An analysis on the stability of a state dependent delay differential equation

Abstract In this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term τ(u(t))=a+bu(t)c+bu(t).$\tau (u(t)) = \frac{{a + bu(t)}}{{c + bu(t)}}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.


Introduction
Delay differential equations (DDE) have been used in many fields for a long time. However, state-dependent delay differential equations (SDDE) are used to make more realistic modelling in the systems whose delay varies according to the internal effects of the system. For example, the length of time to maturity is taken as constant delay in a simple population dynamics model, see in [1]. In [2], it was observed that the length of time to maturity of Antarctic whales and seals alter according to the state of the population and it was analyzed by using a mathematical model with SDDE in [3]. In addition, mathematical models with SDDE appear in many fields such as physics, control theory, neural network, medicine, biology etc., see Section 2 in [4] as a review.
In this paper, we consider the following type of SDDE where A 0 , A 1 2 R and .u.t // > 0 for all t 2 R C . To analyze stability of solution of equation (1), we use the following characteristic equation where h is an independent real valued parameter which is in the range of .u.t //. By this way, we analyze the stability of equation (1) by using the stability analysis of certain linear delay differential equations with constant delay which has the characteristic equation (2).
In the general case, the characteristic roots j , j D 1; 2; , of equation (1) are obtained by solving the characteristic equation (2) where j is a complex number. If the characteristic roots have negative real parts, i.e., Re. j / < 0 for all j D 1; 2; then the solution of (1) is asymptotically stable and if at least one of the characteristic roots have positive real parts, i.e., Re. j / > 0 for some j D 1; 2; then the solution of (1) is unstable. We attempt to determine the stability and instability regions of the system in parameter space .A 0 ; A 1 / by using D-partition method. The method is originated from paper [33]. It is well explained in [34][35][36][37][38] and analysis are conducted. Let's consider the characteristic equation g. ; A 0 ; A 1 / in two parameters for equation (1). D-partition method is based on fact that the roots of the characteristic equation are continuos functions of the parameters A 0 and A 1 . When varying the parameters, j change continuously in complex plane and at the point where the stability changes, one j crosses the imaginary axis. In this method parameter space is divided into regions with the hypersurfaces. These hypersurfaces are called the D-curves. The points of the D-curves correspond to pure imaginary roots or zero root of the characteristic equation. Moreover, in each region in the parameter space determined by the D-curves, the characteristic equation has the same number of roots with positive real part. Thus, finding the number of roots with positive real parts for specific point is enough to find the number of roots with positive real parts the region including this specific point.
In order to obtain D-curves, pure imaginary number D i ! is substituted in characteristic equation g. ; A 0 ; A 1 /. Equating to zero the real and imaginary parts, we have Hence, by making use of (3) and (4), parametric equations can be written as where ! is a parameter and ranges from 1 to 1. These curves and singular solutions of equations (3) and (4) constitute D-curves. We use Rekasius transform where h, T 2 R and for p 2 Z in addition to D-partition method. In 1980, Rekasius [39] proposed the transformation (5) for DDEs. Later, Thowsen [40] did exact calculations by taking the square of right hand side of (5) since the transformation (5) transform a circle to a semi-circle which leads some mistakes. However, Hertz et al. [41] did exact calculations by considering two singular cases: Olgaç and Sipahi [42,43] studied a method using Rekasius transform for DDE with constant delay. We establish the existence and uniqueness of solutions of equation (1)

Existence and uniqueness of solution
In this section, we consider the following type of SDDE where A 0 , A 1 2 R C , a; b; c; d 2 R such that a and c are nonzero and at least one of b or d is nonzero.
In order to guarantee the positivity of delay term .u.t //, we need to put some restrictions on the range of parameter values under consideration.
Theorem 2.1. Let A 0 , A 1 2 R C and .u.t // be delay function of equation (6). The delay differential equation has a unique solution u.
