# American Institute of Mathematical Sciences

July  2022, 9(3): 421-450. doi: 10.3934/jcd.2022009

## Bistability, bifurcations and chaos in the Mackey-Glass equation

 1 Department of Mathematics, University of California, San Diego, La Jolla, California 92093, USA 2 Departments of Mathematics & Statistics, and, Physiology, McGill University, Montreal, Quebec H3A 0B9, Canada

Received  February 2022 Revised  March 2022 Published  July 2022 Early access  April 2022

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.

Citation: Valentin Duruisseaux, Antony R. Humphries. Bistability, bifurcations and chaos in the Mackey-Glass equation. Journal of Computational Dynamics, 2022, 9 (3) : 421-450. doi: 10.3934/jcd.2022009
##### References:
 [1] F. A. Bartha, T. Krisztin and A. Vígh, Stable periodic orbits for the Mackey-Glass equation, J. Differential Equations, 296 (2021), 15-49.  doi: 10.1016/j.jde.2021.05.052. [2] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), 317-346.  doi: 10.1016/0025-5564(94)00078-E. [3] A. Bellen, N. Guglielmi, S. Maset and M. Zennaro, Recent trends in the numerical solution of retarded functional differential equations, Acta Numer., 18 (2009), 1-110.  doi: 10.1017/S0962492906390010. [4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, The Clarendon Press, Oxford University Press, New York, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001. [5] M. Bosschaert, DDE-Biftool Tutorials, Online, (2022), https://sites.google.com/a/uhasselt.be/maikel-bosschaert. [6] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations. A Numerical Approach with MATLAB, SpringerBriefs in Electrical and Computer Engineering, Springer, New York, 2015. doi: 10.1007/978-1-4939-2107-2. [7] D. Breda and E. Van Vleck, Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math., 126 (2014), 225-257.  doi: 10.1007/s00211-013-0565-1. [8] 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. [9] M. Craig, A. R. Humphries and M. C. Mackey, A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of $\mathrm{G - CSF}$/neutrophil binding and internalization, Bull. Math. Biol., 78 (2016), 2304-2357.  doi: 10.1007/s11538-016-0179-8. [10] D. C. De Souza and A. R. Humphries, Dynamics of a mathematical hematopoietic stem-cell population model, SIAM J. Appl. Dyn. Syst., 18 (2019), 808-852.  doi: 10.1137/18M1165086. [11] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21.  doi: 10.1145/513001.513002. [12] J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Phys. D, 4 (1981/82), 366-393.  doi: 10.1016/0167-2789(82)90042-2. [13] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., 19 (1978), 25-52.  doi: 10.1007/BF01020332. [14] L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Annals of the New York Academy of Sciences, 316 (1979), 214-235.  doi: 10.1111/j.1749-6632.1979.tb29471.x. [15] K. Gopalsamy, S. I. Trofimchuk and N. R. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Math. J., 50 (1998), 3-12.  doi: 10.1007/BF02514684. [16] C. Grebogi, E. Ott and J. A. Yorke, Chaotic attractors in crisis, Phys. Rev. Lett., 48 (1982), 1507-1510.  doi: 10.1103/PhysRevLett.48.1507. [17] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [18] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [19] 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. [20] F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, Applied Mathematical Sciences, 126. