# American Institute of Mathematical Sciences

January  2020, 13(1): 85-104. doi: 10.3934/dcdss.2020005

## Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

 1 Department of Mathematics and Statistics, Queen's University, Kingston, ON, Canada K7L 3N6 2 Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 0B9

* Corresponding author: A. R. Humphries

Received  March 2017 Revised  January 2018 Published  February 2019

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.

Citation: F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 85-104. doi: 10.3934/dcdss.2020005
##### References:
 [1] D. I. Barnea, A method and new results for stability and instability of autonomous functional differential equations, SIAM J. Appl. Math, 17 (1969), 681-697.  doi: 10.1137/0117064. [2] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford Science Publications, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001. [3] R. C. Calleja, A. R. Humphries and B. Krauskopf, Resonance phenomena in a scalar delay differential equation with two state-dependent delays, SIAM J. Appl. Dyn. Syst., 16 (2017), 1474-1513.  doi: 10.1137/16M1087655. [4] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750. [5] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, vol. 110 of Applied Mathematical Sciences, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-4206-2. [6] R. D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Phys., 21 (1963), 122-142.  doi: 10.1016/0003-4916(63)90227-6. [7] L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, 1973. [8] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, vol. 184 of Applied Mathematical Sciences, Springer-Verlag, 2013. [9] I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst. Ser. A, 18 (2007), 773-791.  doi: 10.3934/dcds.2007.18.773. [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7. [11] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Cañada, P. Drábek and A. Fonda), vol. 3, Elsevier/North Holland, 2006,435–545. doi: 10.1016/S1874-5725(06)80009-X. [12] N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Soc., 25 (1950), 226-232.  doi: 10.1112/jlms/s1-25.3.226. [13] A. R. Humphries, D. A. Bernucci, R. Calleja, N. Homayounfar and M. Snarski, Periodic solutions of a singularly perturbed delay differential equation with two state-dependent delays, J. Dyn. Diff. Eqs., 28 (2016), 1215-1263.  doi: 10.1007/s10884-015-9484-4. [14] A. R. Humphries, O. DeMasi, F. M. G. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state dependent delays, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 2701-2727.  doi: 10.3934/dcds.2012.32.2701. [15] A. R. Humphries and F. M. G. Magpantay, Lyapunov-Razumikhin techniques for state-dependent delay differential equations, 2017, arXiv: 1507.00141v3. [16] T. Insperger and Stépán, Semi-Discretization for Time-Delay Systems, vol. 178 of Applied Mathematical Sciences, Springer-Verlag, 2011. doi: 10.1007/978-1-4614-0335-7. [17] T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283.  doi: 10.1007/s11071-006-9068-2. [18] G. Kozyreff and T. Erneux, Singular Hopf bifurcation in a differential equation with large state-dependent delay, Proc. Roy. Soc. A, 470 (2013), 20130596, 16 pp. doi: 10.1098/rspa.2013.0596. [19] T. Krisztin, On stability properties for one-dimensional functional differential equations, Funkcialaj Ekvacioj, 34 (1991), 241-256. [20] T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, J. Dyn. Diff. Eqs., 13 (2001), 453-522.  doi: 10.1023/A:1016635223074. [21] F. M. G. Magpantay, N. Kosovalic and J. Wu, An age-structured population model with state-dependent delay: Derivation and numerical integration, SIAM J. Num. Anal., 5 (2014), 735-756.  doi: 10.1137/120903622. [22] J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Diff. Eqns., 72 (1988), 270-315.  doi: 10.1016/0022-0396(88)90157-X. [23] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, Ⅰ., Arch. Ration. Mech. Anal., 120 (1992), 99-146.  doi: 10.1007/BF00418497. [24] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: Ⅱ., J. Reine Angew. Math., 477 (1996), 129-197.  doi: 10.1515/crll.1996.477.129. [25] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: Ⅲ, Discrete Contin. Dyn. Syst. Ser. A, 189 (2003), 640-692.  doi: 10.1016/S0022-0396(02)00088-8. [26] J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differetnial equations, J. Diff. Eqns., 250 (2011), 4037-4084.  doi: 10.1016/j.jde.2010.10.024. [27] A. Myshkis, Razumikhin's method in the qualitative theory of processes with delay, J. Appl. Math. Stoch. Anal., 8 (1995), 233-247.  doi: 10.1155/S1048953395000219. [28] B. S. Razumikhin, An application of Lyapunov method to a problem on the stability of systems with a lag, Autom. Remote Control, 21 (1960), 740-748. [29] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, Springer, 2011. doi: 10.1007/978-1-4419-7646-8. [30] E. Stumpf, Local stability analysis of differential equations with state-dependent delay, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 3445-3461.  doi: 10.3934/dcds.2016.36.3445. [31] H.-O. Walther, Stable periodic motion of a system using echo for position control, J. Dyn. Diff. Eqns., 15 (2003), 143-223.  doi: 10.1023/A:1026161513363. [32] J. A. Yorke, Asymptotic stability for one dimensional differential-delay equations, J. Diff. Eqns., 7 (1970), 189-202.  doi: 10.1016/0022-0396(70)90132-4.

