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

Figure 3.  (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.

Figure 4.  (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} $.

Figure 5.  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

Figure 1.  (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.

Figure 2.  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