SDG workshop - Kingscliff
 
tuesday Speaker: Joachim Worthington (University of Sydney) Title: Instability of certain equilibrium solutions of the Euler fluid equations on the torus Abstract: The 2D Euler fluid equations in vorticity formulation are an infinite dimensional Hamiltonian system. We compute the (in)stability of equilibrium solutions of this PDE on the torus by using a Poisson structure preserving truncation due to Zeitlin to a system of n ODEs, and then consider the limit that n goes to infinity. Extending work of Li [2008] we show that most of the equilibria with vorticity cos(m_1 x + m_2 y) are unstable. After block decomposition this can be understood by analysing a singularly perturbed eigenvalue problem. *** Speaker: Theodore Vo (University of Sydney) Title: Folded Saddle-Node Canards of Type I Abstract: The canard phenomenon occurs generically in singular perturbation problems with at least two slow variables. Folded node canards, folded saddle canards, and their bifurcations have been studied extensively. The folded saddle-node of type I (FSN I) is the codimension-1 bifurcation that gives rise to folded nodes and folded saddles and has been observed in various applications, such as the forced Van der Pol oscillator and in models of neural excitability. Their dynamics however, are not completely understood. In this work, we analyze the local dynamics near a FSN I by combining methods from geometric singular perturbation theory (blow-up), and the theory of dynamic bifurcations (analytic continuation into the plane of complex time). We prove the existence of canards and faux canards near the FSN I, and study the associated bifurcation delay. *** Speaker: Ludmila Manic (Macquarie University) Title: Semi-Infinite LP Problems Related to Problems of Optimal Control of Singularly Perturbed Dynamical Systems Abstract: A near optimal control of a dynamical system can be constructed via a solution of semi-infinite(SI) linear programming (LP) problem. If the dynamical system is singularly perturbed then the optimal value of the family of the corresponding semi-infinite LP problem is discontinuous at the zero value of the parameter. We construct a modified (augmented ) SILP problem which defines the true limit of the optimal value of this family and allows one to construct a near optimal control of the SP dynamics. *** Speaker: Kerry-Lyn Roberts (University of Sydney) Title: Respiratory rhythm generation in the preBoetzinger complex Abstract: It is known that a network of synaptically coupled neurons, located in a region of the brain stem known as the pre-Boetzinger complex (preBoetC), generates respiratory rhythms in mammals. We study a model of two coupled preBoetC neurons by Butera et al (1999) and identify the complex bursting and spiking patterns which arise under variation of two key parameters. By using geometric singular perturbation theory and the technique of averaging over two slow variables, we are able to identify the bifurcations that lead to the transitions between the different regions of activity. We interpret the preBoetC model as one possible unfolding of a codimension three bifurcation, which locally organises the observed regions of bursting and spiking behaviour. *** Speaker: Eric Kwok (UNSW) Title: The Isoperimetric Inequality on Dynamical Systems Abstract: Invariant and almost-invariant sets are objects of interest in the study of transport and mixing processes in dynamical systems. A natural extension to almost-invariant sets are minimally dispersive sets called the coherent sets. A recent development has found success in identifying coherent sets via constructing a linear operator from the transport operator (Perron-Frobenius operator) and diffusion operators. In the present paper, we define a new operator without diffusion, and study the geometry of its eigenvectors. *** Speaker: John Maclean (University of Sydney) Title: Convergence of higher order schemes for the Projective Integration method for stiff ordinary differential equations Abstract: Recently, numerical methods to deal with multi-scale systems have received much attention, including the projective integration method within the equation-free framework and the heterogeneous multiscale method. These powerful methods have successfully been applied to a wide range of problems, including modelling of water in nanotubes, micelle formation, chemical kinetics and climate modelling. We present convergence results for higher-order formulations of the projective integration method for dissipative deterministic multi-scale systems. We briefly discuss the elements of the proof and conclude by discussing the implications of our result when selecting a numerical method. *** Speaker: John Mitry (University of Sydney) Title: Faux Canards: what are they good for? Abstract: A recent study of folded saddle singularities has propositioned the puzzling question of faux canards, their importance and their behaviour. We present a short overview of some results relating to the behaviour of faux canards in addition to some situations in which they prove practically relevant. If time permits, I will also elaborate on faux canards in the case of a folded node. *** Speaker: Sebastian Boie (University of Auckland) Title: Quasi-steady-state reductions in the analysis of biophysical models Abstract: Models of biophysical processes often evolve on very different time scales and are commonly formulated as singularly perturbed problems. Modellers frequently use a reduction technique called quasi-steady-state reduction (QSSR) as a first step to simplify the analysis of a model. In QSSR, a subset of fast variables are assumed to equilibriate instantaneously, thus effectively reducing the dimension of a given problem. Sometimes this reduction preserves the key features of the dynamics but in other cases bifurcations may be lost or new bifurcations introduced. We show how the geometry of the critical manifold determines whether QSSR introduces qualitative changes to the dynamics.