. ; Proof. For the proof, the following four cases are considered.
which is a contradiction. Case 3: Let's prove that u.t / 2 .L 0 ; M 0 / for all t > 0 when < 0 < , which is a contradiction. In a similar way, if u.t 0 / D M 0 " for " > 0, then u 0 .t 0 / 0. On the other hand which is a contradiction since u.t 0 / and A 0 " tend to and 0 respectively when " tends to 0.
which is a contradiction since u.t 0 / and A 0 " tend to and 0 respectively when " tends to 0. In a similar way, if which is a contradiction. As a result, there is no such t 0 2 R C and (8) holds. Since are Lipschitz with respect to each of their argument, local existence and uniqueness of the solution u.t / follows from Driver [5].
and .u.t // be delay function of equation (6). The delay differential equation (7) has a unique solution u.
and D max Proof. For the proof, following two cases are considered. Case 1: Let's prove that u.t / 2 .L 0 ; M 0 / for all t > 0 when 0 < Ä or < 0 < , which is a contradiction. In a similar way, if u.t 0 / D M 0 " for " > 0, then u 0 .t 0 / 0. On the other hand which is a contradiction since u.t 0 / and A 0 " tend to and 0 respectively when " tends to 0. Case 2: Let's prove that u.t / 2 .L 0 ; M 0 / for all t > 0 when Ä < 0 or < 0 < , which is a contradiction since u.t 0 / and A 0 " tend to and 0 respectively when " tends to 0. In a similar way, if u.t 0 / D M 0 , then u 0 .t 0 / 0. On the other hand which is a contradiction. As a result, there is no such t 0 2 R C and (9) holds. Since are Lipschitz with respect to each of their argument, local existence and uniqueness of the solution u.t / follows from Driver [5].
These results allow us to do stability analysis of solution of (1) by using the range of .u.t // which is obtained by Theorems 2.1 and 2.2.
Furthermore, if the delay function .u.t // has a complicated form, then OE1=1 Padé approximation for .u.t // can be obtained and the stability analysis can be done by using rough range of .u.t // which is obtained by the range of the solution u.t/ approximately by the help of Theorems 2.1 and 2.2. The same can be done, if the delay function .u.t// is not known exactly but some of its suitable values are obtained by some experiments or a heuristic method.

Stability analysis
In this section, we firstly consider the stability of equation (1) with the delay function .u.t // which has an upper bound for all t 2 R C , i.e., there exist at least one M 1 2 R such that 0 < .u.t // < M 1 for all t 2 R C . In this case, the value of delay of equation (1) varies in interval .0; M 1 / while t is varying. The independent parameter h of the characteristic equation (2) takes values in the interval .0; M 1 /. As a part of the D-partition method, we have this straight line is a line forming the boundary of the D-partition and is denoted by C . Substituting D i! and equating to zero the real and imaginary parts in characteristic equation (2), we find the following equations Solving the above equations for A 0 and A 1 , the following parametric curve equations are obtained Since A 0 .!; h/ and A 1 .!; h/ are even with respect to !, it is sufficient to take ! 2 .0; 1/. Equations (13) In addition, the following limits can be obtained for k 2 N f0g Proof. Intersection of C 0 .h/ and C is obvious from (15). For the second part of Lemma 3.1, suppose that if C k .h/ and C has intersection points there exist ! 2 J k for equations (13)- (14) which satisfies equation (10). By using equations (13)- (14) in equation (10)  There is no solution ! 2 J k for k 2 N f0g which is a contradiction.
Proof. Suppose that there exist an intersection point. It means that, there exist These equalities imply that from equation (13) and (14). For n 2 N, ! 1 h 0 ¤ ! 2 h 0 C 2n is obtained from the left equality in (16) because of ! 1 ¤ ! 2 : In addition, left and right equalities in (16) lead to cos.! 1 h 0 / D cos.! 2 h 0 / which is a contradiction.
Lemma 3.3. The curve C k .h 0 / intersects the line A 0 D 0 exactly once for h 0 2 R C . Moreover, the intersection point .0; P k / satisfies the following inequalities P k < P kC2 for k D 2n, n 2 N P kC2 < P k for k D 2n C 1, n 2 N: is obtained by substituting ! D C2k 2h 0 in A 1 .!; h 0 /. This completes the proof.
