We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We present the bifurcation diagram in the parameter plane of the two feedback strengths showing how periodic orbits bifurcate from a curve of Hopf bifurcations and disappear along a curve where both period and amplitude grow beyond bound as the orbits become saw-tooth shaped. We then 'switch on' the delay within the state-dependent feedback term, reflected by a parameter $ b>0 $. Our main conclusion is that the new parameter $ b $ has an immediate effect: as soon as $ b>0 $ the bifurcation diagram for $ b = 0 $ changes qualitatively and, specifically, the nature of the limiting saw-tooth shaped periodic orbits changes. Moreover, we show — numerically and through center manifold analysis — that a degeneracy at $ b = 1/3 $ of an equilibrium with a double real eigenvalue zero leads to a further qualitative change and acts as an organizing center for the bifurcation diagram. Our results demonstrate that state dependence in delayed feedback terms may give rise to new dynamics and, moreover, that the observed dynamics may change significantly when the state-dependent feedback depends on past states of the system. This is expected to have implications for models arising in different application contexts, such as models of human balancing and conceptual climate models of delayed action oscillator type.
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Figure 1. Bifurcation diagram of system (3) with $ b = 0 $ in the $ (\alpha, \beta) $-plane. Panel (a) shows the stability diagram of the equilibrium $ u\equiv0 $ with the curves Z (black) of a simple zero eigenvalue and H of a pair of complex conjugate eigenvalues with zero real part, which meet at the DZ (green diamond) of double zero eigenvalues and bound the region where (3) is linearly stable (shaded); also shown for reference is the line $ \alpha = \beta $ (black dotted). The associated Hopf bifurcation of (3) along the curve H has a generalized Hopf bifurcation point GH (brown square), where its criticality changes from supercritical (blue) to subcritical (cyan). Panel (b) is an enlargement that also shows a curve of folds of periodic orbits F (red), a curve M (black dashed) of periodic orbits with a point that attains the minimum value $ -1 $, which has a corner at MF (magenta circle) where there is a fold periodic orbit with minimum value $ -1 $. The blue shaded regions indicate the (co)existence of periodic orbits, and the blue crosses mark the locations of the periodic orbits shown in Fig. 2
Figure 2. Periodic solutions of system (3) for $ b = 0 $ shown as time series of $ u(t) $ and of $ u(t-1-u(t)) $; panel (a) shows the stable periodic orbit for $ \alpha = -0.1 $ and $ \beta = -1.7 $, and panel (b) the unstable orbit for $ \alpha = 0.3 $ and $ \beta = -1.1 $. These parameter values are denoted by blue crosses in Fig. 1
Figure 3. Periodic orbits of system (3) for $ b = 0 $ as $ \alpha $ increases to $ 0 $ for different fixed values of $ \beta<-1 $ as shown. Panel (a) shows their asymptotic profiles in a waterfall plot synchronized to their fast segments, and panel (b) shows them in the $ (u(t), u(t-1-u(t))) $-plane with enlargements near the highlighted two corners; all panels are scaled by the period $ T $, the periodic orbits have been continued in to $ T = 5\cdot10^{2} $, and the arrows in panel (b) indicate the time-scale separation along the orbit
Figure 4. Continuation of eventually stable periodic orbits of system (3) for $ b = 0 $ in $ \alpha $ for different fixed values of $ \beta<-1 $; compare with Fig. 3. Illustrated on a logarithmic scale for $ \alpha $ are the asymptotic behavior of the period in panel (a), the minimum in panel (b), the amplitude in panel (c), and the rescaled period $ T \cdot \alpha / (1+\beta) $ in panel (d); circles indicate Hopf bifurcations and triangles fold points
Figure 5. Periodic orbits of system (3) for $ b = 0 $ as $ \beta $ increases to $ -1 $ for different fixed values of $ 0\leq\alpha<1 $, shown as in Fig. 3; here the periodic orbits for $ 0\leq\alpha\leq0.2 $ have been continued to $ T = 5\cdot10^{2} $ and those for $ 0.3\leq\alpha\leq0.9 $ to $ \beta = -1-10^{-5} $
Figure 6. Continuation of unstable periodic orbits of system (3) for $ b = 0 $ in $ \beta $ for different fixed values of $ \alpha $; compare with Fig. 5. Illustrated on a logarithmic scale for $ \alpha $ are the asymptotic behavior of the period in panel (a), the minimum in panel (b), the amplitude in panel (c), and the rescaled period $ T(1-\alpha) $ in panel (d); circles indicate Hopf bifurcations
