# American Institute of Mathematical Sciences

April  2020, 13(4): 1269-1290. doi: 10.3934/dcdss.2020073

## Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

 1 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraẞe 8-10, 1040 Vienna, Austria 2 Faculty of Mathematics, Technical University of Munich, Boltzmannstraẞe 3, 85748 Garching bei München, Germany

* Corresponding author: Annalisa Iuorio

Received  May 2017 Revised  February 2018 Published  April 2019

Fund Project: The first author is supported by FWF grant W1245.

We investigate a singularly perturbed, non-convex variational problem arising in material science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters. This allows us to observe the influence of the parameters on the behavior of the orbits and to study their interplay in the minimization process. Our results confirm previous analytical results such as the asymptotics of the period of minimizers predicted by Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits.

Citation: Annalisa Iuorio, Christian Kuehn, Peter Szmolyan. Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1269-1290. doi: 10.3934/dcdss.2020073
##### References:
 [1] R. Abeyaratne, K. Bhattacharya and J. K. Knowles, Strain-energy functions with multiple local minima: Modeling phase transformations using finite thermoelasticity, in Nonlinear Elasticity: Theory and Applications (London Math. Soc. Lecture Note Ser.), Cambridge Univ. Press, 283 (2001), 433-490. doi: 10.1017/CBO9780511526466.013. [2] R. Abeyaratne and J. K. Knowles, Evolution of Phase Transitions: A Continuum Theory, Cambridge Univ. Press, Cambridge, 2006.  doi: 10.1017/CBO9780511547133. [3] J. M. Ball, Mathematical models of martensitic microstructure, Material Science and Engineering: A, 378 (2004), 61-69.  doi: 10.1016/j.msea.2003.11.055. [4] J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), 13-52.  doi: 10.1007/BF00281246. [5] K. Bhattacharya, Comparison of the geometrically nonlinear and linear theories of martensitic transformation, Continuum Mechanics and Thermodynamics, 5 (1993), 205-242.  doi: 10.1007/BF01126525. [6] K. Bhattacharya, Microstructure of Martensite: Why it Forms and How it Gives Rise to the Shape-Memory Effect, volume 2, Oxford Univ. Press, 2003. [7] P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh-Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393-3441.  doi: 10.1137/140999177. [8] A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693.  doi: 10.1137/070682654. [9] B. Dacorogna, Introduction to the Calculus of Variations, 3rd edition, Imperial College Press, London, 2015. [10] M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046106, 13pp. doi: 10.1063/1.4827026. [11] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), 739-765.  doi: 10.1088/0951-7715/23/3/017. [12] E. Doedel, AUTO: a program for the automatic bifurcation analysis of autonomous systems, in Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing (Vol. Ⅰ (Winnipeg, Man., 1980)), 30 (1981), 265-284. [13] G. Dolzmann, Variational Methods for Crystalline Microstructure-analysis and Computation, Lecture Notes in Mathematics, 1803. Springer-Verlag, Berlin, 2003. doi: 10.1007/b10191. [14] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Amer. Math. Soc., 121 (1996), x+100 pp. doi: 10.1090/memo/0577. [15] J. L. Ericksen, Equilibrium of bars, J. Elasticity (Special issue dedicated to A.E. Green), 5 (1975), 191-201. doi: 10.1007/BF00126984. [16] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [17] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, The Bulletin of Mathematical Biophysics, 17 (1955), 257-278.  