Portfolio

My research centers on dynamical systems, with a particular focus on chaos theory, bifurcation theory, and the application of numerical methods to vector fields and maps.

Academic Background

In 2018, I completed my professional degree in Ingeniería Civil Matemática (equivalent to a MSc in Applied Mathematics) at the Universidad Técnica Federico Santa María in Valparaíso, Chile, under the supervision of Pablo Aguirre. This formative period cultivated my passion for mathematics and provided the foundation for my doctoral research.

Following this, I continued my academic journey at the University of Auckland, where I pursued a PhD in Applied Mathematics supervised by Hinke M. Osinga and Bernd Krauskopf Hinke M. Osinga and Bernd Krauskopf. I successfully completed my doctorate in 2024.

Research Philosophy and Vision

I firmly believe in the critical role of dynamical systems in understanding the mysteries of complex phenomena that surround us. Thus, one of my driving motivations is to develop sophisticated numerical methods to not only unravel but also aim to make abstract concepts and complex dynamics accessible through visualisation. By bridging the gap between theory and visual representation, I hope to contribute to a broader appreciation and understanding of the mathematical structures that underpin so much of the world around us.

PhD Research

My doctoral research focuses on a type of chaos known as wild chaos. Unlike classical chaos, wild chaos has specific robustness properties and emerges in diffeomorphisms of dimension three or higher. The primary objective of my work is to uncover the geometric mechanisms that give rise to this abstract form of chaos in an explicit system.

An essential ingredient in this context is a blender: a geometric object that appears to be higher-dimensional. What does a blender look like? The one-dimensional manifold of a blender ‘looks’ as a surface. The figure below illustrates the computation of the one-dimensional manifolds for two fixed points in a three-dimensional Hénon-like family. As parameters vary, we observe the destruction of the blender: the surface-like structure thins and eventually transitions into a Cantor set. By tracking the $2^{11}$ intersection points of these manifolds with a plane, we can computationally capture the exact moment the blender loses its defining topological properties.

Intersection points of one-dimensional manifolds as a function of xi

The destruction of a blender in a three-dimensional Henon-like family as parameters change. Plotted are the one-dimensional manifolds of two different fixed points and their 2^11 intersection points with a plane.

This figure was generated using a MATLAB algorithm I developed for the computation of one-dimensional manifolds of maps. This algorithm is capable of handling extremely complex manifold structures with a high degree of accuracy and efficiency. By using this computational approach, we can topologically characterise the emergence of a blender and provide critical insights into how these objects lose their defining properties as system parameters evolve.

Joint work with: Hinke M. Osinga and Bernd Krauskopf

Associated Publication: Item #3 in the Publications and Preprints section.

Code Demo: https://github.com/dcjulio/Computing-1D-manifolds-in-maps

Publications and Preprints

Constructing pseudo-orbits on invariant manifolds of maps as seeds for boundary value problems
Dana C'Julio, Sanaz Amani, Sam Doak, Bernd Krauskopf & Hinke M. Osinga,
Preprint, University of Auckland, 2026.
Mind the gaps: Emergence of blenders in a three-dimensional Hénon-like map with different orientation properties
Dana C'Julio, Bernd Krauskopf & Hinke M. Osinga,
Preprint, University of Auckland, 2025.
Computing parametrised large intersection sets of 1D invariant manifolds
Dana C'Julio, Bernd Krauskopf & Hinke M. Osinga,
Numerical Algorithms 96(3), pp. 1079–1108, 2024; in Collection ANODE 2023 – In honour of John Butcher's 90th birthday.
Preprint
Finding Strategies to Regulate Propagation and Containment of Dengue via Invariant Manifold Analysis
Dana Contreras-Julio, Pablo Aguirre, José Mujica & Olga Vasilieva,
SIAM Journal on Applied Dynamical Systems, 19, pp. 1392-1437, 2020.
Selected for inclusion in SIAM Epidemiology Collection.
Preprint
Allee thresholds and basins of attraction in a predation model with double Allee effect,
Dana Contreras-Julio & Pablo Aguirre,
Mathematical Methods in the Applied Sciences, 41, pp. 2699-2714, 2018.
Preprint

Posters

Blender Hunting: the search for wild chaos (2023)

SGS Research Showcase, University of Auckland, New Zealand.
Awarded the Judges’ Choice Second Place in the Academic Poster category.

Three-dimensional horseshoes and orientation reversal in wild chaos (2021)

SIAM Conference on Applications of Dynamical Systems
Awarded the Red Sock Award for the best poster presentation.

Chaos in the double pendulum (2016)

Festival de las matematicas, Valparaiso, Chile.

Global Dynamics and bifurcations in a predator-prey model with double Allee effect (2015)

XLI Semana de la Matematica, Valparaiso, Chile.
Awarded the best poster presentation.