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1828-9. [21] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete Contin. Dyn. Syst., 32 (2012), 2701-2727.  doi: 10.3934/dcds.2012.32.2701. [22] K. Ikeda and K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29 (1987), 223-235.  doi: 10.1016/0167-2789(87)90058-3. [23] T. Insperger and G. Stépán, Semi-Discretization for Time-Delay Systems. Stability and Engineering Applications, Applied Mathematical Sciences, 178. Springer, New York, 2011. doi: 10.1007/978-1-4614-0335-7. [24] L. Junges and J. A. Gallas, Intricate routes to chaos in the Mackey-Glass delayed feedback system, Physics Letters A, 376 (2012), 2109-2116.  doi: 10.1016/j.physleta.2012.05.022. [25] J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., Springer, Berlin, 730 (1979), 204-227. [26] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [27] Y. A. Kuznetsov, H. G. E. Meijer and L. Van Veen, The fold-flip bifurcation, J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2253-2282.  doi: 10.1142/S0218127404010576. [28] S. Lepri, G. Giacomelli, A. Politi and F. T. Arecchi, High-dimensional chaos in delayed dynamical systems, Physica D, 70 (1994), 235-249.  doi: 10.1016/0167-2789(94)90016-7. [29] M. C. Mackey, Periodic hematological disorders: Quintessential examples of dynamical diseases, Chaos, 30 (2020), 063123, 8 pp. doi: 10.1063/5.0006517. [30] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326. [31] Mathworks, MATLAB 2020a, Mathworks, Natick, Massachusetts, 2020. [32] B. Mensour and A. Longtin, Power spectra and dynamical invariants for delay-differential and difference equations, Physica D, 113 (1998), 1-25.  doi: 10.1016/S0167-2789(97)00185-1. [33] R. D. Nussbaum, Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.  doi: 10.1307/mmj/1029001104. [34] L. Pujo-Menjouet, Blood cell dynamics: Half of a century of modelling, Math. Model. Nat. Phenom., 11 (2016), 92-115.  doi: 10.1051/mmnp/201611106. [35] J. Rankin and H. M. Osinga, Parameter-dependent behaviour of periodic channels in a locus of boundary crisis, European Physical Journal Special Topics, 226 (2017), 1739-1750.  doi: 10.1140/epjst/e2017-70048-x. [36] C. Robert, K. T. Alligood, E. Ott and J. A. Yorke, Explosions of chaotic sets, Physica D, 144 (2000), 44-61.  doi: 10.1016/S0167-2789(00)00074-9. [37] G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890. [38] S. Schirm and M. Scholz, A biomathematical model of human erythropoiesis and iron metabolism, Scientific Reports, 10, (2020), 8602. doi: 10.1038/s41598-020-65313-5. [39] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL Manual - Bifurcation Analysis of Delay Differential Equations, 2015, Eprint, arXiv: 1406.7144 [math.DS]. [40] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [41] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015. [42] H.-O. Walther, The impact on mathematics of the paper ''Oscillation and Chaos in Physiological Control Systems" by Mackey and Glass in Science, 1977, (2020), Eprint, arXiv: 2001.09010 [math.DS]. [43] J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007), 2483-2498.  doi: 10.1088/0951-7715/20/11/002. [44] S. Wieczorek, B. Krauskopf and D. Lenstra, Unnested islands of period doublings in an injected semiconductor laser, Phys. Rev. E, 64 (2001), 056204, 9 pp. doi: 10.1103/PhysRevE.64.056204.

