The Modi ed Chain Method for Systems of Delay Di erential Equations with Applications Theses of a PhD Dissertation Beáta Krasznai Supervisor: Professor Mihály Pituk University of Pannonia Faculty of Information Technology Doctoral School of Information Science and Technology 2015.
1 Introduction In many biological, physical, chemical, technological and economical processes changing in time the speed of the change depends not only on the present state, but also depends on the previous state. This can be modelled using delay di erential equations [9], [13], [19], [22], [27], [31], [33]. The e ect of the delay makes the analysis di cult, because the phase space of the system is in nite dimensional, contrary with ordinary di erential equations, which lead only to nite dimensional problems. In order to overcome this di culty, Repin [34] suggested a method, the so-called chain method, where the solutions of the delay di erential equation are approximated by the solutions of a higher dimensional system of ordinary di erential equations. For further results related to the chain method, see the works by Banks [2], Banks and Kappel [3] and Janushevskij [24]. The recent investigations by Gedeon and Hines [10, 11], Demidenko [8] and Koch [25] show that the chain method is still being studied. Based on some properties of compartmental systems, Gy½ori [16, 17] initiated a new modi cation of the chain method. This modi cation is signi cant because Gy½ori s approximation converges not only on nite intervals, but under certain conditions it converges to the solution of the delay equation on the whole half-line. 2 The Aim and the Structure of the Thesis In the thesis we have extended the modi ed chain method studied earlier for scalar equations to a more general class of delay di erential systems. In Chapter 2 we de ne the approximating system of ordinary di erential equations and we prove the convergence of the method for the solutions of the delay di erential equation corresponding to su ciently smooth initial functions rst on nite time intervals and then under certain additional hypotheses we prove the convergence on the whole interval [0; 1) : In the case of su ciently smooth initial functions we also give an estimate for the order of convergence. 1
In Chapter 3 we prove the convergence of the modi ed chain method for the solutions of the delay equation corresponding to merely continuous initial functions. In contrast with the approximation theorems given in Chapter 2, in this general case we are not able to give an estimate for the order of convergence. In Chapter 4 we apply the modi ed chain method to concrete delay di erential equations. The examples include a pharmacokinetic model equation, a model of sh population, a model of neural networks with two di erent activation functions and further equations illustrating the accuracy of the convergence of the numerical approximation. In these examples the approximating systems of ordinary di erential equations have been solved by algorithm ode45 from program Matlab. In the Appendix we extend the stability criteria obtained in Chapter 3 to a more general class of delay di erential equations. Applying a comparison principle we obtain a new global exponential stability criterion for a model equation with several delays describing a neural network. 3 New Scienti c Results Consider the delay di erential equation x 0 (t) = Ax (t) + f (x (t)) + g (x (t )) ; t 0; (1) where > 0; A 2 R dd and f; g : R d! R d are continuous functions. For every continuous function : [ ; 0]! R d Eq. (1) has a solution x with initial data x (t) = (t) ; t 2 [ ; 0] : (2) We assume that f and g are Lipschitz continuous, that is, there exist constants L; K > 0 such that for every u; v 2 R d we have kf (u) f (v)k L ku vk (3) 2
and kg (u) g (v)k K ku vk ; (4) where kk is any norm on R d : The solution x of the initial value problem (1) component y 0 of the system of ordinary di erential equations with initial values y 0 0 = Ay 0 + f (y 0 ) + n y n; y 0 1 = g (y 0 ) y 0 j = n y j 1 (2) will be approximated by the n y 1; (5) n y j; 2 j n; y 0 (0) = (0) and y j (0) = (j 1) Z n g ( (s)) ds; 1 j n: (6) j n The next theorem shows that if the solution of the delay di erential equation is su ciently smooth on the initial interval then on any nite interval it can be approximated by the component y 0 of the solution of the initial value problem (5) (6) : The theorem provides an estimate for the order of convergence, too. Theorem 2.1. Suppose that condition (3) and (4) hold. Let : [ di erentiable and let 0 : [ ; 0]! R d be Lipschitz continuous and ; 0]! R d be 0 (0 ) = A (0) + f ( (0)) + g ( ( )) : (7) Assume also that g : R d! R d is di erentiable and g 0 : R d! R dd is Lipschitz continuous on compact subsets of R d : Let x be the solution of the initial value T problem (1) (2) and for each n 2 let y [n] = y [n] 0 ; y [n] 1 ; :::; y n [n] be the solution of the initial value problem (5) (6) : Then for every T > 0 there exists C > 0 3