doi: 10.1007/BF02477753. [18] V. Gelfreich and L. Lerman, Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system, Nonlinearity, 15 (2002), 447-457.  doi: 10.1088/0951-7715/15/2/312. [19] A. Giuliani and S. Müller, Striped periodic minimizers of a two-dimensional model for martensitic phase transitions, Comm. Math. Phys., 309 (2012), 312-339.  doi: 10.1007/s00220-011-1374-y. [20] M. Grinfeld and G. J. Lord, Bifurcations in the regularized Ericksen bar model, Journal of Elasticity, 90 (2008), 161-173.  doi: 10.1007/s10659-007-9137-x. [21] J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Syst., 8 (2009), 854-879.  doi: 10.1137/080741999. [22] J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit, DCDS-S, 2 (2009), 851-872.  doi: 10.3934/dcdss.2009.2.851. [23] J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153.  doi: 10.1137/090758404. [24] J. Guckenheimer and D. LaMar, Periodic orbit continuation in multiple time scale systems, in Understanding Complex Systems: Numerical continuation methods for dynamical systems, Springer, (2007), 253-267. doi: 10.1007/978-1-4020-6356-5_8. [25] J. Guckenheimer and P. Meerkamp, Bifurcation analysis of singular Hopf bifurcation in $\mathbb{R}^3$, SIAM J. Appl. Dyn. Syst., 11 (2012), 1325-1359.  doi: 10.1137/11083678X. [26] T. J. Healey and H. Kielhöfer, Global continuation via higher-gradient regularization and singular limits in forced one-dimensional phase transitions, SIAM J. Math. Anal., 31 (2000), 1307-1331.  doi: 10.1137/S0036141098340065. [27] T. J. Healey and U. Miller, Two-phase equilibria in the anti-plane shear of an elastic solid with interfacial effects via global bifurcation, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 463 (2007), 1117-1134. doi: 10.1098/rspa.2006.1807. [28] C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Springer, Berlin, Heidelberg, 1609 (1995), 44-118. doi: 10.1007/BFb0095239. [29] C. K. R. T. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, Patterns and Dynamics in Reactive Media (Minneapolis, MN, 1989), IMA Vol. Math. Appl., 37, Springer, New York, 1991,101-115. doi: 10.1007/978-1-4612-3206-3_7. [30] A. G. Khachaturyan, Theory of structural transformations in solids, Courier Corporation, 2013. [31] A. G. Khachaturyan and G. Shatalov, Theory of macroscopic periodicity for a phase transition in the solid state, Soviet Phys. JETP, 29 (1969), 557-561. [32] R. V. Kohn and S. Müller, Surface energy and microstructure in coherent phase transitions, Comm. Pure Appl. Math., 47 (1994), 405-435.  doi: 10.1002/cpa.3160470402. [33] I. Kosiuk and P. Szmolyan, Scaling in singular perturbation problems: Blowing-up a relaxation oscillator, SIAM J. Appl. Dyn. Syst., 10 (2011), 1307-1343.  doi: 10.1137/100814470. [34] K. U. Kristiansen, Computation of saddle-type slow manifolds using iterative methods, SIAM J. Appl. Dyn. Syst., 14 (2015), 1189-1227.  doi: 10.1137/140961948. [35] M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equat., 133 (1997), 49-97.  doi: 10.1006/jdeq.1996.3198. [36] C. Kuehn, Multiple Time Scale Dynamics, Vol. 191, Springer, Berlin, 2015. doi: 10.1007/978-3-319-12316-5. [37] C. Kuehn and P. Szmolyan, Multiscale geometry of the Olsen model and non-classical relaxation oscillations, J. Nonlinear Sci., 25 (2015), 583-629.  doi: 10.1007/s00332-015-9235-z. [38] S. Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. Partial Differential Equations, 1 (1993), 169-204.  