# Student Feature - Andrus Giraldo Munoz

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Kia Ora,

My research focusses on understanding the underlying geometry that guides the qualitative behaviour of different mathematical models described by differential equations. To achieve this, I employ theoretical and numerical techniques [9] to compute invariant manifolds in phase space to understand how they interact as different parameters are varied. During my PhD, under the supervision of Professors Bernd Krauskopf and Hinke Osinga at the University of Auckland, I studied how the stable and unstable manifolds of saddle equilibria and periodic orbits interact at the moment a system exhibits a homoclinic flip bifurcation [15,16]. This global bifurcation is a phenomenon that can only occur in three- or higher-dimensional systems of differential equations.

To illustrate what a homoclinic flip bifurcation is, consider a three-dimensional system of differential equations that possesses a real saddle equilibrium with one unstable and two stable eigenvalues, so that it possesses a one-dimensional unstable manifold (red curves in the animation) and a two-dimensional stable manifold (dark-blue surface in the animation). At a homoclinic bifurcation, one of the curves that forms the unstable manifold is contained in the stable manifold. In this case, the stable manifold generically closes on itself in an orientable or a non-orientable way. At a special point in parameter plane, one can transition between the two cases, and this is called a homoclinic flip bifurcation. Such point of transition has attracted a lot of attention, because it is an organizing centre for the creation of saddle periodic orbits and exotic behaviour, e.g., Smale horseshoe and chaos [2,3,4,5,6,12].

Phase portraits of a three-dimensional system [7] at different parameter values. The middle row shows phase portraits at a moment of an orientable ($$\mathbf{H_o}$$), a nonorientable ($$\mathbf{H_t}$$), and a homoclinic flip ($$\mathbf{H_t}$$) bifurcations. The panels 1 and 2 show the situation when the orientable homoclinic bifurcation is perturbed in parameters, while panels 3 and 4 are for the nonorientable case. Here, the stable manifold is rendered as a dark-blue surface and the unstable manifold is represented by the red curves. Also shown are: the stable (cyan surface) and unstable (orange surface) manifolds of saddle periodic orbits (green curves), and the unstable (red surface) and stable (cyan curve) of a saddle focus equilibrium. For more information refer to [15,16].

At the moment, I am a Research Fellow at the University of Auckland where I work alongside Professors Bernd Krauskopf and Neil Broderick on the dynamics that arises on two coupled Photonic Crystal (PhC) nanocavities. These devices have gained a lot of interest as an experimental framework to study quantum effects due to their operation with a low number of photons. In particular, Dr. Yacomotti and his collaborators in Paris have designed and manufactured such devices. They have experimentally shown the existence of symmetry breaking when PhC nanocavities are turned into nanolasers [10,13].

Due to the low number of photons at which these devices operate, one can model them as an open quantum system where photons can escape from the device [14]. However, extracting relevant information from the open quantum system is a challenging task due to its high-dimensionality. For this reason, a semiclassical approximation [14] of the open quantum system by a four-dimensional system of differential equation is often used, although, for such approximation to be valid, the number of photons inside the device is “high”. We performed a bifurcation analysis of the differential equation model for the PhC nanocavities, and found regions in parameter plane with bistability, multi-stability with periodic orbits, symmetry breaking and chaos. The interactions between certain invariant manifolds and homoclinic bifurcations of wild Shilnikov type [1,11] are part of the overall picture. We also simulated the open quantum system formulation with a quantum trajectory jump algorithm [8] and found that the semiclassical formulation explains the behaviour of the quantum system even when the number of photons in the PhC nanocavities is “small”. Furthermore, these results provide a good indication that this behaviour should be well within the range of future experiments. I presented this work in my poster presentation at Snowbird 2019.

To end this student feature on a more personal note, I am from Medellin, Colombia; and I have lived in New Zealand since 2014. Beyond being enthusiastic about dynamical systems, I love to dance salsa.

If you meet me in a conference, please do not hesitate to approach me.

Muchas gracias por su atención.

Andrus

#### References:

[1] L. P. Shilnikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., 6 (1965), pp. 163–166.
[2] E. Yanagida, Branching of double pulse solutions from single pulse solutions in nerve axon equations, J. Differential Equations, 66 (1987), pp. 243–262.
[3] M. Kisaka, H. Kokubu, and H. Oka, Bifurcations to n-homoclinic orbits and n-periodic orbits in vector fields, J. Dynam. Differential Equations, 5 (1993), pp. 305–357.
[4] A. J. Homburg, H. Kokubu, and M. Krupa, The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynam. Systems, 14 (1994), pp. 667–693.
[5] B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, PhD thesis, University of Stuttgart, Stuttgart, Germany, 1993.
[6] V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynam. Systems, 16 (1996), pp. 1071–1086.
[7] B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynam. Differential Equations, 9 (1997), pp. 269–288.
[8] H. J. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields, Springer Science & Business Media, 2007.
[9] B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems, B. Krauskopf, H. M. Osinga, and J. Galan-Vioque, eds., Springer Netherlands, 2007, pp. 117–154.
[10] M. Brunstein, Nonlinear dynamics in III-V semiconductor photonic crystal nano-cavities, Ph.D. thesis, Université Paris Sud - Paris XI (2011)
[11] R. Barrio, A. Shilnikov, and L. Shilnikov, Kneadings, Symbolic dynamics and painting Lorenz chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012).
[12] P. Aguirre, B. Krauskopf, and H. M. Osinga, Global invariant manifolds near homoclinic orbits to a real saddle: (Non)orientability and flip bifurcation, SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 1803–1846.
[13] P. Hamel, S. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson, and A. M. Yacomotti, Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers, NaturePhotonics 9 (2015).
[14] B. Cao, K. W. Mahmud, and M. Hafezi, Two coupled nonlinear cavities in a driven-dissipative environment, Phys. Rev. A 94, 063805 (2016).
[15] A. Giraldo, B. Krauskopf, and H. M. Osinga, Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 640–686.
[16] A. Giraldo, B. Krauskopf, and H. M. Osinga, Cascades of global bifurcations and chaos near a homoclinic flip bifurcation: a case study, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2784–2829.

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