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##### References:
 [1] D. I. Barnea, A method and new results for stability and instability of autonomous functional differential equations, SIAM J. Appl. Math, 17 (1969), 681-697.  doi: 10.1137/0117064. [2] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford Science Publications, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001. [3] R. C. Calleja, A. R. Humphries and B. Krauskopf, Resonance phenomena in a scalar delay differential equation with two state-dependent delays, SIAM J. Appl. Dyn. Syst., 16 (2017), 1474-1513.  doi: 10.1137/16M1087655. [4] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750. [5] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, vol. 110 of Applied Mathematical Sciences, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-4206-2. [6] R. D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Phys., 21 (1963), 122-142.  doi: 10.1016/0003-4916(63)90227-6. [7] L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, 1973. [8] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, vol. 184 of Applied Mathematical Sciences, Springer-Verlag, 2013. [9] I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst. Ser. A, 18 (2007), 773-791.  doi: 10.3934/dcds.2007.18.773. [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7. [11] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Cañada, P. Drábek and A. Fonda), vol. 3, Elsevier/North Holland, 2006,435–545. doi: 10.1016/S1874-5725(06)80009-X. [12] N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Soc., 25 (1950), 226-232.  doi: 10.1112/jlms/s1-25.3.226. [13] A. R. Humphries, D. A. Bernucci, R. Calleja, N. Homayounfar and M. Snarski, Periodic solutions of a singularly perturbed delay differential equation with two state-dependent delays, J. Dyn. Diff. Eqs., 28 (2016), 1215-1263.  doi: 10.1007/s10884-015-9484-4. [14] A. R. Humphries, O. DeMasi, F. M. G. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state dependent delays, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 2701-2727.  doi: 10.3934/dcds.2012.32.2701. [15] A. R. Humphries and F. M. G. Magpantay, Lyapunov-Razumikhin techniques for state-dependent delay differential equations, 2017, arXiv: 1507.00141v3. [16] T. Insperger and Stépán, Semi-Discretization for Time-Delay Systems, vol. 178 of Applied Mathematical Sciences, Springer-Verlag, 2011. doi: 10.1007/978-1-4614-0335-7. [17] T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283.  doi: 10.1007/s11071-006-9068-2. [18] G. Kozyreff and T. Erneux, Singular Hopf bifurcation in a differential equation with large state-dependent delay, Proc. Roy. Soc. A, 470 (2013), 20130596, 16 pp. doi: 10.1098/rspa.2013.0596. [19] T. Krisztin, On stability properties for one-dimensional functional differential equations, Funkcialaj Ekvacioj, 34 (1991), 241-256. [20] T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, J. Dyn. Diff. Eqs., 13 (2001), 453-522.  doi: 10.1023/A:1016635223074. [21] F. M. G. Magpantay, N. Kosovalic and J. Wu, An age-structured population model with state-dependent delay: Derivation and numerical integration, SIAM J. Num. Anal., 5 (2014), 735-756.  doi: 10.1137/120903622. [22] J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Diff. Eqns., 72 (1988), 270-315.  doi: 10.1016/0022-0396(88)90157-X. [23] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, Ⅰ., Arch. Ration. Mech. Anal., 120 (1992), 99-146.  doi: 10.1007/BF00418497. [24] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: Ⅱ., J. Reine Angew. Math., 477 (1996), 129-197.  doi: 10.1515/crll.1996.477.129. [25] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: Ⅲ, Discrete Contin. Dyn. Syst. Ser. A, 189 (2003), 640-692.  doi: 10.1016/S0022-0396(02)00088-8. [26] J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differetnial equations, J. Diff. Eqns., 250 (2011), 4037-4084.  doi: 10.1016/j.jde.2010.10.024. [27] A. Myshkis, Razumikhin's method in the qualitative theory of processes with delay, J. Appl. Math. Stoch. Anal., 8 (1995), 233-247.  