doi: 10.1007/BF01191616. [39] S. Müller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems, Springer, Berlin, Heidelberg, 1713 (1999), 85-210. doi: 10.1007/BFb0092670. [40] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235. [41] P. Pedregal, Variational Methods in Nonlinear Elasticity, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719529. [42] M. Pitteri and G. Zanzotto, Continuum Models for Phase Transitions and Twinning in Crystals, Chapman & Hall/CRC, Boca Raton, Florida, 2003. [43] A. L. Roytburd, Martensitic transformation as a typical phase transformation in solids, Solid state physics, 33 (1978), 317-390.  doi: 10.1016/S0081-1947(08)60471-3. [44] C. Soto-Trevino, A geometric method for periodic orbits in singularly-perturbed systems, in Multiple-time-scale Dynamical Systems, Springer, New York, 22 (2001), 141-202. doi: 10.1007/978-1-4613-0117-2_6. [45] L. Truskinovsky and G. Zanzotto, Finite-scale microstructures and metastability in one-dimensional elasticity, Meccanica, 30 (1995), 577-589.  doi: 10.1007/BF01557086. [46] L. Truskinovsky and G. Zanzotto, EricksenaAZs bar revisited: Energy wiggles, Journal of the Mechanics and Physics of Solids, 44 (1996), 1371-1408.  doi: 10.1016/0022-5096(96)00020-8. [47] K. T. Tsaneva-Atanasova, H. M. Osinga, T. Riess and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theor. Biol., 264 (2010), 1133-1146.  doi: 10.1016/j.jtbi.2010.03.030. [48] A. Vainchtein, T. Healey, P. Rosakis and L. Truskinovsky, The role of the spinodal region in one-dimensional martensitic phase transitions, Phys. D, 115 (1998), 29-48.  doi: 10.1016/S0167-2789(97)00224-8. [49] N. K. Yip, Structure of stable solutions of a one-dimensional variational problem, ESAIM Control Optim. Calc. Var., 12 (2006), 721-751.  doi: 10.1051/cocv:2006019.

show all references

##### References:
 [1] R. Abeyaratne, K. Bhattacharya and J. K. Knowles, Strain-energy functions with multiple local minima: Modeling phase transformations using finite thermoelasticity, in Nonlinear Elasticity: Theory and Applications (London Math. Soc. Lecture Note Ser.), Cambridge Univ. Press, 283 (2001), 433-490. doi: 10.1017/CBO9780511526466.013. [2] R. Abeyaratne and J. K. Knowles, Evolution of Phase Transitions: A Continuum Theory, Cambridge Univ. Press, Cambridge, 2006.  doi: 10.1017/CBO9780511547133. [3] J. M. Ball, Mathematical models of martensitic microstructure, Material Science and Engineering: A, 378 (2004), 61-69.  doi: 10.1016/j.msea.2003.11.055. [4] J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), 13-52.  doi: 10.1007/BF00281246. [5] K. Bhattacharya, Comparison of the geometrically nonlinear and linear theories of martensitic transformation, Continuum Mechanics and Thermodynamics, 5 (1993), 205-242.  doi: 10.1007/BF01126525. [6] K. Bhattacharya, Microstructure of Martensite: Why it Forms and How it Gives Rise to the Shape-Memory Effect, volume 2, Oxford Univ. Press, 2003. [7] P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh-Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393-3441.  doi: 10.1137/140999177. [8] A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693.  doi: 10.1137/070682654. [9] B. Dacorogna, Introduction to the Calculus of Variations, 3rd edition, Imperial College Press, London, 2015. [10] M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046106, 13pp. doi: 10.1063/1.4827026. [11] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), 739-765.  doi: 10.1088/0951-7715/23/3/017. [12] E. Doedel, AUTO: a program for the automatic bifurcation analysis of autonomous systems, in Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing (Vol. Ⅰ (Winnipeg, Man., 1980)), 30 (1981), 265-284. [13] G. Dolzmann, Variational Methods for Crystalline Microstructure-analysis and Computation, Lecture Notes in Mathematics, 1803. Springer-Verlag, Berlin, 2003. doi: 10.1007/b10191. [14] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Amer. Math. Soc., 121 (1996), x+100 pp. doi: 10.1090/memo/0577. [15] J. L. Ericksen, Equilibrium of bars, J. Elasticity (Special issue dedicated to A.E. Green), 5 (1975), 191-201. doi: 10.1007/BF00126984. [16] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [17] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, The Bulletin of Mathematical Biophysics, 17 (1955), 257-278.  