such that for every n 2; sup x (t) t2[0;t ] y [n] 0 (t) C n : (8) In particular, y [n] 0! x uniformly on every nite subinterval of [0; 1) as n! 1: In the next theorem we show that under certain assumptions the approximation described in Theorem 2.1 is uniform on the whole interval [0; 1) : Moreover, we obtain a similar estimate for the order of convergence as before. Theorem 2.2. If in addition to the hypotheses of Theorem 2.1 we assume that x is bounded on [0; 1) and L + K < (A) ; (9) where (A) is the Lozinskii measure of matrix A; then the constant C in inequality (8) is independent of T; that is for every n 2 sup x (t) t2[0;1) y [n] 0 (t) C n : (10) In particular, y [n] 0! x uniformly on the whole interval [0; 1) as n! 1: The Lozinskii measure (A) of matrix A 2 R dd is de ned by ke + Ak 1 (A) = lim ;!0+ where E 2 R dd is the unit matrix and kk is the induced matrix norm. In the next two approximation theorems, we omit the smoothness assumption on the initial function. The approximation in this general case is also uniform on both nite and in nite intervals. However, we are not able to give an estimate for the order of convergence. Theorem 3.1. Suppose that the hypotheses of Theorem 2.1 are satis ed except the conditions on the initial function : [ ; 0]! R d which is assumed to be merely 4
continuous. Then using the notations of Theorem 2.1, we have that y [n] 0! x uniformly on every nite subinterval of [0; 1) : Theorem 3.2. If in addition to the hypotheses of Theorem 3.1 we assume that x is bounded on [0; 1) and (9) holds, then y [n] 0! x uniformly on [0; 1) as n! 1: Our approximation theorems have been applied to concrete delay di erential equations. Our rst example is a model equation describing the pharmacokinetics of propofol in the human body. The state variables in the model are the masses of propofol in the intravascular blood, in muscle and in fat. The second example is a model of sh population and the third example is a model equation describing neural networks with two activation functions in which we have examined the error of the approximation. In order to prove the above approximation theorems, we have established estimates on the distance of two solutions of the delay equation. These estimates are interesting in their own rights because they imply new stability criteria. Theorem 3.4. Suppose (3) and (4) hold. Then under condition (9) every solution of Eq. (1) is globally exponentially stable. Theorem 3.10. Suppose conditions (3) ; (4) and (9) hold. If f + g is bounded (11) or L + K < 1 ka 1 k ; (12) then Eq. (1) has a globally exponentially stable equilibrium. 5
4 Theses The most important results of the dissertation are summarizes in the following theses. Thesis 1 I have extended the modi ed chain method known for scalar delay di erential equations to a more general class of systems of delay di erential equations. I have proved the convergence of the method for the solutions of the delay equation corresponding to su ciently smooth initial functions and I have established an estimate for the order of convergence. 1.1. I have given su cient conditions under which the modi ed chain method is convergent on every nite interval for the solutions of the delay equation corresponding to su ciently smooth initial functions. I have established an estimate for the order of convergence, too (see Theorem 2.1). 1.2. I have given explicit su cient conditions under which the modi ed chain method is convergent on the whole interval [0; 1) for the solutions of the delay equation corresponding to su ciently smooth initial functions. I have given an estimate for the order of convergence, too (see Theorem 2.2). Thesis 2 Using the above results, I have proved the convergence of the modi ed chain method for all solutions of the delay equation corresponding to continuous initial functions. I have applied the approximation theorems to various model equations. 2.1. I have given su cient conditions under which the modi ed chain method is convergent on every nite interval for the solutions of the delay equation corresponding to continuous initial functions (see Theorem 3.1). 2.2. I have given explicit su cient conditions under which the modi ed chain method is convergent on the whole interval [0; 1) for the solutions of the delay equation corresponding to continuous initial functions (see Theorem 3.2). 6
