90 research outputs found
Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution
We analyze a large system of globally coupled phase oscillators whose natural
frequencies are bimodally distributed. The dynamics of this system has been the
subject of long-standing interest. In 1984 Kuramoto proposed several
conjectures about its behavior; ten years later, Crawford obtained the first
analytical results by means of a local center manifold calculation.
Nevertheless, many questions have remained open, especially about the
possibility of global bifurcations. Here we derive the system's complete
stability diagram for the special case where the bimodal distribution consists
of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott
and Antonsen, we show that in this case the infinite-dimensional problem
reduces exactly to a flow in four dimensions. Depending on the parameters and
initial conditions, the long-term dynamics evolves to one of three states:
incoherence, where all the oscillators are desynchronized; partial synchrony,
where a macroscopic group of phase-locked oscillators coexists with a sea of
desynchronized ones; and a standing wave state, where two counter-rotating
groups of phase-locked oscillators emerge. Analytical results are presented for
the bifurcation boundaries between these states. Similar results are also
obtained for the case in which the bimodal distribution is given by the sum of
two Gaussians.Comment: 28 pages, 7 figures; submitted to Phys. Rev. E Added comment
Existence of hysteresis in the Kuramoto model with bimodal frequency distributions
We investigate the transition to synchronization in the Kuramoto model with
bimodal distributions of the natural frequencies. Previous studies have
concluded that the model exhibits a hysteretic phase transition if the bimodal
distribution is close to a unimodal one, due to the shallowness the central
dip. Here we show that proximity to the unimodal-bimodal border does not
necessarily imply hysteresis when the width, but not the depth, of the central
dip tends to zero. We draw this conclusion from a detailed study of the
Kuramoto model with a suitable family of bimodal distributions.Comment: 9 pages, 5 figures, to appear in Physical Review
Bifurcations of piecewise smooth ļ¬ows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
On the existence of hysteresis in the Kuramoto model with bimodal frequency distributions
We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one, due to the shallowness the central dip. Here we show that proximity to the unimodal-bimodal border does not necessarily imply hysteresis when the width, but not the depth, of the central dip tends to zero. We draw this conclusion from a detailed study of the Kuramoto model with a suitable family of bimodal distributions
Integrability, localisation and bifurcation of an elastic conducting rod in a uniform magnetic field
The classical problem of the buckling of an elastic rod in a magnetic ĀÆeld is investigated
using modern techniques from dynamical systems theory. The Kirchhoff equations,
which describe the static equilibrium equations of a geometrically exact rod under end
tension and moment are extended by incorporating the evolution of a fixed external
vector (in the direction of the magnetic field) that interacts with the rod via a Lorentz
force. The static equilibrium equations (in body cordinates) are found to be noncanonical
Hamiltonian equations. The Poisson bracket is generalised and the equilibrium equations
found to sit, as the third member, in a family of rod equations in generalised magnetic
fields. When the rod is linearly elastic, isotropic, inextensible and unshearable the equations
are completely integrable and can be generated by a Lax pair.
The isotropic system is reduced using the Casimirs, via the Euler angles, to a four-dimensional
canonical system with a first integral provided the magnetic field is not
aligned with the force within the rod at any point as the system losses rank. An energy
surface is specified, defning three-dimensional flows. Poincare sections then show closed
curves.
Through Mel'nikov analysis it is shown that for an extensible rod the presence of a
magnetic field leads to the transverse intersection of the stable and unstable manifolds
and the loss of complete integrability. Consequently, the system admits spatially chaotic
solutions and a multiplicity of multimodal homoclinic solutions exist. Poincare sections
associated with the loss of integrability are displayed.