doi: 10.1007/BF02477753. [18] V. Gelfreich and L. Lerman, Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system, Nonlinearity, 15 (2002), 447-457.  doi: 10.1088/0951-7715/15/2/312. [19] A. Giuliani and S. Müller, Striped periodic minimizers of a two-dimensional model for martensitic phase transitions, Comm. Math. Phys., 309 (2012), 312-339.  doi: 10.1007/s00220-011-1374-y. [20] M. Grinfeld and G. J. Lord, Bifurcations in the regularized Ericksen bar model, Journal of Elasticity, 90 (2008), 161-173.  doi: 10.1007/s10659-007-9137-x. [21] J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Syst., 8 (2009), 854-879.  doi: 10.1137/080741999. [22] J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit, DCDS-S, 2 (2009), 851-872.  doi: 10.3934/dcdss.2009.2.851. [23] J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153.  doi: 10.1137/090758404. [24] J. Guckenheimer and D. LaMar, Periodic orbit continuation in multiple time scale systems, in Understanding Complex Systems: Numerical continuation methods for dynamical systems, Springer, (2007), 253-267. doi: 10.1007/978-1-4020-6356-5_8. [25] J. Guckenheimer and P. Meerkamp, Bifurcation analysis of singular Hopf bifurcation in $\mathbb{R}^3$, SIAM J. Appl. Dyn. Syst., 11 (2012), 1325-1359.  doi: 10.1137/11083678X. [26] T. J. Healey and H. Kielhöfer, Global continuation via higher-gradient regularization and singular limits in forced one-dimensional phase transitions, SIAM J. Math. Anal., 31 (2000), 1307-1331.  doi: 10.1137/S0036141098340065. [27] T. J. Healey and U. Miller, Two-phase equilibria in the anti-plane shear of an elastic solid with interfacial effects via global bifurcation, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 463 (2007), 1117-1134. doi: 10.1098/rspa.2006.1807. [28] C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Springer, Berlin, Heidelberg, 1609 (1995), 44-118. doi: 10.1007/BFb0095239. [29] C. K. R. T. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, Patterns and Dynamics in Reactive Media (Minneapolis, MN, 1989), IMA Vol. Math. Appl., 37, Springer, New York, 1991,101-115. doi: 10.1007/978-1-4612-3206-3_7. [30] A. G. Khachaturyan, Theory of structural transformations in solids, Courier Corporation, 2013. [31] A. G. Khachaturyan and G. Shatalov, Theory of macroscopic periodicity for a phase transition in the solid state, Soviet Phys. JETP, 29 (1969), 557-561. [32] R. V. Kohn and S. Müller, Surface energy and microstructure in coherent phase transitions, Comm. Pure Appl. Math., 47 (1994), 405-435.  doi: 10.1002/cpa.3160470402. [33] I. Kosiuk and P. Szmolyan, Scaling in singular perturbation problems: Blowing-up a relaxation oscillator, SIAM J. Appl. Dyn. Syst., 10 (2011), 1307-1343.  doi: 10.1137/100814470. [34] K. U. Kristiansen, Computation of saddle-type slow manifolds using iterative methods, SIAM J. Appl. Dyn. Syst., 14 (2015), 1189-1227.  doi: 10.1137/140961948. [35] M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equat., 133 (1997), 49-97.  doi: 10.1006/jdeq.1996.3198. [36] C. Kuehn, Multiple Time Scale Dynamics, Vol. 191, Springer, Berlin, 2015. doi: 10.1007/978-3-319-12316-5. [37] C. Kuehn and P. Szmolyan, Multiscale geometry of the Olsen model and non-classical relaxation oscillations, J. Nonlinear Sci., 25 (2015), 583-629.  