Figure 7. Bifurcation diagram of system (3) in the $ (\alpha, \beta) $-plane for (a) $ b = 0 $, (b) $ b = 0.01 $, (c1) $ b = 0.1 $ with enlargement (c2); and (d1) $ b = 0.2 $ with enlargement (d2). Shown are the curves Z (black) of simple zero eigenvalue and H of Hopf bifurcations (blue when supercritical, cyan when subcritical) of $ u = 0 $, the curve F (red) of fold bifurcation of periodic orbits, and the loci M (black dashed) and L (green dashed); also shown are the points DZ (green diamond) of double zero eigenvalues, GH (brown square) of generalized Hopf bifurcation, CP (purple triangle) of cusp bifurcation on F, and MF (magenta circle) of fold periodic orbit with minimum value $ -1 $. The stability region of $ u = 0 $ is shaded gray, and blue shading indicates (co)existence of periodic orbits
Figure 8. Periodic orbits of the system (3) for $ b = 0.1 $ as $ \alpha $ approaches the locus M for the different fixed values of $ \beta = -1.2 $ down to $ \beta = -2 $. Panel (a) shows their (rescaled) profiles at M synchronized to their fast segments, and panel (b) shows them in the $ (u(t-1-u(t-b))/T, u(t)/T) $-plane; compare with Fig. 3. Panel (c) shows the continuations in $ \alpha $ of the periodic orbits, represented by the period $ T $, from the Hopf bifurcation H (circles), possibly via fold points (triangles) to the locus M (squares); the same branches are shown in panel (d) as a function of the amplitude $ A $ of the periodic orbits; compare with Fig. 4
Figure 9. Periodic orbits of system (3) for $ b = 0.1 $ as $ \beta $ approaches M for ten values of $ \alpha = 0 $ up to $ \alpha = 0.9 $. Panels (a) and (b) show the periodic orbits at M as (rescaled) profiles and in the $ (u(t-1-u(t-b))/T, u(t)/T) $-plane, respectively; compare with Fig. 5. Panels (c) and (d) show their period $ T $ and amplitude $ A $ when continued in $ \beta $ from H (circles), via fold points (triangles) to M (squares) and beyond (lighter curves) up to $ T = 100 $. Panels (e) and (f) show the periodic orbits with $ T = 100 $ as profiles and in the $ (u(t-1-u(t-b))/T, u(t)/T) $-plane, respectively
Figure 10. Bifurcation diagram of system (3) in the $ (\alpha, \beta) $-plane (a1) for $ b = 0.5 $, and (a2) an enlargement near the point DZ; compare with Fig. 7. The inset panel (b) shows the branches of generalized Hopf bifurcation points GH along the Hopf curve H in the $ (\alpha, b) $-plane; the brown square highlights the point GH for $ b = 0.5 $ shown in panels (a1) and (a2)
Figure 11. Illustration of the truncated dynamics on the center manifold near DZ when the parameter $ p $ is small and negative ($ -1\ll p<0 $). Panels (a) and (c) show the slow manifold $ v = y/(q+2y) $, which is a small perturbation of the line of equilibria at $ p = 0 $ and the fixed point at $ (0, 0) $. Panel (b) shows the level curves of the potential $ V $ in (19) for the case $ q = 0 $ and $ b = 1/3 $, where the truncated system (18) is conservative
Figure 12. Bifurcation diagrams near the point DZGH. Panel (a) is the two-parameter bifurcation diagram in the $ (p, q) $-plane of the fifth-order expansion on the center manifold for $ b = 1/3 $; shown are the curves Z of zero-eigenvalue equilibria, H of Hopf bifurcations and F of fold periodic orbits, which intersect at the point DZGH. Panel (b) shows the family of fold periodic orbits along F in the $ (y, y') $-plane as a function of the parameter $ p $ (with $ q $ varying accordingly, but not shown). The three-parameter bifurcation diagram near DZGH of the fifth-order expansion in $ (b, p, q) $-space near $ (1/3, 0, 0) $ is shown in panel (c), and that of the full DDE (3) in $ (b, \alpha, \beta) $-space near $ (1/3, 1, -1) $ is shown in panel (d); both consist of the surfaces Z, H and F, and the curves DZ and GH that meet at the point DZGH
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Bifurcation diagram of system (3) with
Periodic solutions of system (3) for
Periodic orbits of system (3) for
Continuation of eventually stable periodic orbits of system (3) for
Periodic orbits of system (3) for
Continuation of unstable periodic orbits of system (3) for
Bifurcation diagram of system (3) in the
Periodic orbits of the system (3) for
Periodic orbits of system (3) for
Bifurcation diagram of system (3) in the
Illustration of the truncated dynamics on the center manifold near DZ when the parameter
Bifurcation diagrams near the point DZGH. Panel (a) is the two-parameter bifurcation diagram in the