277 research outputs found
Multi-agent decision-making dynamics inspired by honeybees
When choosing between candidate nest sites, a honeybee swarm reliably chooses
the most valuable site and even when faced with the choice between near-equal
value sites, it makes highly efficient decisions. Value-sensitive
decision-making is enabled by a distributed social effort among the honeybees,
and it leads to decision-making dynamics of the swarm that are remarkably
robust to perturbation and adaptive to change. To explore and generalize these
features to other networks, we design distributed multi-agent network dynamics
that exhibit a pitchfork bifurcation, ubiquitous in biological models of
decision-making. Using tools of nonlinear dynamics we show how the designed
agent-based dynamics recover the high performing value-sensitive
decision-making of the honeybees and rigorously connect investigation of
mechanisms of animal group decision-making to systematic, bio-inspired control
of multi-agent network systems. We further present a distributed adaptive
bifurcation control law and prove how it enhances the network decision-making
performance beyond that observed in swarms
Hopf bifurcation with non-semisimple 1:1 resonance
A generalised Hopf bifurcation, corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance), is analysed using a normal form approach. This bifurcation has linear codimension-3, and a centre subspace of dimension 4. The four-dimensional normal form is reduced to a three-dimensional system, which is normal to the group orbits of a phase-shift symmetry. There may exist 0, 1 or 2 small-amplitude periodic solutions. Invariant 2-tori of quasiperiodic solutions bifurcate from these periodic solutions. The authors locate one-dimensional varieties in the parameter space 1223 on which the system has four different codimension-2 singularities: a Bogdanov-Takens bifurcation a 1322 symmetric cusp, a Hopf/Hopf mode interaction without strong resonance, and a steady-state/Hopf mode interaction with eigenvalues (0, i,-i)
Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems
We consider Hamiltonian systems near equilibrium that can be (formally) reduced to one degree of freedom. Spatio-temporal symmetries play a key role. The planar reduction is studied by equivariant singularity theory with distinguished parameters. The method is illustrated on the conservative spring-pendulum system near resonance, where it leads to integrable approximations of the iso-energetic Poincaré map. The novelty of our approach is that we obtain information on the whole dynamics, regarding the (quasi-) periodic solutions, the global configuration of their invariant manifolds, and bifurcations of these.
A reversible bifurcation analysis of the inverted pendulum
The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincaré map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.
Resonances in a spring-pendulum: algorithms for equivariant singularity theory
A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.
Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance
We present a general review of the bifurcation sequences of periodic orbits
in general position of a family of resonant Hamiltonian normal forms with
nearly equal unperturbed frequencies, invariant under
symmetry. The rich structure of these classical systems is investigated with
geometric methods and the relation with the singularity theory approach is also
highlighted. The geometric approach is the most straightforward way to obtain a
general picture of the phase-space dynamics of the family as is defined by a
complete subset in the space of control parameters complying with the symmetry
constraint. It is shown how to find an energy-momentum map describing the phase
space structure of each member of the family, a catastrophe map that captures
its global features and formal expressions for action-angle variables. Several
examples, mainly taken from astrodynamics, are used as applications.Comment: 36 pages, 10 figures, accepted on International Journal of
Bifurcation and Chaos. arXiv admin note: substantial text overlap with
arXiv:1401.285
A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem
We consider the overdamped limit of two-dimensional double well systems
perturbed by weak noise. In the weak noise limit the most probable
fluctuational path leading from either point attractor to the separatrix (the
most probable escape path, or MPEP) must terminate on the saddle between the
two wells. However, as the parameters of a symmetric double well system are
varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the
bifurcation point in parameter space, the activation kinetics of the system
become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a
system at the bifurcation point, by using the Maslov-WKB method to construct an
approximation to the quasistationary probability distribution of the system
that is valid in a boundary layer near the separatrix. The approximation is a
formal asymptotic solution of the Smoluchowski equation. Our analysis relies on
the development of a new scaling theory, which yields `critical exponents'
describing weak-noise behavior near the saddle, at the bifurcation point.Comment: LaTeX, 60 pages, 24 Postscript figures. Uses epsf macros to include
the figures. A file in `uufiles' format containing the figures is separately
available at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/figures.uu
and a Postscript version of the whole paper (figures included) is available
at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/paperF.p
An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry
We present a general analysis of the bifurcation sequences of periodic orbits
in general position of a family of reversible 1:1 resonant Hamiltonian normal
forms invariant under symmetry. The rich structure of these
classical systems is investigated both with a singularity theory approach and
geometric methods. The geometric approach readily allows to find an
energy-momentum map describing the phase space structure of each member of the
family and a catastrophe map that captures its global features. Quadrature
formulas for the actions, periods and rotation number are also provided.Comment: 22 pages, 3 figures, 1 tabl
Homoclinic puzzles and chaos in a nonlinear laser model
We present a case study elaborating on the multiplicity and self-similarity
of homoclinic and heteroclinic bifurcation structures in the 2D and 3D
parameter spaces of a nonlinear laser model with a Lorenz-like chaotic
attractor. In a symbiotic approach combining the traditional parameter
continuation methods using MatCont and a newly developed technique called the
Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics on fast
parallel computing hardware with graphics processing units (GPUs), we exhibit
how specific codimension-two bifurcations originate and pattern regions of
chaotic and simple dynamics in this classical model. We show detailed
computational reconstructions of key bifurcation structures such as Bykov
T-point spirals and inclination flips in 2D parameter space, as well as the
spatial organization and 3D embedding of bifurcation surfaces, parametric
saddles, and isolated closed curves (isolas).Comment: 28 pages, 23 figure
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