doi: 10.1155/S1048953395000219. [28] B. S. Razumikhin, An application of Lyapunov method to a problem on the stability of systems with a lag, Autom. Remote Control, 21 (1960), 740-748. [29] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, Springer, 2011. doi: 10.1007/978-1-4419-7646-8. [30] E. Stumpf, Local stability analysis of differential equations with state-dependent delay, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 3445-3461.  doi: 10.3934/dcds.2016.36.3445. [31] H.-O. Walther, Stable periodic motion of a system using echo for position control, J. Dyn. Diff. Eqns., 15 (2003), 143-223.  doi: 10.1023/A:1026161513363. [32] J. A. Yorke, Asymptotic stability for one dimensional differential-delay equations, J. Diff. Eqns., 7 (1970), 189-202.  doi: 10.1016/0022-0396(70)90132-4.
(a) The green region is the parameter region where the zero solution was proven to be locally asymptotically stable using Lyapunov-Razumikhin techniques in [15] (denoted as the set $\{P(1,0,3)<1\}$). The blue curve shows the boundary of the region where we numerically evaluate $M_{1,\infty} = N_{1,\infty} = 0$ and the red curve shows the boundary of the set where $M_{2,\infty} = N_{2,\infty} = 0$. The steady state of (2) is guaranteed to be globally asymptotically stable to the left of the red curve. The brown curve shows the locus of a fold bifurcation of periodic orbits computed using DDE-Biftool. To the right of the brown curve there are co-existing non-trivial periodic orbits and the steady-state cannot be globally asymptotically stable. Other parameters are fixed at $a = c = 1$. (b) Bifurcation diagram at $\mu = -0.25$ as $\sigma$ is varied showing a subcritical Hopf bifurcation from the steady state and the amplitude of the resulting periodic orbits. There is a small interval of $\sigma$ values for which bistability occurs, and the steady-state is only locally asymptotically stable.
(a) & (c): Numerically simulated solutions $u(t)$ to (2) with $\varphi (t) = 0.99N_0$ for all $t\leq0$, along with the bounds $M_{k,n}$ and $N_{k,n}$ from Theorem 3.6 with $k = 1$ and $k = 2$. In (a), $(\mu,\sigma) = (-0.25,-1.75)\not\in \mathop \Sigma_{w}$ and the steady state solution is unstable. The solution $u(t)$ converges to a periodic orbit. In (c), $(\mu,\sigma) = (-2,-2.8)\in \mathop \Sigma_{w}$ and $u(t)$ converges to the steady state. (b) & (d): For the same values of $\mu$ as in (a) and (c) we vary $\sigma$ and plot the bounds $[M_{k,\infty},N_{k,\infty}]$ on the persistent dynamics from Theorem 3.7 for $k = 1$ and $k = 2$ along with $[\min_t u(t),\max_t u(t)]$ for the periodic orbit created at the Hopf bifurcation when $\sigma$ crosses the boundary of $\mathop \Sigma_{w}$.
Relative improvement in the lower and upper bounds of periodic solutions to (2) using the $k = 1$ and $k = 2$ generalised Lyapunov-Razumikhin technique. The yellow region matches the region of global asymptotic stability indicated by the blue (for $k = 1$) and red (for $k = 2$) curves in Figure 3
(a) The analytic stability region $\Sigma_\star$ in the $(\mu,\sigma)$ plane, divided into the delay-independent cone $\mathop \Sigma _{\Delta}$, and the delay-dependent wedge $\mathop \Sigma_{w}$ and cusp $\mathop \Sigma_{c}$. (b) Sample dynamics using parameter values in the $\mathop \Sigma _{\Delta}$ (upper panel) and $\mathop \Sigma_{w}$ (lower panel). Both state-dependent ($c = 1$) and constant delay ($c = 0$) solutions are shown in each case. The initial function is the constant function $\varphi(t) = 1$ for all the examples.
Lower bounds on $\delta$, with $a = c = 1$ for parameter values $(\mu,\sigma)$ in parts of the wedge $\mathop \Sigma_{w}$ so that the ball $\{\varphi :\|\varphi \|<\delta\}$ is contained in the basin of attraction of the steady state. These bounds were derived using the Lyapunov-Razumikhin techniques presented in [15]. The SDDE is initially integrated through $k\tau$ time units so that bounds can be established on derivatives of the solution. The value of $k\in\{1,2,3\}$ which results in the largest bound is indicated
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