97 research outputs found
An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning
Manifold-learning techniques are routinely used in mining complex
spatiotemporal data to extract useful, parsimonious data
representations/parametrizations; these are, in turn, useful in nonlinear model
identification tasks. We focus here on the case of time series data that can
ultimately be modelled as a spatially distributed system (e.g. a partial
differential equation, PDE), but where we do not know the space in which this
PDE should be formulated. Hence, even the spatial coordinates for the
distributed system themselves need to be identified - to emerge from - the data
mining process. We will first validate this emergent space reconstruction for
time series sampled without space labels in known PDEs; this brings up the
issue of observability of physical space from temporal observation data, and
the transition from spatially resolved to lumped (order-parameter-based)
representations by tuning the scale of the data mining kernels. We will then
present actual emergent space discovery illustrations. Our illustrative
examples include chimera states (states of coexisting coherent and incoherent
dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics,
arising in partial differential equations and/or in heterogeneous networks. We
also discuss how data-driven spatial coordinates can be extracted in ways
invariant to the nature of the measuring instrument. Such gauge-invariant data
mining can go beyond the fusion of heterogeneous observations of the same
system, to the possible matching of apparently different systems
Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback
Interaction via pulses is common in many natural systems, especially
neuronal. In this article we study one of the simplest possible systems with
pulse interaction: a phase oscillator with delayed pulsatile feedback. When the
oscillator reaches a specific state, it emits a pulse, which returns after
propagating through a delay line. The impact of an incoming pulse is described
by the oscillator's phase reset curve (PRC). In such a system we discover an
unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic
regular spiking solution bifurcates with several multipliers crossing the unit
circle at the same parameter value. The number of such critical multipliers
increases linearly with the delay and thus may be arbitrary large. This
bifurcation is accompanied by the emergence of numerous "jittering" regimes
with non-equal interspike intervals (ISIs). Each of these regimes corresponds
to a periodic solution of the system with a period roughly proportional to the
delay. The number of different "jittering" solutions emerging at the
bifurcation point increases exponentially with the delay. We describe the
combinatorial mechanism that underlies the emergence of such a variety of
solutions. In particular, we show how a periodic solution exhibiting several
distinct ISIs can imply the existence of multiple other solutions obtained by
rearranging of these ISIs. We show that the theoretical results for phase
oscillators accurately predict the behavior of an experimentally implemented
electronic oscillator with pulsatile feedback
Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions
Synchronization is an essential collective phenomenon in networks of
interacting oscillators. Twisted states are rotating wave solutions in ring
networks where the oscillator phases wrap around the circle in a linear
fashion. Here, we analyze Hopf bifurcations of twisted states in ring networks
of phase oscillators with nonpairwise higher-order interactions. Hopf
bifurcations give rise to quasiperiodic solutions that move along the
oscillator ring at nontrivial speed. Because of the higher-order interactions,
these emerging solutions may be stable. Using the Ott--Antonsen approach, we
continue the emergent solution branches which approach anti-phase type
solutions (where oscillators form two clusters whose phase is apart) as
well as twisted states with a different winding number.Comment: 24 pages, 8 figure
Transition from chimera/solitary states to traveling waves
We study numerically the spatiotemporal dynamics of a ring network of
nonlocally coupled nonlinear oscillators, each represented by a two-dimensional
discrete-time model of the classical van der Pol oscillator. It is shown that
the discretized oscillator exhibits a richer behavior, combining the
peculiarities of both the original system and its own dynamics. Moreover, a
large variety of spatiotemporal structures is observed in the network of
discrete van der Pol oscillators when the discretization parameter and the
coupling strength are varied. Such regimes as the coexistence of multichimera
state/traveling wave and solitary state are revealed for the first time and
studied in detail. It is established that the majority of the observed
chimera/solitary states, including the newly found ones, are transient towards
the purely traveling wave mode. The peculiarities of the transition process and
the lifetime (transient duration) of the chimera structures and the solitary
state are analyzed depending on the system parameters, observation time,
initial conditions, and influence of external noise
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