doi: 10.1007/s00332-015-9235-z. [38] S. Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. Partial Differential Equations, 1 (1993), 169-204.  doi: 10.1007/BF01191616. [39] S. Müller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems, Springer, Berlin, Heidelberg, 1713 (1999), 85-210. doi: 10.1007/BFb0092670. [40] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235. [41] P. Pedregal, Variational Methods in Nonlinear Elasticity, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719529. [42] M. Pitteri and G. Zanzotto, Continuum Models for Phase Transitions and Twinning in Crystals, Chapman & Hall/CRC, Boca Raton, Florida, 2003. [43] A. L. Roytburd, Martensitic transformation as a typical phase transformation in solids, Solid state physics, 33 (1978), 317-390.  doi: 10.1016/S0081-1947(08)60471-3. [44] C. Soto-Trevino, A geometric method for periodic orbits in singularly-perturbed systems, in Multiple-time-scale Dynamical Systems, Springer, New York, 22 (2001), 141-202. doi: 10.1007/978-1-4613-0117-2_6. [45] L. Truskinovsky and G. Zanzotto, Finite-scale microstructures and metastability in one-dimensional elasticity, Meccanica, 30 (1995), 577-589.  doi: 10.1007/BF01557086. [46] L. Truskinovsky and G. Zanzotto, EricksenaAZs bar revisited: Energy wiggles, Journal of the Mechanics and Physics of Solids, 44 (1996), 1371-1408.  doi: 10.1016/0022-5096(96)00020-8. [47] K. T. Tsaneva-Atanasova, H. M. Osinga, T. Riess and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theor. Biol., 264 (2010), 1133-1146.  doi: 10.1016/j.jtbi.2010.03.030. [48] A. Vainchtein, T. Healey, P. Rosakis and L. Truskinovsky, The role of the spinodal region in one-dimensional martensitic phase transitions, Phys. D, 115 (1998), 29-48.  doi: 10.1016/S0167-2789(97)00224-8. [49] N. K. Yip, Structure of stable solutions of a one-dimensional variational problem, ESAIM Control Optim. Calc. Var., 12 (2006), 721-751.  doi: 10.1051/cocv:2006019.
Schematic representation of simple laminates microstructures as periodic solutions with $X \in [0,1]$. (a) Microstructures in one space dimension: austenite (A) and martensite (M) alternate, while the transition area is shown in gray. (b) Structure in space of the variable $u_X$, whose values $\pm 1$ represent the two different phases of the material of width of order $\mathcal{O}(\varepsilon^\alpha)$, with $\alpha = 1/3$ for minimizers (as shown in [38]) and $\alpha = 0$ for other critical points. The width of the transition interval is of order $\mathcal{O}(\varepsilon)$.
Critical manifold ${\mathcal C}_0$ in $(w,u,v)$-space. The magenta dashed lines are the fold lines ${\mathcal L}_\pm$. The blue solid curves correspond to $\mathcal{C}_{0,l}^\mu$ and $\mathcal{C}_{0,r}^\mu$, i.e., the intersection of $\mathcal{C}_{0,l}$ and $\mathcal{C}_{0,r}$ and the hypersurface $H(u,v,w,z) = \mu$ for $\mu = 0$.
$\mathcal{C}_{0,l}^\mu$ and $\mathcal{C}_{0,r}^\mu$ in $(w,u)$-space with $\mu = 0$; cf. Figure 2.
Fast flow in the $(w,z)$-space for 20. Equilibria are marked with blue dots and the stable and unstable manifold trajectories in green. The heteroclinic fast connections are indicated with double arrows.
Singular periodic orbit $\gamma_0^\mu$ for a fixed value of $\mu$ ($\mu = 0$), obtained by composition of slow (blue) and fast (green) pieces. (a) Orbit in $(w,z,u)$-space. (b) Orbit in the $(w,u,v)$-space. The fast pieces are indicated via dashed lines to illustrate the fact we are here considering their projection in $(w,u,v)$, while they actually occur in the $(w,z)$-plane. Consequently, they do not intersect ${\mathcal C}_{0,m}$.
