Bistability, bifurcations and chaos in the Mackey-Glass equation

Valentin Duruisseaux; Antony R. Humphries

doi:10.3934/jcd.2022009

Previous Article
Computation of nonreciprocal dynamics in nonlinear materials
JCD Home
This Issue
Next Article
Determining the global manifold structure of a continuous-time heterodimensional cycle

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

Bistability, bifurcations and chaos in the Mackey-Glass equation

Valentin Duruisseaux ^1, and Antony R. Humphries ^2,

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.

Keywords: Delay-differential equation, Mackey-Glass equation, Lyapunov exponents, bifurcations, bistability, subcritical period-doubling, chaos.

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

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.

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Ahmed Elhassanein. Complex dynamics of a forced discretized version of the Mackey-Glass delay differential equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 93-105. doi: 10.3934/dcdsb.2015.20.93
[2]	Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191
[3]	Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[4]	Urszula Foryś, Jan Poleszczuk. A delay-differential equation model of HIV related cancer--immune system dynamics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 627-641. doi: 10.3934/mbe.2011.8.627
[5]	Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555
[6]	Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092
[7]	Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[8]	Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537
[9]	L. Dieci, M. S Jolly, Ricardo Rosa, E. S. Van Vleck. Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 555-580. doi: 10.3934/dcdsb.2008.9.555
[10]	Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
[11]	John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044
[12]	Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[13]	Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations and Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
[14]	J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653
[15]	Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[16]	Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633
[17]	P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[18]	Pasquale Palumbo, Simona Panunzi, Andrea De Gaetano. Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 399-424. doi: 10.3934/dcdsb.2007.7.399
[19]	Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332
[20]	Alexandre Boritchev. Decaying turbulence for the fractional subcritical Burgers equation. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2229-2249. doi: 10.3934/dcds.2018092

Impact Factor:

Tools

Metrics

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

American Institute of Mathematical Sciences

Bistability, bifurcations and chaos in the Mackey-Glass equation

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Bistability, bifurcations and chaos in the Mackey-Glass equation

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors