Figure 1.
(a)-(b) Time series and phase portrait of a $ 1^5 $ MMO pattern observed in system (5) for $ \beta_1 = 0.35 $ , $ h = 1.65 $ and the other parameter values as in (6)
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Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem
Department of Mathematics, Georgia College & State University, Milledgeville, GA 31061, USA |
We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one fast (prey dynamics) and two slow variables (dynamics of the predators). The model exhibits a variety of rich and interesting dynamics, including, but not limited to mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistent oscillatory state. More interestingly, in a neighborhood of singular Hopf bifurcation, long lasting transient dynamics in the form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from geometric singular perturbation theory. The numerical continuation study reveals the rich bifurcation structure in the system, including the existence of codimension-two bifurcations such as fold-Hopf and generalized Hopf bifurcations.
Keywords:
Mixed-mode oscillations,
singular Hopf bifurcation,
long transients,
geometric singular perturbation,
basins of attraction,
chaotic dynamics.
Mathematics Subject Classification:
Primary: 37G35, 34E17; Secondary: 92D40, 92D25.
Citation:
Susmita Sadhu.
Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020342
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T. R. Baumgartner, A. Soutar and V. Bartrina, Reconstruction of the history of Pacific sardine and Northern anchovy populations over the past two millennia from sediments of the Santa Barbara basin, California, CalCOFl Rep., Vol. 33, 1992. Google Scholar |
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B. Braaksma,
Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlinear Sci., 8 (1998), 457-490.
doi: 10.1007/s003329900058. |
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M. Brøns, T. J. Kaper and H. G. Rotstein, Introduction to focus issue: Mixed mode oscillations: Experiment, computation, and analysis, Chaos, 18 (2008), 015101.
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M. Brøns and R. Kaasen,
Canards and mixed-mode oscillations in a forest pest model, Theoretical Population Biology, 77 (2010), 238-242.
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M. Brøns, M. Krupa and M. Wechselberger,
Mixed mode oscillations due to the generalized canard phenomenon, Fields Institute Communications, 49 (2006), 39-63.
|
| [8] |
M. Casimir, History of outbreaks of the Australian plague locust, Chortoicetes terminifera (Walk.), between $1933$ and $1959$ and analyses of the influence of rainfall in these outbreaks, Aust. J. Agric. Res., 13 (1962), 670-700. Google Scholar |
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R. Curtu and J. Rubin,
Interaction of canard and singular Hopf mechanisms in a neural model, SIAM J. Appl. Dyn Syst., 10 (2011), 1443-1479.
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Food chain chaos due to junction-fold point, Chaos, 11 (2001), 514-525.
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Food chain chaos due to Shilnikovs orbit, Chaos, 12 (2002), 533-538.
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M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger,
Mixed-mode oscillations with multiple time scales, SIAM Review, 54 (2012), 211-288.
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Singular Hopf bifurcation in systems with two slow variables, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1355-1377.
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The competitive exclusion principle, Science, 131 (1960), 1292-1297.
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F. M. Hilker and H. Malchow,
Strange periodic attractors in a prey-predator system with infected prey, Math. Pop. Studies, 13 (2006), 119-134.
doi: 10.1080/08898480600788568. |
| [23] |
S. B. Hsu, S. P. Hubbell and P. Waltman,
Competing predators, SIAM J. Appl. Math., 35 (1978), 617-625.
doi: 10.1137/0135051. |
| [24] |
M. T. M. Koper,
Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol- Duffing model with a cross-shaped phase diagram, Physica D, 80 (1995), 72-74.
doi: 10.1016/0167-2789(95)90061-6. |
| [25] |
E. Korpim$\ddot{\mathrm{a}}$ki, P. R. Brown, J. Jacob and R. P. Pech, The puzzles of population cycles and outbreaks of small mammals solved?, BioScience, 54 (2004), 1071–1079. https://academic.oup.com/bioscience/article/54/12/1071/329290. Google Scholar |
| [26] |
M. Krupa, N. Popović and N. Kopell,
Mixed-mode oscillations in three time-scale systems: A prototypical example, SIAM J. Appl. Dyn. Syst., 7 (2008), 361-420.
doi: 10.1137/070688912. |
| [27] |
M. Krupa and M. Wechselberger,
Local analysis near a folded saddle-node singularity, J. of Differential Equations, 248 (2010), 2841-2888.
doi: 10.1016/j.jde.2010.02.006. |
| [28] |
C. Kuehn, Multiple Time Scale Dynamics, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12316-5. |
| [29] |
M. Kuwamura and H. Chiba, Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators, Chaos, 19 (2009), 043121, 10 pp.
doi: 10.1063/1.3270262. |
| [30] |
Y. A. Kuznetsov, Elements of Applied BifurcationTheory, Springer-Verlag, New York, 1998. |
| [31] |
B. Letson, J. Rubin and T. Vo,
Analysis of interacting local oscillation mechanisms in three-timescale systems, SIAM J. Appl. Math., 77 (2017), 1020-1046.
doi: 10.1137/16M1088429. |
| [32] |
W. Liu, D. Xiao and Y. Yi,
Relaxation oscillations in a class of predator-prey systems, J. Diff. Equ., 188 (2003), 306-331.
doi: 10.1016/S0022-0396(02)00076-1. |
| [33] |
J. Maselko and H. L. Swinney,
Complex periodic oscillation and Farey arithmetic in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 85 (1986), 6430-6441.
doi: 10.1063/1.451473. |
| [34] |
R. McGehee and R. A. Armstrong,
Some mathematical problems concerning the ecological principle of competitive exclusion, J. Diff. Eq., 23 (1977), 30-52.
doi: 10.1016/0022-0396(77)90135-8. |
| [35] |
A. Y. Morozov, M. Banerjee and S. V. Petrovskii, Long term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, J. Theo. Biol., 396 (2016), 116-124. Google Scholar |
| [36] |
A. Morozov, K. Abbott, K. Cuddington, T. Francisd, G. Gellnere, A. Hastings, Y.-C. Laig, S. Petrovskii, K. Scranton and M. LouZeeman, Long transients in ecology: Theory and applications, Physics of Life Reviews, 32 (2020, ) 1–40.
doi: 10.1016/j.plrev.2019.09.004. |
| [37] |
J. Mujica, B. Krauskopf and H. M. Osinga,
Tangencies between global invariant manifolds and slow manifolds near a singular Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 17 (2018), 1395-1431.
doi: 10.1137/17M1133452. |
| [38] |
S. Muratori and S. Rinaldi,
Remarks on competitive coexistence, SIAM J. Applied Math., 49 (1989), 1462-1472.
doi: 10.1137/0149088. |
| [39] |
A. B. Peet, P. A. Deutsch and E. Peacock-Lpez,
Complex dynamics in a three-level trophic system with intraspecies interaction, J. Theor. Biol., 232 (2005), 491-503.
doi: 10.1016/j.jtbi.2004.08.028. |
| [40] |
J.-C. Poggiale, C. Aldebert, B. Girardot and B. W. Kooi,
Analysis of a predatorprey model with specific time scales: a geometrical approach proving the occurrence of canard solutions, J. of Math. Bio., 80 (2020), 39-60.
doi: 10.1007/s00285-019-01337-4. |
| [41] |
L. C. Pontryagin, Asymptotic behavior of solutions of systems of differential equations with a small parameter in the derivatives of highest order, Izv. Akad. Nauk. SSSR Ser. Math., 21 (1957), 605–626 (in Russian). |
| [42] |
S. Rinaldi and S. Muratori, Slow-fast limit cycles in predator-prey models, Ecological Modelling, 61 (1992), 287-308. Google Scholar |
| [43] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
| [44] |
H. G. Rotstein, M. Wechselberger and N. Kopell,
Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II stellate cell model, SIAM J Applied Dynamical Systems, 7 (2008), 1582-1611.
doi: 10.1137/070699093. |
| [45] |
S. Ruan, A. Ardito, P. Ricciardi and D. L. DeAngelis,
Coexistence in competition models with density-dependent mortality, C. R. Biologies, 330 (2007), 845-854.
doi: 10.1016/j.crvi.2007.10.004. |
| [46] |
S. Sadhu, Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem, Chaos, 27(2017), 3, 033108.
doi: 10.1063/1.4977553. |
| [47] |
S. Sadhu and S. Chakraborty Thakur,
Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations, bistability and sensitivity to parameters, Ecological Complexity, 32 (2017), 196-208.
doi: 10.1016/j.ecocom.2016.08.007. |
| [48] |
S. Sadhu and C. Kuehn, Stochastic mixed-mode oscillations in a three-species predator-prey model, Chaos, 28 (2018), 3, 033606.
doi: 10.1063/1.4994830. |
| [49] |
S. Sadhu, Analysis of long term transient dynamics near singular Hopf bifurcation in a two-timescale ecosystem, submitted. Google Scholar |
| [50] |
S. Schecter,
Persistent unstable equilibria and closed orbits of a singularly perturbed equation, J. Diff. Eqns., 60 (1985), 131-141.
doi: 10.1016/0022-0396(85)90124-X. |
| [51] |
M. Scheffer and S. R. Carpenter,
Catastrophic regime shifts in ecosystems: Linking theory to observation, Trends in Ecol. and Evol., 18 (2003), 648-656.
doi: 10.1016/j.tree.2003.09.002. |
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M. Scheffer, Critical Transitions in Nature and Society, 16 Princeton University Press (2009). Google Scholar |
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P. Szmolyan and M. Wechselberger,
Canards in $\mathbb{R}^3$, Journal of Differential Equations, 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
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M. Wechselberger,
Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139.
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D. E. Wright, Analysis of the development of major plagues of the Australian plague locust Chortoicetes terminifera (Walker) using a simulation model, Aust. J. Ecol., 12 (1987), 423-437. Google Scholar |
show all references
References:
| [1] |
S. Ai and S. Sadhu,
The entry-exit theorem and relaxation oscillations in slow-fast planar systems, Journal of Diff. Eq., 268 (2020), 7220-7249.
doi: 10.1016/j.jde.2019.11.067. |
| [2] |
C. Asaro and L. A. Chamberlin, Outbreak History $(1953-2014)$ of Spring Defoliators Impacting Oak-Dominated Forests in Virginia, with Emphasis on Gypsy Moth (Lymantria dispar L.) and Fall Cankerworm (Alsophila pometaria Harris), American Entomologist, (2015) 174–185. Google Scholar |
| [3] |
T. R. Baumgartner, A. Soutar and V. Bartrina, Reconstruction of the history of Pacific sardine and Northern anchovy populations over the past two millennia from sediments of the Santa Barbara basin, California, CalCOFl Rep., Vol. 33, 1992. Google Scholar |
| [4] |
B. Braaksma,
Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlinear Sci., 8 (1998), 457-490.
doi: 10.1007/s003329900058. |
| [5] |
M. Brøns, T. J. Kaper and H. G. Rotstein, Introduction to focus issue: Mixed mode oscillations: Experiment, computation, and analysis, Chaos, 18 (2008), 015101.
doi: 10.1063/1.2903177. |
| [6] |
M. Brøns and R. Kaasen,
Canards and mixed-mode oscillations in a forest pest model, Theoretical Population Biology, 77 (2010), 238-242.
doi: 10.1016/j.tpb.2010.02.003. |
| [7] |
M. Brøns, M. Krupa and M. Wechselberger,
Mixed mode oscillations due to the generalized canard phenomenon, Fields Institute Communications, 49 (2006), 39-63.
|
| [8] |
M. Casimir, History of outbreaks of the Australian plague locust, Chortoicetes terminifera (Walk.), between $1933$ and $1959$ and analyses of the influence of rainfall in these outbreaks, Aust. J. Agric. Res., 13 (1962), 670-700. Google Scholar |
| [9] |
R. Curtu and J. Rubin,
Interaction of canard and singular Hopf mechanisms in a neural model, SIAM J. Appl. Dyn Syst., 10 (2011), 1443-1479.
doi: 10.1137/110823171. |
| [10] |
S. L. T. de Souza, I. L. Caldas, R. L. Viana and J. M. Balthazar,
Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboxes, Chaos, Solitons & Fractals, 21 (2004), 763-772.
doi: 10.1016/j.chaos.2003.12.096. |
| [11] |
B. Deng,
Food chain chaos due to junction-fold point, Chaos, 11 (2001), 514-525.
doi: 10.1063/1.1396340. |
| [12] |
B. Deng and G. Hines,
Food chain chaos due to Shilnikovs orbit, Chaos, 12 (2002), 533-538.
doi: 10.1063/1.1482255. |
| [13] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger,
Mixed-mode oscillations with multiple time scales, SIAM Review, 54 (2012), 211-288.
doi: 10.1137/100791233. |
| [14] |
B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, 2002.
doi: 10.1137/1.9780898718195. |
| [15] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq., 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
| [16] |
J. Guckenheimer,
Singular Hopf bifurcation in systems with two slow variables, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1355-1377.
doi: 10.1137/080718528. |
| [17] |
J. Guckenheimer and P. Meerkamp,
Unfoldings of singular hopf bifurcation, SIAM J. Appl. Dyn. Syst., 11 (2012), 1325-1359.
doi: 10.1137/11083678X. |
| [18] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
| [19] |
J. Guckenheimer and I. Lizarraga, Shilnikov homoclinic bifurcation of mixed-mode oscillations, SIAM J. Appl. Dyn. Syst., 14 (2015), 764-786.
doi: 10.1137/140972007. |
| [20] |
G. Hardin,
The competitive exclusion principle, Science, 131 (1960), 1292-1297.
doi: 10.1126/science.131.3409.1292. |
| [21] |
A. Hastings, K. C. Abbott, K. Cuddington, T. Francis, G. Gellner, Y-C. Lai, A. Morozov, S. Petrovskii, K. Scranton and M. L. Zeeman, Transient phenomena in ecology, Science, 07 (2018). Google Scholar |
| [22] |
F. M. Hilker and H. Malchow,
Strange periodic attractors in a prey-predator system with infected prey, Math. Pop. Studies, 13 (2006), 119-134.
doi: 10.1080/08898480600788568. |
| [23] |
S. B. Hsu, S. P. Hubbell and P. Waltman,
Competing predators, SIAM J. Appl. Math., 35 (1978), 617-625.
doi: 10.1137/0135051. |
| [24] |
M. T. M. Koper,
Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol- Duffing model with a cross-shaped phase diagram, Physica D, 80 (1995), 72-74.
doi: 10.1016/0167-2789(95)90061-6. |
| [25] |
E. Korpim$\ddot{\mathrm{a}}$ki, P. R. Brown, J. Jacob and R. P. Pech, The puzzles of population cycles and outbreaks of small mammals solved?, BioScience, 54 (2004), 1071–1079. https://academic.oup.com/bioscience/article/54/12/1071/329290. Google Scholar |
| [26] |
M. Krupa, N. Popović and N. Kopell,
Mixed-mode oscillations in three time-scale systems: A prototypical example, SIAM J. Appl. Dyn. Syst., 7 (2008), 361-420.
doi: 10.1137/070688912. |
| [27] |
M. Krupa and M. Wechselberger,
Local analysis near a folded saddle-node singularity, J. of Differential Equations, 248 (2010), 2841-2888.
doi: 10.1016/j.jde.2010.02.006. |
| [28] |
C. Kuehn, Multiple Time Scale Dynamics, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12316-5. |
| [29] |
M. Kuwamura and H. Chiba, Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators, Chaos, 19 (2009), 043121, 10 pp.
doi: 10.1063/1.3270262. |
| [30] |
Y. A. Kuznetsov, Elements of Applied BifurcationTheory, Springer-Verlag, New York, 1998. |
| [31] |
B. Letson, J. Rubin and T. Vo,
Analysis of interacting local oscillation mechanisms in three-timescale systems, SIAM J. Appl. Math., 77 (2017), 1020-1046.
doi: 10.1137/16M1088429. |
| [32] |
W. Liu, D. Xiao and Y. Yi,
Relaxation oscillations in a class of predator-prey systems, J. Diff. Equ., 188 (2003), 306-331.
doi: 10.1016/S0022-0396(02)00076-1. |
| [33] |
J. Maselko and H. L. Swinney,
Complex periodic oscillation and Farey arithmetic in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 85 (1986), 6430-6441.
doi: 10.1063/1.451473. |
| [34] |
R. McGehee and R. A. Armstrong,
Some mathematical problems concerning the ecological principle of competitive exclusion, J. Diff. Eq., 23 (1977), 30-52.
doi: 10.1016/0022-0396(77)90135-8. |
| [35] |
A. Y. Morozov, M. Banerjee and S. V. Petrovskii, Long term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, J. Theo. Biol., 396 (2016), 116-124. Google Scholar |
| [36] |
A. Morozov, K. Abbott, K. Cuddington, T. Francisd, G. Gellnere, A. Hastings, Y.-C. Laig, S. Petrovskii, K. Scranton and M. LouZeeman, Long transients in ecology: Theory and applications, Physics of Life Reviews, 32 (2020, ) 1–40.
doi: 10.1016/j.plrev.2019.09.004. |
| [37] |
J. Mujica, B. Krauskopf and H. M. Osinga,
Tangencies between global invariant manifolds and slow manifolds near a singular Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 17 (2018), 1395-1431.
doi: 10.1137/17M1133452. |
| [38] |
S. Muratori and S. Rinaldi,
Remarks on competitive coexistence, SIAM J. Applied Math., 49 (1989), 1462-1472.
doi: 10.1137/0149088. |
| [39] |
A. B. Peet, P. A. Deutsch and E. Peacock-Lpez,
Complex dynamics in a three-level trophic system with intraspecies interaction, J. Theor. Biol., 232 (2005), 491-503.
doi: 10.1016/j.jtbi.2004.08.028. |
| [40] |
J.-C. Poggiale, C. Aldebert, B. Girardot and B. W. Kooi,
Analysis of a predatorprey model with specific time scales: a geometrical approach proving the occurrence of canard solutions, J. of Math. Bio., 80 (2020), 39-60.
doi: 10.1007/s00285-019-01337-4. |
| [41] |
L. C. Pontryagin, Asymptotic behavior of solutions of systems of differential equations with a small parameter in the derivatives of highest order, Izv. Akad. Nauk. SSSR Ser. Math., 21 (1957), 605–626 (in Russian). |
| [42] |
S. Rinaldi and S. Muratori, Slow-fast limit cycles in predator-prey models, Ecological Modelling, 61 (1992), 287-308. Google Scholar |
| [43] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
| [44] |
H. G. Rotstein, M. Wechselberger and N. Kopell,
Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II stellate cell model, SIAM J Applied Dynamical Systems, 7 (2008), 1582-1611.
doi: 10.1137/070699093. |
| [45] |
S. Ruan, A. Ardito, P. Ricciardi and D. L. DeAngelis,
Coexistence in competition models with density-dependent mortality, C. R. Biologies, 330 (2007), 845-854.
doi: 10.1016/j.crvi.2007.10.004. |
| [46] |
S. Sadhu, Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem, Chaos, 27(2017), 3, 033108.
doi: 10.1063/1.4977553. |
| [47] |
S. Sadhu and S. Chakraborty Thakur,
Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations, bistability and sensitivity to parameters, Ecological Complexity, 32 (2017), 196-208.
doi: 10.1016/j.ecocom.2016.08.007. |
| [48] |
S. Sadhu and C. Kuehn, Stochastic mixed-mode oscillations in a three-species predator-prey model, Chaos, 28 (2018), 3, 033606.
doi: 10.1063/1.4994830. |
| [49] |
S. Sadhu, Analysis of long term transient dynamics near singular Hopf bifurcation in a two-timescale ecosystem, submitted. Google Scholar |
| [50] |
S. Schecter,
Persistent unstable equilibria and closed orbits of a singularly perturbed equation, J. Diff. Eqns., 60 (1985), 131-141.
doi: 10.1016/0022-0396(85)90124-X. |
| [51] |
M. Scheffer and S. R. Carpenter,
Catastrophic regime shifts in ecosystems: Linking theory to observation, Trends in Ecol. and Evol., 18 (2003), 648-656.
doi: 10.1016/j.tree.2003.09.002. |
| [52] |
M. Scheffer, Critical Transitions in Nature and Society, 16 Princeton University Press (2009). Google Scholar |
| [53] |
P. Szmolyan and M. Wechselberger,
Canards in $\mathbb{R}^3$, Journal of Differential Equations, 177 (2001), 419-453.
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Figure 2.
(a) The critical manifold $ \mathcal{M} = {\Pi}\cup S $ . The folded node is marked by a cyan dot. (b) Reduced dynamics of system (11) zoomed near the folded node singularity (cyan) projected on the manifold $ S $ . The black curves are trajectories of system (11). Also shown are the ordinary singularity (black dot), weak eigendirection (green), the strong eigendirection (red) and the singular funnel. Here $ h = 0.8 $ and the other parameter values are as in (6)
Figure 3.
One-parameter bifurcation of the desingularized system (12) as $ h $ is varied and the other parameter values are as in (6). The $ y $ -axis represents the maximum of $ x $ . The black curves are ordinary singularities (namely, the coexistent equilibria and the boundary equilibria $ E_{xy} $ (horizontal line) and $ E_{xz} $ ) and the red curve represents the folded singularities. These curves intersect at FSN II points (green dots). The stable parts of a branch are shown by solid curves, while the unstable parts by the dashed curves. The folded singularities exist in the positive octant for $ h>0.5958 $
Figure 4.
A one-parameter bifurcation diagram in $ h $ for $ \beta_1 = 0.25 $ and other parameter values as in (6). The red (black) represent stable (unstable) branches of equilibria, where $ E^* $ denotes the coexistent equilibrium, $ E_{xy} $ and $ E_{xz} $ are the boundary equilibria on the $ xy $ and $ xz $ planes respectively. The green (blue) circles represent stable (unstable) limit cycles. HB: Hopf bifurcation, PD: period doubling, SNL: saddle node bifurcation of limit cycles. See the next two figures for a detailed view of the inset. The plot was generated in XPPAUT [14]
Figure 6.
Numerical continuation of periodic orbits of system (5) as $ h $ is varied. Here, $ T $ represents the period. Green (blue) circles represent stable (unstable) orbits. The bottom curve shows the evolution of the family of orbits $ \Gamma_h $ , and the curves above are continuations of the successive sequence of families of period-doubled Hopf orbits (see text for details). $ \text{PD}_1 $ : $ 1 $ st period-doubled bifurcation, $ \text{PD}_2 $ : $ 2 $ nd period-doubled bifurcation, SNL: saddle-node bifurcation of limit cycles. All parameter values are as in Figure 4
Figure 7.
(a) Phase portrait of a chaotic MMO orbit. (b) Corresponding time series in $ x $ . (c) The trajectory rotates around the weak eigendirection (red) while getting pushed towards the equilibrium (magenta) along the stable eigendirection (green) of the equilibrium. The folded node is marked by a cyan dot. Here $ \beta_1 = 0.25 $ , $ h = 0.8 $ and the other parameter values are as in (6)
Figure 8.
(a)-(b) Phase space and time series of a MMO orbit of signature $ 1^{11} $ for $ h = 0.633 $ , $ \beta_1 = 0.1977 $ and the other parameter values as in (6). The SAOs rotate around the weak eigendirection (in red) and are induced by the rotational properties of the weak canard (in blue). The folded node is marked by a cyan dot
Figure 9.
(a) A chaotic MMO orbit rotating around the weak eigendirection (red) after leaving the equilibrium (magenta) along its unstable eigendirection. The folded node is shown in cyan. (b) Corresponding time series in $ x $ . Note the amplitudes and epochs of small oscillations between two large oscillations vary by orders of magnitude. Here $ \beta_1 = 0.1825 $ , $ h = 0.2648 $ and the other parameter values are as in (7)
Figure 10.
Time series in $ x $ demonstrating long transient dynamics in the form of MMOs past supercritical Hopf bifurcation as $ h $ is varied. The initial condition chosen is $ (0.01, 0.01, 0.12) $ . Note the duration of the transients varies with $ h $ . Here $ \beta_1 = 0.25 $ and the other parameter values are as in (6)
Figure 11.
Transient basins of attraction for $ h = 0.785 $ restricted to the plane $ x = 0.3428 $ computed over the interval $ [0, N] $ . Blue dots lie in the basin of short transients and the red dots in the basin of long transients. Here $ \beta_1 = 0.25 $ and the other parameter values are as in (6). (a) $ N = 2000 $ . (b) $ N = 5000 $
Figure 12.
(a) A trajectory exhibiting MMOs while approaching $ \Gamma_{h} $ (blue). The trajectory tightly wraps around the weak eigendirection $ v_w $ (in red) and approaches $ E_1^* $ (marked as a magenta dot), inducing SAOs in the MMOs. The folded node singularity is marked as the cyan dot and the stable eigendirection of $ E_1^* $ in green. (b) Corresponding time series in $ x $ . (c) A zoomed view of the SAOs between two LAOs. Note the amount of time spent near the folded node and the equilibrium. (d) Time series of $ \Gamma_{h} $ , the periodic stable attractor. Here, $ \beta_1 = 0.25 $ , $ h = 0.785 $ and the other parameter values are as in (6)
Figure 13.
The Poincaré section $ \Sigma_h $ drawn transverse to the manifold $ S $ . Also, shown are the strong and weak eigendirections $ v_s $ and $ v_w $ respectively, along with folded node singularity in cyan. All the parameter values are as in Figure 11
Figure 14.
(a) $ (x, z) $ coordinates of intersection of a trajectory starting at $ (0.01, 0.01, 0.12) $ with the cross-section $ \Sigma_h: = 2.94x+ 1.3y -1.52z = 0.588 $ such that $ dx/ds<0 $ . (b) A zoomed view of the region $ a_2 $ near the folded node
Figure 15.
(a)-(b) Phase portrait and time series of a trajectory starting out at $ (0.27795, 0.4252, 0.0598) $ exhibits relaxation oscillations as transient dynamics and eventually asymptotes to $ E_{xz} = (0.357, 0, 0.615) $ for $ \beta_1 = 0.1923 $ , $ h = 0.2649 $ and other parameter values as in (7)
Figure 16.
A two-parameter bifurcation diagram in $ (h, \beta_1) $ space. The other parameter values are as in (6). FH: fold-Hopf bifurcation (marked by black dot), SN: saddle-node bifurcation (red curve), H: Hopf bifurcation (blue curve and the vertical line), FSN II bifurcation (purple curve), PD: period-doubling bifurcation (yellow curve), BR: branch curve (cyan). The equilibria lying to the right of BR are biologically feasible. The plot was generated in XPPAUT [14]
Figure 17.
(a)-(b) Phase space and time series of a trajectory for $ h = 0.615 $ . The trajectory approaches $ E_1^* $ along $ W^s(E_1^*) $ and spirals out along $ W^u(E_1^*) $ as it eventually settles down to the periodic orbit (red), which emerges through a Hopf bifurcation. Here $ \beta_1 = 0.1977 $ and the other parameter values are as in (6)
Figure 18.
Bifurcation diagram in $ h $ with $ \beta_1 = 0.35 $ and other parameter values as in (6). Here $ x_{max} $ is the maximum value of $ x $
Figure 19.
A two-parameter bifurcation diagram in $ (h, \beta_1) $ space. The other parameter values are as in (7). FH: fold-Hopf bifurcation (zero-pair), SN: saddle-node bifurcation of equilibria (red curve), H: Hopf bifurcation (blue curve), TR: torus bifurcation (magenta curve), BR: branch curve (cyan curve), GH: generalized Hopf bifurcation. Region 1: a unique coexistent equilibrium exists, which is unstable, Region 2i: existence of two coexistent equilibria, both unstable, Region 2ii: two coexistent equilibria exist, one of which is locally asymptotically stable, Region 3: a unique coexistent equilibrium exists, which is locally asymptotically stable. The plot was generated in XPPAUT [14]
Figure 20.
(a)-(b) Transient temporal dynamics of system (5) asymptotically approaching to the equilibrium $ E_{xz} = (0.5225, 0, 0.536) $ . Here $ \beta_1 = 0.3001 $ , $ h = 0.4767 $ and the other parameter values are as in (7). The chosen values lie in region 2i in Figure 19
Figure 21.
Bistable dynamics exhibited by system (5) at $ \beta_1 = 0.1923 $ and $ h = 0.2648 $ . The system either approaches a periodic orbit (red) born out of torus bifurcation in (a) or to the equilibrium $ E_{xz} = (0.357, 0, 0.615) $ in (b). The other parameter values are as in (7). The initial conditions in (a) are $ (0.2995, 0.1175, 0.4154) $ and $ (0.34279, 0.104, 0.346) $ in (b)
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