2.3. I have applied the approximation theorems to a model equation describing the pharmacokinetics of propofol in the human body, to a model of sh population and to model equations from neural networks. Thesis 3 I have established new global exponential stability criteria for the solutions of delay di erential equations. 3.1. I have proved a su cient condition under which every solution of the delay equation is globally exponentially stable (see Theorem 3.4). 3.2. I have given a su cient condition under which the delay equation has a globally exponentially stable equilibrium (see Theorem 3.10 and Theorem A.2). 5 Publications with Citations (P1) Krasznai, B., Gy½ori, I. and Pituk, M., "The modi ed chain method for a class of delay di erential equations arising in neural networks.", Mathematical and Computer Modelling 51:5-6 (2010), 452-460. (Impact factor = 1.103) 1. Demidenko, G. V., "Systems of di erential equations of higher dimension and delay equations.", Siberian Mathematical Journal, 53:6 (2012), 1021-1028. (P2) Krasznai, B., Gy½ori, I. and Pituk, M., "Positive decreasing solutions of higher-order nonlinear di erence equations.", Advances in Di erence Equations, Article ID: 973432 (2010). (Impact factor = 0.892) 2. Diblík, J., Hlaviµcková I., "Asymptotic upper and lower estimates of a class of positive solutions of a discrete linear equation with a single delay.", Abstract and Applied Analysis., Article ID: 764351 (2012). 3. Peics, H., "Positive solutions of second-order linear di erence equation with variable delays.", Advances in Di erence Equations, Article ID: 82 (2013). 7
(P3) Krasznai, B., "Stability criteria for delay di erential equations. Recent Advances in Delay Di erential and Di erence Equations.", Springer Proceedings in Mathematics and Statistics 94, Springer, New York, (2014), 161-171. (ISBN: 978-3-319-08250-9) 6 International Conference Talks (E1) Krasznai, B.: The modi ed chain method for a class of delay di erential equations arising in neural networks. Workshop of the Committee on Mathematical Analysis and Application of the Hungarian Academy of Sciences, Széchenyi István University, Gy½or, May 22, 2009. (E2) Krasznai, B.: Approximation of delay di erential equations by the modi ed chain method., Workshop on Delay Di erential and Di erence Equations, Veszprém, July 17-18, 2014. References [1] Altrichter, M., Horváth, G., Pataki, B., Strausz, G., Takács, G., & Valyon, J., "Neurális hálózatok." Panem, Budapest (2006). [2] Banks, H. T., "Delay systems in biological models: Approximation techniques." Nonlinear Systems and Applications (1977), 21-38. [3] Banks, H. T., and Kappel F., "Spline approximations for functional di erential equations." Journal of Di erential Equations 34:3 (1979), 496-522. [4] Bogár, L., "Aneszteziológia és intenzív terápia." Medicina Könyvkiadó, Budapest (2009). [5] Cao, J., and Zhou. D., "Stability analysis of delayed cellular neural networks." Neural Networks 11:9 (1998), 1601-1605. 8
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[18] Gy½ori, I., and Eller., J., "Compartmental systems with pipes." Mathematical Biosciences 53:3 (1981), 223-247. [19] Gy½ori, I., and Ladas, G. E., "Oscillation Theory of Delay Di erential Equations with Applications". Oxford University Press, New York (1991). [20] Gy½ori, I., and Turi, J., "Uniform approximation of a nonlinear delay equation on in- nite intervals." Nonlinear Analysis: Theory, Methods & Applications 17:1 (1991), 21-29. [21] Haddad, W. M., Chellaboina, V., and Hui, Q., "Nonnegative and Compartmental Dynamical Systems." Princeton University Press, Prinston (2010). [22] Hale, J. K., and Verduyn-Lunel, S. M., "Introduction to Functional Fi erential Equations." Springer, New York (2013). [23] Idels, L., and Kipnis, M., "Stability criteria for a nonlinear nonautonomous system with delays." Applied Mathematical Modelling 33:5 (2009), 2293-2297. [24] Janushevski, P. T. "Control of Object with Delays." (in Russian) Nauka, Moscow (1978). [25] Koch, G., et al., "Multi-response model for rheumatoid arthritis based on delay di erential equations in collagen-induced arthritic mice treated with an anti-gm- CSF antibody." Journal of Pharmacokinetics and Pharmacodynamics 39:1 (2012), 55-65. [26] Koch, G., "Modeling of Pharmacokinetics and Pharmacodynamics with Application to Cancer and Arthritis." PhD Thesis, Universitat Kostanz, Germany (2012). [27] Kolmanovskii, V., and Myshkis, A., "Applied Theory of Functional Di erential Equations". Springer, New York (1992). [28] Krasznai, B., "Stability criteria for delay di erential equations. Recent Advances in Delay Di erential and Di erence Equations.", Springer Proceedings in Mathematics and Statistics 94, Springer, New York, (2014), 161-171. 10
[29] Krasznai, B., Gy½ori, I. and Pituk, M., "The modi ed chain method for a class of delay di erential equations arising in neural networks.", Mathematical and Computer Modelling 51:5-6 (2010), 452-460. [30] Krasznai, B., Gy½ori, I. and Pituk, M., "Positive decreasing solutions of higherorder nonlinear di erence equations.", Advances in Di erence Equations, Article ID: 973432, (2010). [31] Kuang, Y., "Delay Di erential Equations: with Applications in Population Dynamics". Academic Press, San Diego (1993). [32] Mohamad, S., and Gopalsamy, K., "Exponential stability of continuous-time and discrete-time cellular neural networks with delays." Applied Mathematics and Computation 135:1 (2003), 17-38. [33] Smith, H., "An Introduction to Delay Di erential Equations with Applications to the Life Sciences". Springer, New York (2010). [34] Repin, Y. M., "On the approximate replacement of systems with lag by ordinary dynamical systems." Journal of Applied Mathematics and Mechanics 29:2 (1965), 254-264. 11