Studies In networks of coupled oscillators often assume that every unit is coupled
globally (with equal strength), or then, only
locally (only via their nearest neighbors). But what happens in the intermediate, nonlocal case? This question can be addressed by introducing a coupling strength that attenuates depending on the distance of the oscillators in a network, e.g.,
We consider phase oscillator systems of the Kuramoto type, i.e.,
The entities are here defined as follows:
If we have a finite set of populations corresponding to spatial locations on a ring, we find so-called
chimera states:
The x-axis represents the spatial coordinate (or population), and the y-axis corresponds to the phase of individual oscillators. Populations are represented by the 'slots' delineated by red vertical lines, inhabited by a number of oscillators. One can clearly distinguish that the oscillators form
two groups, one of which is
phase-locked (left and right), and the other one is oscillating in an
incoherent fashion (center). Surprisingly, this state of coexisting synchronized and desynchronized oscillators is stable! It can be shown to be born through a saddle-node bifurcation. For many of these systems we also see that such a state may undergo a further transition characterized by a supercritical Hopf. These states are referred to as breathing chimeras due to their periodic reappearance.
Questions:
- Do chimera states exist on the topology of a line segment or the infinite line, if we have a continuum of oscillators?
- More generally, can we characterize the types of topologies (or networks, respectively) on which chimeras exist?
- A puzzling type of chimera state appears on 2D lattices with rotating spiral waves, where the topological defect in the center is replaced with a region of incoherence. How can we explain that?
- Chimera states appear very generic - in which systems of physics, biology and neurology do we find them?
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Impinging jet on flat
surface. The flow separation occurs after the jump, and is indicated by
the smaller of the two circles; the larger one indicates the roller
vortex structure as it may appear for Type II jumps.
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A stationary hydraulic jump forms when a liquid jet with circular cross
section impinges on a flat surface and a steady open surface flow is
established. Liquid jet impingement is not only of particular interest
for the cooling of technical devices, but also of fundamental physical
interest. The flow changes from supercritical to subcritical at a
critical radius where typically a circular jump is formed and flow
separation occurs. It is observed that the jump undergoes a transition
from this circular structure (Type I) to more complex structures (Type
II) featuring a roller similar to a stationary breaking wave. As flow
parameters are changed, the symmetry may be broken spontaneously and
polygons form (reported in
Ellegaard
et al., Nature 392, 767
(1998)).
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Transition from so-called
Type I jump with separation vortex to Type II jump with roller
structure at the surface.
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Matching
of polygonal hydraulic jump shapes
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A phenomenological ODE model of the Type II jump was developed and
analysed
(with T.Bohr, S.Watanabe
and RUC). The forces that act on the roller such as viscosity,
hydrostatic pressure and surface tension, are incorporated. We find
that the model reproduces circular as well as polygonal structures, of
both convex and concave shapes, in agreement with measured phase
diagrams. Coexisting solutions for constant flow parameters are also
found which indicate the possibility of hysteretical effects as
observed in experiments.
Below, four examples of
polygonal jumps are displayed and compared with shapes that are
obtained as a result of the theoretical model that I have been working
on during my Master's thesis. The pictures taken from experiments (
C.
Ellegaard et al.) are shown on the left; they are taken from
below the
impact plate and thus correspond to the projection of the vertical jet
axis onto the horizontal plane.
