124,688 research outputs found
Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators
We describe a flexible and modular delayed-feedback nonlinear oscillator that
is capable of generating a wide range of dynamical behaviours, from periodic
oscillations to high-dimensional chaos. The oscillator uses electrooptic
modulation and fibre-optic transmission, with feedback and filtering
implemented through real-time digital-signal processing. We consider two such
oscillators that are coupled to one another, and we identify the conditions
under which they will synchronize. By examining the rates of divergence or
convergence between two coupled oscillators, we quantify the maximum Lyapunov
exponents or transverse Lyapunov exponents of the system, and we present an
experimental method to determine these rates that does not require a
mathematical model of the system. Finally, we demonstrate a new adaptive
control method that keeps two oscillators synchronized even when the coupling
between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special
theme issue to accompany 2009 International Workshop on Delayed Complex
Systems
Experimental Study of the Sampled Labyrinth Chaos
In this paper, some new numerical as well as experimental results connected with the so-called labyrinth chaos are presented. This very unusual chaotic motion can be generated by mathematical model involving the scalar goniometrical functions which makes a three-dimensional autonomous dynamical system strongly nonlinear. Final circuitry implementation with analog core and digital parts can be used for modeling Brownian motion. From the viewpoint of generating chaotic motion by some electronic circuit, first step is to solve problems associated with the two-port nonlinear transfer functions synthesis. In the case of labyrinth chaos the finite dynamical range of the input variables introduced by the used active elements usually limits the performance greatly, similarly as it holds for the multi-grid spiral attractors. This paper shows an elegant way how to remove these obstacles by using uni-versal multiple-port with internal digital signal processing
Temporality-induced chaos in the Kuramoto Model
Switched dynamical systems have been extensively studied in engineering literature in the context of system control. In these systems, the dynamical laws change between different subsystems depending on the environment, a process that is known to produce emergent behaviors---notably chaos. These dynamics are analogous to those of temporal networks, in which the network topology changes over time, thereby altering the dynamics on the network. It stands to reason that temporal networks may therefore produce emergent chaos and other exotic behaviors unanticipated in static networks, yet concrete examples remain elusive. Here, we present a minimal example of a networked system in which temporality produces chaotic dynamics not possible in any static subnetwork alone. Specifically, we consider a variant of the famous Kuramoto model, in which the network topology alternates between different configurations in response to the phase dynamics. We show under certain conditions this can produce a strange attractor, and we verify the presence of chaos by analyzing its geometrical properties. Our results provide new insights on the consequences of temporality for network dynamics, and acts as a proof of concept for a novel mechanism behind generating chaotic dynamics in networks
No-slip Billiards
We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. A general theory of rigid body collisions in is developed, which returns the known dimension two model as a special case but generalizes to higher dimensions. We give new results on periodicity and boundedness of orbits which suggest that a class of billiards (including all polygons) is not ergodic. Computer generated phase portraits demonstrate non-ergodic features, suggesting chaotic no-slip billiards cannot easily be constructed using the common techniques for generating chaos in standard billiards. However, Sinai type dispersing billiards, which are always ergodic in the case of standard billiards, appear to be ergodic above a certain curvature threshold
Generating a Fractal Butterfly Floquet Spectrum in a Class of Driven SU(2) Systems
A scheme for generating a fractal butterfly Floquet spectrum, first proposed
by Wang and Gong [Phys. Rev. A {\bf 77}, 031405(R) (2008)], is extended to
driven SU(2) systems such as a driven two-mode Bose-Einstein condensate. A new
class of driven systems without a link with the Harper model context is shown
to have an intriguing butterfly Floquet spectrum. The found butterfly spectrum
shows remarkable deviations from the known Hosftadter's butterfly. In addition,
the level crossings between Floquet states of the same parity and between
Floquet states of different parities are studied and highlighted. The results
are relevant to studies of fractal statistics, quantum chaos, coherent
destruction of tunneling, as well as the validity of mean-field descriptions of
Bose-Einstein condensates.Comment: 11 pages, 9 figures, relatively large size, a full-length report of
the findings in arXiv 0906.225
Wiener Chaos and the Cox-Ingersoll-Ross model
In this we paper we recast the Cox--Ingersoll--Ross model of interest rates
into the chaotic representation recently introduced by Hughston and Rafailidis.
Beginning with the ``squared Gaussian representation'' of the CIR model, we
find a simple expression for the fundamental random variable X. By use of
techniques from the theory of infinite dimensional Gaussian integration, we
derive an explicit formula for the n-th term of the Wiener chaos expansion of
the CIR model, for n=0,1,2,.... We then derive a new expression for the price
of a zero coupon bond which reveals a connection between Gaussian measures and
Ricatti differential equations.Comment: 27 page
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