Bifurcations of relative equilibria
M Krupa - SIAM journal on mathematical analysis, 1990 - SIAM
This paper discusses the dynamics and bifurcation theory of equivariant dynamical systemsnear
relative equilibria, that is, group orbits invariant under the flow of an equivariant vector …
relative equilibria, that is, group orbits invariant under the flow of an equivariant vector …
Extending geometric singular perturbation theory to nonhyperbolic points---fold and canard points in two dimensions
M Krupa, P Szmolyan - SIAM journal on mathematical analysis, 2001 - SIAM
The geometric approach to singular perturbation problems is based on powerful methods
from dynamical systems theory. These techniques have been very successful in the case of …
from dynamical systems theory. These techniques have been very successful in the case of …
Relaxation oscillation and canard explosion
M Krupa, P Szmolyan - Journal of Differential Equations, 2001 - Elsevier
We give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed
planar vector fields. The transition from small Hopf-type cycles to large relaxation cycles…
planar vector fields. The transition from small Hopf-type cycles to large relaxation cycles…
Robust heteroclinic cycles
M Krupa - Journal of Nonlinear Science, 1997 - Springer
One phenomenon in the dynamics of differential equations which does not typically occur in
systems without symmetry is heteroclinic cycles. In symmetric systems, cycles can be robust …
systems without symmetry is heteroclinic cycles. In symmetric systems, cycles can be robust …
Mixed mode oscillations due to the generalized canard phenomenon
Mixed mode oscillations combine features of small oscillations and large oscillations of
relaxation type. We describe a mechanism for mixed mode oscillations based on the presence of …
relaxation type. We describe a mechanism for mixed mode oscillations based on the presence of …
Asymptotic stability of heteroclinic cycles in systems with symmetry
M Krupa, I Melbourne - Ergodic Theory and Dynamical Systems, 1995 - cambridge.org
Systems possessing symmetries often admit heteroclinic cycles that persist under perturbations
that respect the symmetry. The asymptotic stability of such cycles has previously been …
that respect the symmetry. The asymptotic stability of such cycles has previously been …
Mixed-mode oscillations in three time-scale systems: a prototypical example
Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a
combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode …
combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode …
Local analysis near a folded saddle-node singularity
M Krupa, M Wechselberger - Journal of Differential Equations, 2010 - Elsevier
Folded saddle-nodes occur generically in one parameter families of singularly perturbed
systems with two slow variables. We show that these folded singularities are the organizing …
systems with two slow variables. We show that these folded singularities are the organizing …
Extending slow manifolds near transcritical and pitchfork singularities
M Krupa, P Szmolyan - Nonlinearity, 2001 - iopscience.iop.org
We consider the dynamics of singularly perturbed differential equations near points where
the critical manifold has a transcritical or a pitchfork singularity. Our main tool is the recently …
the critical manifold has a transcritical or a pitchfork singularity. Our main tool is the recently …
Complement factor H genotypes impact risk of age-related macular degeneration by interaction with oxidized phospholipids
…, M Bedell, MH Nelson, F Lu, M Krupa… - Proceedings of the …, 2012 - National Acad Sciences
The rs1061170T/C variant encoding the Y402H change in complement factor H (CFH) has
been identified by genome-wide association studies as being significantly associated with …
been identified by genome-wide association studies as being significantly associated with …