show all references

##### References:
 [1] F. A. Bartha, T. Krisztin and A. Vígh, Stable periodic orbits for the Mackey-Glass equation, J. Differential Equations, 296 (2021), 15-49.  doi: 10.1016/j.jde.2021.05.052. [2] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), 317-346.  doi: 10.1016/0025-5564(94)00078-E. [3] A. Bellen, N. Guglielmi, S. Maset and M. Zennaro, Recent trends in the numerical solution of retarded functional differential equations, Acta Numer., 18 (2009), 1-110.  doi: 10.1017/S0962492906390010. [4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, The Clarendon Press, Oxford University Press, New York, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001. [5] M. Bosschaert, DDE-Biftool Tutorials, Online, (2022), https://sites.google.com/a/uhasselt.be/maikel-bosschaert. [6] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations. A Numerical Approach with MATLAB, SpringerBriefs in Electrical and Computer Engineering, Springer, New York, 2015. doi: 10.1007/978-1-4939-2107-2. [7] D. Breda and E. Van Vleck, Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math., 126 (2014), 225-257.  doi: 10.1007/s00211-013-0565-1. [8] 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. [9] M. Craig, A. R. Humphries and M. C. Mackey, A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of $\mathrm{G - CSF}$/neutrophil binding and internalization, Bull. Math. Biol., 78 (2016), 2304-2357.  doi: 10.1007/s11538-016-0179-8. [10] D. C. De Souza and A. R. Humphries, Dynamics of a mathematical hematopoietic stem-cell population model, SIAM J. Appl. Dyn. Syst., 18 (2019), 808-852.  doi: 10.1137/18M1165086. [11] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21.  doi: 10.1145/513001.513002. [12] J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Phys. D, 4 (1981/82), 366-393.  doi: 10.1016/0167-2789(82)90042-2. [13] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., 19 (1978), 25-52.  doi: 10.1007/BF01020332. [14] L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Annals of the New York Academy of Sciences, 316 (1979), 214-235.  doi: 10.1111/j.1749-6632.1979.tb29471.x. [15] K. Gopalsamy, S. I. Trofimchuk and N. R. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Math. J., 50 (1998), 3-12.  doi: 10.1007/BF02514684. [16] C. Grebogi, E. Ott and J. A. Yorke, Chaotic attractors in crisis, Phys. Rev. Lett., 48 (1982), 1507-1510.  doi: 10.1103/PhysRevLett.48.1507. [17] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [18] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [19] 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. [20] F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, Applied Mathematical Sciences, 126. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1828-9. [21] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete Contin. Dyn. Syst., 32 (2012), 2701-2727.  doi: 10.3934/dcds.2012.32.2701. [22] K. Ikeda and K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29 (1987), 223-235.  doi: 10.1016/0167-2789(87)90058-3. [23] T. Insperger and G. Stépán, Semi-Discretization for Time-Delay Systems. Stability and Engineering Applications, Applied Mathematical Sciences, 178. Springer, New York, 2011. doi: 10.1007/978-1-4614-0335-7. [24] L. Junges and J. A. Gallas, Intricate routes to chaos in the Mackey-Glass delayed feedback system, Physics Letters A, 376 (2012), 2109-2116.  doi: 10.1016/j.physleta.2012.05.022. [25] J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., Springer, Berlin, 730 (1979), 204-227. [26] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [27] Y. A. Kuznetsov, H. G. E. Meijer and L. Van Veen, The fold-flip bifurcation, J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2253-2282.  doi: 10.1142/S0218127404010576. [28] S. Lepri, G. Giacomelli, A. Politi and F. T. Arecchi, High-dimensional chaos in delayed dynamical systems, Physica D, 70 (1994), 235-249.  doi: 10.1016/0167-2789(94)90016-7. [29] M. C. Mackey, Periodic hematological disorders: Quintessential examples of dynamical diseases, Chaos, 30 (2020), 063123, 8 pp. doi: 10.1063/5.0006517. [30] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326. [31] Mathworks, MATLAB 2020a, Mathworks, Natick, Massachusetts, 2020. [32] B. Mensour and A. Longtin, Power spectra and dynamical invariants for delay-differential and difference equations, Physica D, 113 (1998), 1-25.  doi: 10.1016/S0167-2789(97)00185-1. [33] R. D. Nussbaum, Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.  doi: 10.1307/mmj/1029001104. [34] L. Pujo-Menjouet, Blood cell dynamics: Half of a century of modelling, Math. Model. Nat. Phenom., 11 (2016), 92-115.  doi: 10.1051/mmnp/201611106. [35] J. Rankin and H. M. Osinga, Parameter-dependent behaviour of periodic channels in a locus of boundary crisis, European Physical Journal Special Topics, 226 (2017), 1739-1750.  doi: 10.1140/epjst/e2017-70048-x. [36] C. Robert, K. T. Alligood, E. Ott and J. A. Yorke, Explosions of chaotic sets, Physica D, 144 (2000), 44-61.  doi: 10.1016/S0167-2789(00)00074-9. [37] G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890. [38] S. Schirm and M. Scholz, A biomathematical model of human erythropoiesis and iron metabolism, Scientific Reports, 10, (2020), 8602. doi: 10.1038/s41598-020-65313-5. [39] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL Manual - Bifurcation Analysis of Delay Differential Equations, 2015, Eprint, arXiv: 1406.7144 [math.DS]. [40] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [41] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015. [42] H.-O. Walther, The impact on mathematics of the paper ''Oscillation and Chaos in Physiological Control Systems" by Mackey and Glass in Science, 1977, (2020), Eprint, arXiv: 2001.09010 [math.DS]. [43] J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007), 2483-2498.  doi: 10.1088/0951-7715/20/11/002. [44] S. Wieczorek, B. Krauskopf and D. Lenstra, Unnested islands of period doublings in an injected semiconductor laser, Phys. Rev. E, 64 (2001), 056204, 9 pp. doi: 10.1103/PhysRevE.64.056204.
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
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
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
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 $2\times10^{-4}$
(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 $\tau$
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
(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
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) $n = 11.215$
(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
(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
Chaotic attractor dimension for the Mackey-Glass equation (11) computed using (10)
(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
(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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