6,378 research outputs found
Koopman Operator and its Approximations for Systems with Symmetries
Nonlinear dynamical systems with symmetries exhibit a rich variety of
behaviors, including complex attractor-basin portraits and enhanced and
suppressed bifurcations. Symmetry arguments provide a way to study these
collective behaviors and to simplify their analysis. The Koopman operator is an
infinite dimensional linear operator that fully captures a system's nonlinear
dynamics through the linear evolution of functions of the state space.
Importantly, in contrast with local linearization, it preserves a system's
global nonlinear features. We demonstrate how the presence of symmetries
affects the Koopman operator structure and its spectral properties. In fact, we
show that symmetry considerations can also simplify finding the Koopman
operator approximations using the extended and kernel dynamic mode
decomposition methods (EDMD and kernel DMD). Specifically, representation
theory allows us to demonstrate that an isotypic component basis induces block
diagonal structure in operator approximations, revealing hidden organization.
Practically, if the data is symmetric, the EDMD and kernel DMD methods can be
modified to give more efficient computation of the Koopman operator
approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out
the development, we discuss the effect of measurement noise
Relative Periodic Solutions of the Complex Ginzburg-Landau Equation
A method of finding relative periodic orbits for differential equations with
continuous symmetries is described and its utility demonstrated by computing
relative periodic solutions for the one-dimensional complex Ginzburg-Landau
equation (CGLE) with periodic boundary conditions. A relative periodic solution
is a solution that is periodic in time, up to a transformation by an element of
the equation's symmetry group. With the method used, relative periodic
solutions are represented by a space-time Fourier series modified to include
the symmetry group element and are sought as solutions to a system of nonlinear
algebraic equations for the Fourier coefficients, group element, and time
period. The 77 relative periodic solutions found for the CGLE exhibit a wide
variety of temporal dynamics, with the sum of their positive Lyapunov exponents
varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8.
Preliminary work indicates that weighted averages over the collection of
relative periodic solutions accurately approximate the value of several
functionals on typical trajectories.Comment: 32 pages, 12 figure
Collective Current Rectification
We consider a network of coupled underdamped ac-driven dynamical units
exposed to a heat bath. The topology of connections defines the
absence/presence of certain spatial symmetries, which in turn cause
nonzero/zero value of a mean dc-output. We discuss dynamical mechanisms of the
rectification and identify dc-current reversals with
synchronization/desynchronization transitions in the network dynamics.Comment: 3 fig
Many-body Quantum Chaos and Entanglement in a Quantum Ratchet
We uncover signatures of quantum chaos in the many-body dynamics of a
Bose-Einstein condensate-based quantum ratchet in a toroidal trap. We propose
measures including entanglement, condensate depletion, and spreading over a
fixed basis in many-body Hilbert space which quantitatively identify the region
in which quantum chaotic many-body dynamics occurs, where random matrix theory
is limited or inaccessible. With these tools we show that many-body quantum
chaos is neither highly entangled nor delocalized in the Hilbert space,
contrary to conventionally expected signatures of quantum chaos.Comment: 6 pages, 3 figures. 1 tabl
Synchronization of chaotic networks with time-delayed couplings: An analytic study
Networks of nonlinear units with time-delayed couplings can synchronize to a
common chaotic trajectory. Although the delay time may be very large, the units
can synchronize completely without time shift. For networks of coupled
Bernoulli maps, analytic results are derived for the stability of the chaotic
synchronization manifold. For a single delay time, chaos synchronization is
related to the spectral gap of the coupling matrix. For networks with multiple
delay times, analytic results are obtained from the theory of polynomials.
Finally, the analytic results are compared with networks of iterated tent maps
and Lang-Kobayashi equations which imitate the behaviour of networks of
semiconductor lasers
Isotropy of Angular Frequencies and Weak Chimeras With Broken Symmetry
The notion of a weak chimeras provides a tractable definition for chimera
states in networks of finitely many phase oscillators. Here we generalize the
definition of a weak chimera to a more general class of equivariant dynamical
systems by characterizing solutions in terms of the isotropy of their angular
frequency vector - for coupled phase oscillators the angular frequency vector
is given by the average of the vector field along a trajectory. Symmetries of
solutions automatically imply angular frequency synchronization. We show that
the presence of such symmetries is not necessary by giving a result for the
existence of weak chimeras without instantaneous or setwise symmetries for
coupled phase oscillators. Moreover, we construct a coupling function that
gives rise to chaotic weak chimeras without symmetry in weakly coupled
populations of phase oscillators with generalized coupling
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