193 research outputs found
Amplified Hopf bifurcations in feed-forward networks
In a previous paper, the authors developed a method for computing normal
forms of dynamical systems with a coupled cell network structure. We now apply
this theory to one-parameter families of homogeneous feed-forward chains with
2-dimensional cells. Our main result is that Hopf bifurcations in such families
generically generate branches of periodic solutions with amplitudes growing
like , , , etc. Such amplified
Hopf branches were previously found by others in a subclass of feed-forward
networks with three cells, first under a normal form assumption and later by
explicit computations. We explain here how these bifurcations arise generically
in a broader class of feed-forward chains of arbitrary length
Center manifolds of coupled cell networks
Dynamical systems with a network structure can display anomalous bifurcations
as a generic phenomenon. As an explanation for this it has been noted that
homogeneous networks can be realized as quotient networks of so-called
fundamental networks. The class of admissible vector fields for these
fundamental networks is equal to the class of equivariant vector fields of the
regular representation of a monoid. Using this insight, we set up a framework
for center manifold reduction in fundamental networks and their quotients. We
then use this machinery to explain the difference in generic bifurcations
between three example networks with identical spectral properties and identical
robust synchrony spaces
Dynamics of Tipping Cascades on Complex Networks
Tipping points occur in diverse systems in various disciplines such as
ecology, climate science, economy or engineering. Tipping points are critical
thresholds in system parameters or state variables at which a tiny perturbation
can lead to a qualitative change of the system. Many systems with tipping
points can be modeled as networks of coupled multistable subsystems, e.g.
coupled patches of vegetation, connected lakes, interacting climate tipping
elements or multiscale infrastructure systems. In such networks, tipping events
in one subsystem are able to induce tipping cascades via domino effects. Here,
we investigate the effects of network topology on the occurrence of such
cascades. Numerical cascade simulations with a conceptual dynamical model for
tipping points are conducted on Erd\H{o}s-R\'enyi, Watts-Strogatz and
Barab\'asi-Albert networks. Additionally, we generate more realistic networks
using data from moisture-recycling simulations of the Amazon rainforest and
compare the results to those obtained for the model networks. We furthermore
use a directed configuration model and a stochastic block model which preserve
certain topological properties of the Amazon network to understand which of
these properties are responsible for its increased vulnerability. We find that
clustering and spatial organization increase the vulnerability of networks and
can lead to tipping of the whole network. These results could be useful to
evaluate which systems are vulnerable or robust due to their network topology
and might help to design or manage systems accordingly.Comment: 22 pages, 12 figure
Recommended from our members
Bifurcation analysis of two coupled Jansen-Rit neural mass models
We investigate how changes in network structure can lead to pathological oscillations similar to those observed in epileptic brain. Specifically, we conduct a bifurcation analysis of a network of two Jansen-Rit neural mass models, representing two cortical regions, to investigate different aspects of its behavior with respect to changes in the input and interconnection gains. The bifurcation diagrams, along with simulated EEG time series, exhibit diverse behaviors when varying the input, coupling strength, and network structure. We show that this simple network of neural mass models can generate various oscillatory activities, including delta wave activity, which has not been previously reported through analysis of a single Jansen-Rit neural mass model. Our analysis shows that spike-wave discharges can occur in a cortical region as a result of input changes in the other region, which may have important implications for epilepsy treatment. The bifurcation analysis is related to clinical data in two case studies
- …