Transversal intersection in the $(w,z,v)$ space between $W_u({\mathcal C}_{0,l})$ (in orange) and $W_s({\mathcal C}_{0,r})$ (in magenta). The blue line represents the critical manifold ${\mathcal C}_0$.
Schematic representation of the SMST algorithm applied to $\mathcal{C}_{0, l}^\mu$ (an analogous situation occurs for $\mathcal{C}_{0, r}^\mu$). The critical manifold is indicated by a dotted blue line, while the red line represents the slow manifold for $\varepsilon = 0.001$. The orange point corresponds to $(0, w_L, 0)$, which actually belongs to both manifolds
Continuation in $\mu$: (a) bifurcation diagram in $(\mu, P)$-space, where two periodic solutions corresponding to $\mu = -0.124$ are marked by crosses; (b) corresponding solutions in $\left(w,z,u \right)$-space: the one on the lower branch (magenta) is almost purely fast, while the one on the upper branch (purple) contains long non-vanishing slow pieces.
Continuation in $\mu$. (a) Zoom on the upper part of the bifurcation diagram in-$(\mu,P)$ space, where two periodic orbits corresponding to $\mu = 0.0025$ are marked by crosses. (a1)-(a2) The orbits are shown in $\left(w,z,u\right)$-space. The periodic orbit on the bottom part of the upper branch (purple) corresponds to analytical expectations with two fast and two slow segments. The periodic orbit on the top part of the upper branch (magenta) includes two new fast "homoclinic excursions". (b) Zoom on the lower part of the bifurcation diagram in $(\mu, P)$-space, where three solutions are marked. (b1) The solutions in phase space all correspond to periodic orbits around the center equilibrium $p_0$; note that the scale in the $u$-coordinate is extremely small so the three periodic orbits almost lie in the hyperplane $\{u = 0\}$.
Continuation in $\varepsilon$: on the left side bifurcation diagrams in $(\varepsilon, P)$ are shown, on the right the corresponding solutions in $\left( w, z, u \right)$-space are displayed. (a) $\mu = \mu_l$, (b) $\mu = \mu_c$, (c) $\mu = \mu_r$.
Illustration of two-parameter continuation. (a) Three different bifurcation diagrams have been computed, each starting from a solution at $\mu = 0$ for three different values of $\varepsilon = 0.1,0.01,10^{-5}$ (red, green, blue). It is already visible and confirmed by the computation that the sequence of leftmost fold points on each branch converges to $\mu = -1/8$ as $\varepsilon\rightarrow 0$. However, the period scaling law of the orbits precisely at these fold points, which is shown in (b) as three dots corresponding to the three folds in (a) and a suitable interpolation (black line), does not converge as $\mathcal{O}(\varepsilon^{1/3})$ (grey reference line with slope $\frac13$).
Possible fits of the form $P \simeq \varepsilon^{\alpha}$ for the numerical data computed with $\mu = \mu_l$ (black line): $\alpha = 2$, blue; $\alpha = 1/3$, green; $\alpha = 1$, red.
Parabola-shaped diagram obtained by fixing $\varepsilon = 0.001$ and numerically computing the value of the functional $\mathcal{I}^{\varepsilon}$ along the solutions computed via continuation in $\texttt{AUTO}$. The plot presents a minimum, and the value of $P$ corresponding to $\varepsilon$ where this minimum is realized is recorded in order to check the period law 3.
Comparison between the values of $P$ minimizing $\mathcal{I}^{\varepsilon}$ for several discrete values in the range $I_{\varepsilon}$ (red circles) and the period law 3 (black line). (a) Zoom on the range $\left[ 10^{-7}, 10^{-2} \right]$, where it is expected that large values of $\varepsilon$ tend to deviate from the $\mathcal{O}(\varepsilon^{1/3})$ leading-order scaling, while for low values the scaling the scaling agrees. (b) The same plot as in (a) on a log-log scale.
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