Martin Wechselberger

Martin Wechselberger

Professor

Research & Grants

My research program focuses on the development of geometric methods for multi-timescale dynamical systems. By combining Geometric Singular Perturbation Theory (GSPT) with techniques from algebraic and differetial geometry, I aim to uncover the mechanisms behind complex rhythms and critical transitions in nature.

Current Research Grants

  • Geometric methods to detect rate-induced tipping events in genuine multi-timescale models (DP260100522)
    • Role: First Investigator (Lead)
    • Summary: This project develops a novel geometric framework for rate-induced tipping (R-tipping) to predict ecosystem collapses and identify early warning signals for climate-related transitions.
  • Determining endocrine-mediated plastic responses to transient heat waves (DP250101953)
    • Role: Chief Investigator
    • Summary: An interdisciplinary project determining animal resilience to heat waves by uncovering endocrine-mediated mechanisms and constructing predictive mathematical models of the environment-endocrine-phenotype axis.
  • A coordinate-independent theory for multiple time-scale dynamical systems (DP220101817)
    • Role: First Investigator (Lead)
    • Summary: Formulation of a coordinate-independent GSPT to analyse systems where standard global timescale splittings are unavailable.

Past ARC Funding (Lead Investigator)

  • A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models (DP200102130)
    • Role: Lead Investigator
    • Summary: Developing a geometric framework to explain the existence and stability of shocks in reaction-nonlinear diffusion models using GSPT.
  • A geometric theory for non-standard relaxation oscillators (DP180103022)
    • Role: Lead Investigator
    • Summary: This project investigates “non-standard” relaxation oscillators where the classical GSPT framework fails due to the presence of a non-uniform timescale spliting.
  • Geometric methods in mathematical physiology (Future Fellowship - FT120100309)
    • Role: Future Fellow (Lead)
    • Summary: This fellowship supported foundational work in the geometry of canards and folded singularities in arbitrary dimensions.
  • A geometric theory for travelling waves in advection-reaction-diffusion models (DP110102775)
    • Role: Lead Investigator
    • Summary: Mathematical analysis of complex cell migration patterns with sharp interfaces by dentifying geometric features of such multiple scales problems.

International Collaboration

I have been a key collaborator on four Marsden Grants (New Zealand), facilitating international excellence in the dynamical systems community through long-standing partnerships with the University of Auckland (Vivien Kirk, James Sneyd).

  • Modelling calcium dynamics in living animals: multiple time and space scales in theory and practice (20-UOA-074)
  • Surprisingly slow dynamics in calcium models: where are the slow time scales? (15-UOA-188)
  • Dynamics of multiscale excitable systems: applications to calcium dynamics and neuroscience (11-UOA-113)
  • Designing experiments using mathematics: nonlinear dynamics and calcium oscillations (08-UOA-028)