Homoclinic solutions are computed and post-buckling paths found using continutaion
methods. The rods buckle in a Hamiltonian-Hopf bifurcation about a periodic
solution. A codimension-two point, which describes a double Hamiltonian-Hopf bifurcation,
determines whether straight rods buckle into localised configurations at either two
critical values of the magnetic field, a single critical value or do not buckle at all. The
codimension-two point is found to be an organising centre for primary and multimodal
solutions
First-order phase transitions in the Kuramoto model with compact bimodal frequency distributions
The Kuramoto model of a network of coupled phase oscillators exhibits a first-order phase transition when the distribution of natural frequencies has a finite flat region at its maximum. First-order phase transitions including hysteresis and bistability are also present if the frequency distribution of a single network is bimodal. In this study, we are interested in the interplay of these two configurations and analyze the Kuramoto model with compact bimodal frequency distributions in the continuum limit. As of yet, a rigorous analytic treatment has been elusive. By combining Kuramoto's self-consistency approach, Crawford's symmetry considerations, and exploiting the Ott-Antonsen ansatz applied to a family of rational distribution functions that converge towards the compact distribution, we derive a full bifurcation diagram for the system's order-parameter dynamics. We show that the route to synchronization always passes through a standing wave regime when the bimodal distribution is compounded by two unimodal distributions with compact support. This is in contrast to a possible transition across a region of bistability when the two compounding unimodal distributions have infinite support
Bifurcation analysis of a paradigmatic model of monsoon prediction
International audienceLocal and global bifurcation structure of the forced Lorenz model in the r-F plane is investigated. The forced Lorenz model is a conceptual model for understanding the influence of the slowly varying boundary forcing like Sea Surface Temperature (SST) on the Indian summer monsoon rainfall variability. Shift in the probability density function between the two branches of the Lorenz attractor as a function of SST forcing is calculated. It is found that the one-dimensional return map (cusp map) splits into two cusps on introduction of forcing
Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation
We study the statistical physics of a surprising phenomenon arising in large
networks of excitable elements in response to noise: while at low noise,
solutions remain in the vicinity of the resting state and large-noise solutions
show asynchronous activity, the network displays orderly, perfectly
synchronized periodic responses at intermediate level of noise. We show that
this phenomenon is fundamentally stochastic and collective in nature. Indeed,
for noise and coupling within specific ranges, an asymmetry in the transition
rates between a resting and an excited regime progressively builds up, leading
to an increase in the fraction of excited neurons eventually triggering a chain
reaction associated with a macroscopic synchronized excursion and a collective
return to rest where this process starts afresh, thus yielding the observed
periodic synchronized oscillations. We further uncover a novel anti-resonance
phenomenon: noise-induced synchronized oscillations disappear when the system
is driven by periodic stimulation with frequency within a specific range. In
that anti-resonance regime, the system is optimal for measures of information
capacity. This observation provides a new hypothesis accounting for the
efficiency of Deep Brain Stimulation therapies in Parkinson's disease, a
neurodegenerative disease characterized by an increased synchronization of
brain motor circuits. We further discuss the universality of these phenomena in
the class of stochastic networks of excitable elements with confining coupling,
and illustrate this universality by analyzing various classical models of
neuronal networks. Altogether, these results uncover some universal mechanisms
supporting a regularizing impact of noise in excitable systems, reveal a novel
anti-resonance phenomenon in these systems, and propose a new hypothesis for
the efficiency of high-frequency stimulation in Parkinson's disease
Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
We analyze the nonlinear dynamics near the incoherent state in a mean-field
model of coupled oscillators. The population is described by a Fokker-Planck
equation for the distribution of phases, and we apply center-manifold reduction
to obtain the amplitude equations for steady-state and Hopf bifurcation from
the equilibrium state with a uniform phase distribution. When the population is
described by a native frequency distribution that is reflection-symmetric about
zero, the problem has circular symmetry. In the limit of zero extrinsic noise,
although the critical eigenvalues are embedded in the continuous spectrum, the
nonlinear coefficients in the amplitude equation remain finite in contrast to
the singular behavior found in similar instabilities described by the
Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both
types of bifurcation are possible and they coincide at a codimension-two Takens
Bogdanov point. The steady-state bifurcation may be supercritical or
subcritical and produces a time-independent synchronized state. The Hopf
bifurcation produces both supercritical stable standing waves and supercritical
unstable travelling waves. Previous work on the Hopf bifurcation in a bimodal
population by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable
travelling waves and stable standing waves, respectively. A comparison to these
previous calculations shows that the prediction of stable travelling waves
results from a failure to include all unstable modes.Comment: 46 pages (Latex), 4 figures available in hard copy from the author
([email protected]); paper submitted to the Journal of Statistical
Physic
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