Qualitative and Analytical Treatment of Nonlinear Dynamical Systems in Neurological Diseases. In: Rajagopal, S., Ramachandran, S., Sundararaman, G., Gadde Venkata, S. (eds) Role of Nutrients in Neurological Disorders. Nutritional Neurosciences. Springer, Singapore. https://doi.org/10.1007/978-981-16-8158-5_4
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