Theorem 3.4. The solution of equation is asymptotically stable, i.e., all the roots of equation have negative real parts, if and only if To find the number of roots with positive real parts in each region in the parameter space determined by the D-curves, we use the following the ideas from [38]. Writing D C i ! with , ! 2 R in characteristic equation g. ; A 0 ; A 1 /, we find two real equations for the real and imaginary parts of . Direction of movement of an element is determined by the following proposition, using Jacobian matrix J defined by J D " @G 1 @A 0 @G 1 @A 1 @G 2 @A 0 @G 2 @A 1 # D0 . Proposition 3.5. The pure imaginary roots enter the right half-plane for parameters sets in the .A 0 ; A 1 / parameter region to the left of the D-curves, when we follow this curve in the direction of increasing !, whenever det.J / < 0 and to the right when det.J / > 0 [38].
Since the determinant of Jacobian matrix of equation (18) satisfies the following inequalities the pure imaginary roots move into the right half-plane when moving away in the parameter space to the left of C 2k .h/ and to the right of C 2kC1 .h/, with "left" and "right" as determined w.r.t. a counter clock-wise tracking of C 2k .h/ and a clock-wise tracking of C 2kC1 .h/ respectively. In Fig. 1 these results are illustrated for h D 1. The curves (13) and (14) and the straight line (10) form the D-partition are shown and the number of roots in the right half plane is indicated for each region. Proof. Suppose that there exists an intersection point for h 1 < h 2 . It means that, there exist ! 1 h 1 , ! 2 h 2 2 .k ; .k C 1/ / such that cos.! 1 h 1 / D cos.! 2 h 2 / is obtained by using (21) and (22) which implies that ! 1 h 1 D ! 2 h 2 because of ! 1 h 1 , ! 2 h 2 2 .k ; .k C 1/ /. Therefore we have ! 1 ¤ ! 2 from the assumption h 1 < h 2 which contradicts (22).
Proof. Taking the derivative of (13) and (14) Fig. 2. S h D f.A 0 ; A 1 /jA 0 ; A 1 2 R and satisfy the conditions .a/ and .b/ for h 2 R C :g Proof. It is clear from Lemma 3.7.
Theorem 3.9. The solution of equation (1) with delay term .u.t // > 0 which satisfies the condition 0 < .u.t // < M 1 for all t 2 R C is asymptotically stable if and only if the following conditions are satisfied: .e a/ 1 Critical delays of an equation are the values at which the qualitative behavior of the system changes. Between any two successive critical values, the behavior of the solution does not change. Now, by using transformation (5) we rewrite the critical delay values of equation (17) in terms of parameter.
Proposition 3.11. D i ! is a root of equation (18) for some h if and only if D i ! is also a root of for some T 0 Proof. Let D i! be a root of equation (18). By using transformation (5) in equation (18), we obtain Multiplying this equation by 1 C i !T and arranging properly , we get which implies that D i ! is a root of equation (23) for h D 2 ! .arctan.!T / C p /. Moreover, the singular cases of Rekasius transform (5) are satisfied for equation (16) and equation (21).
As a result, T D are obtained under the condition jA 0 j < A 1 . Let h n denote least h p value which is greater than 0. Therefore, the solution of equation (17) is stable for h 2 .0; h n / when 0 < A 0 < A 1 . Hence we can state the following result about stability of the solution for the equation (1).
Theorem 3.12. The solution of the equation (1) with delay .u.t // > 0 such that 0 < .u.t // < M 1 for all t 2 R C is asymptotically stable under the conditions jA 0 j < A 1 and M 1 < h n .
Now, we give a stability criterion which is independent from delay for equation (1) by using D-partition method.
Theorem 3.13. If jA 1 j < A 0 , then the solution of equation (1) is asymptotically stable.
Proof. It is obvious from (11) that, jA 1 .!/j jA 0 .!/j for all ! 2 J k . Therefore, all of the D-curves is in this region, i.e., there is no D-curve in the region described by jA 1 j < jA 0 j. Moreover, the half line A 0 > 0 and A 1 D 0 on which the solution equation (1) is asymptotically stable, is in the region described by jA 1 j < A 0 .