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Bistability, bifurcations and chaos in the Mackey-Glass equation

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  • Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In particular a cusp bifurcation of periodic orbits and resulting branches of folds of periodic orbits effectively partition the parameter space into regions where different behaviours are seen. The cusp bifurcation leads directly to bistability between periodic orbits, and subsequently to bistability between a periodic orbit and a chaotic attractor. This leads to two different mechanisms by which the chaotic attractor is destroyed in a global bifurcation with a periodic orbit in either an interior crisis or a boundary crisis. In another part of parameter space a sequence of subcritical period-doublings is found to give rise to bistability between a periodic orbit and a chaotic attractor. Torus bifurcations, and a codimension-two fold-flip bifurcation are also identified, and Lyapunov exponent computations are used to determine chaotic regions and attractor dimension.

    Mathematics Subject Classification: Primary: 34K18, 34K60; Secondary: 34K13, 34K23, 37G15, 37L30, 37N25.


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  • Figure 1.  Bifurcation Diagram for the Mackey-Glass Equation (11) with $ \tau = 2 $ as $ n $ is increased showing the 4 branches of periodic orbits computed using DDE-BIFTOOL, with solid/dashed lines indicating respectively stable/unstable orbits

    Figure 2.  Phase portraits for the Mackey-Glass Equation (11) with $ \tau = 2 $. Stable orbits are coloured blue, while all other colours indicate unstable orbits. For (a)-(d) all orbits computed using DDE-BIFTOOL. For (e) and (f) all DDE-BIFTOOL computed periodic orbits are unstable, and the stable object shown is the result of simulation using $\texttt{dde23}$. On panels (b) and (c) the $ \dot{u} = 0 $ nullcline is also shown

    Figure 3.  Orbit Diagram for the Mackey-Glass Equation (11) with $ \tau = 2 $, with $ n $ varied between $ 4 $ and $ 12 $ in increments of $ 10^{-3} $ between each simulation. See Section 3.2 for further details on the numerical computation

    Figure 4.  Orbit Diagrams for the Mackey-Glass Equation (11) with $ \tau = 2 $. (a) A close up of the dynamics during the period-doubling cascade for $ n\approx8.85 $, with $ n $ increased in increments of $ 5\times10^{-4} $ between each computation. (b) The periodic window near $ n\approx9.7 $, obtained with increments of $ n $ of $ 2\times10^{-4} $

    Figure 5.  (a) Phase portrait of the Mackey-Glass Equation (11) with $ \tau = 2 $ and $ n = 20 $. The stable periodic orbit (shown in blue) was computed using $\texttt{dde23}$, while all the DDE-BIFTOOL computed periodic orbits (all other colours) are unstable. (b) Two-parameter continuation of the first three period-doubling bifurcations in $ n $ and $ \tau $

    Figure 6.  Bifurcation Diagram for the Mackey-Glass Equation (11) showing the first Hopf bifurcation, and the bifurcations from the stable periodic orbit created at the Hopf bifurcation, as well as the bifurcations from the first (stable) period-doubled orbit. Also shown are regions where chaotic dynamics are detected. Several valleys of non-chaotic behaviour are visible in the chaotic regions, and the fold and period-doubling bifurcations that bound one of these valleys are shown. Also indicated are several interesting codimension-two bifurcations and regions of bistability of solutions

    Figure 7.  (a) Enlarged detail of Bifurcation Diagram from Figure 6 near the cusp point at $ (n, \tau) = (15.16, 1.1) $. Phase portraits for (b) $ (n, \tau) = (14.5, 1.3) $ showing two stable and one unstable periodic orbits, (c) $ (n, \tau) = (14, 1.35) $ showing two stable (one period-doubled) and two unstable periodic orbits, and, (d) $ (n, \tau) = (13.4, 1.44) $ showing one stable and two unstable periodic orbits coexisting with a stable chaotic attractor

    Figure 8.  Phase portraits for the Mackey-Glass equation with $ \tau = 2 $ showing the destruction of the chaotic attractor as the fold bifurcation is crossed at $ n = 11.21485 $. The stable dynamics (in blue) is found by simulation. The unstable periodic orbits (all other colours) computed by DDE-BIFTOOL correspond to the left-hand periodic orbit and its period doublings. (a) $ n = 11.1 $ (b and c) $ n = 11.2 $, with integration time tripled in (c). (d) $ n = 11.215 $

    Figure 9.  (a) Enlarged detail of part of the Bifurcation Diagram from Figure 6. (b) DDE-BIFTOOL bifurcation diagram as $ \tau $ is varied with $ n = 8.1 $ fixed, showing period-doubling bifurcations from the first stable period-doubled orbit, and from the resulting period-doubled orbits, and fold bifurcations on these orbits. (c) Orbit Diagram for the Mackey-Glass Equation (11) computed with $\texttt{dde23}$ with $ n = 8.1 $ for increasing $ \tau $ in increments of $ 10^{-4} $, with initial function given by solution at the previous $ \tau $ value. (d) As (c) except using the constant initial function (9) for every simulation. In (c) and (d) parameter values for the bifurcations in (b) are also marked for comparison

    Figure 10.  (a) Enlarged detail of Bifurcation Diagram from Figure 6 with two fold-flip bifurcations indicated. (b) Bifurcation curves near the fold-flip bifurcation FF1, numbered and labelled as in [27]; see text for details

    Figure 11.  Chaotic attractor dimension for the Mackey-Glass equation (11) computed using (10)

    Figure 12.  (a) A detail from Figure 6 indicating the parameter range in the $ (n, \tau) $ plane considered in panels (b) and (c). (b) Rotated surface plot of the Lyapunov dimension for the parameters within the black box in panel (a) near to the valley of periodic behaviour first seen in Figure 6. The surface is coloured red where the Lyapunov dimension $ d>2 $. (c) Surface plot of the Lyapunov dimension for the parameters within the grey box in panel (a) showing many valleys of periodic behaviour

    Figure 13.  (a) For $ n = 7 $, growth of attractor dimension with $ \tau $ (blue line). Also shown is the dimension of the unstable manifold of the steady state $ \xi_1 $ (dotted line). (b) As (a) but for $ n = 18 $

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