N-dimensional dynamical systems exploiting instabilities in full

We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions

Fluid dynamical prediction of changed v1 flow at energies available at the CERN Large Hadron Collider

Substantial collective flow is observed in collisions between lead nuclei at Large Hadron Collider (LHC) as evidenced by the azimuthal correlations in the transverse momentum distributions of the produced particles. Our calculations indicate that the global v1-flow, which at RHIC peaked at negative rapidities (named third flow component or antiflow), now at LHC is going to turn toward forward rapidities (to the same side and direction as the projectile residue). Potentially this can provide a sensitive barometer to estimate the pressure and transport properties of the quark-gluon plasma. Our calculations also take into account the initial state center-of-mass rapidity fluctuations, and demonstrate that these are crucial for v1 simulations. In order to better study the transverse momentum flow dependence we suggest a new "symmetrized" v1S(pt) function, and we also propose a new method to disentangle global v1 flow from the contribution generated by the random fluctuations in the initial state. This will enhance the possibilities of studying the collective Global v1 flow both at the STAR Beam Energy Scan program and at LHC.

Dynamical properties of nonmarkovian stochastic differential equations

We study nonstationary non-Markovian processes defined by Langevin-type stochastic differential equations with an OrnsteinUhlenbeck driving force. We concentrate on the long time limit of the dynamical evolution. We derive an approximate equation for the correlation function of a nonlinear nonstationary non-Markovian process, and we discuss its consequences. Non-Markovicity can introduce a dependence on noise parameters in the dynamics of the correlation function in cases in which it becomes independent of these parameters in the Markovian limit. Several examples are discussed in which the relaxation time increases with respect to the Markovian limit. For a Brownian harmonic oscillator with fluctuating frequency, the non-Markovicity of the process decreases the domain of stability of the system, and it can change an infradamped evolution into an overdamped one.

Closure of the Monte Carlo dynamical equation in the spherical Sherrington-Kirkpatrick model

We study the analytical solution of the Monte Carlo dynamics in the spherical Sherrington-Kirkpatrick model using the technique of the generating function. Explicit solutions for one-time observables (like the energy) and two-time observables (like the correlation and response function) are obtained. We show that the crucial quantity which governs the dynamics is the acceptance rate. At zero temperature, an adiabatic approximation reveals that the relaxational behavior of the model corresponds to that of a single harmonic oscillator with an effective renormalized mass.

Fluid dynamical prediction of changed v1-flow at energies available at the CERN Large Hadron Collider

Substantial collective flow is observed in collisions between lead nuclei at Large Hadron Collider (LHC) as evidenced by the azimuthal correlations in the transverse momentum distributions of the produced particles. Our calculations indicate that the global v1-flow, which at RHIC peaked at negative rapidities (named third flow component or antiflow), now at LHC is going to turn toward forward rapidities (to the same side and direction as the projectile residue). Potentially this can provide a sensitive barometer to estimate the pressure and transport properties of the quark-gluon plasma. Our calculations also take into account the initial state center-of-mass rapidity fluctuations, and demonstrate that these are crucial for v1 simulations. In order to better study the transverse momentum flow dependence we suggest a new"symmetrized" vS1(pt) function, and we also propose a new method to disentangle global v1 flow from the contribution generated by the random fluctuations in the initial state. This will enhance the possibilities of studying the collective Global v1 flow both at the STAR Beam Energy Scan program and at LHC.

Fluid dynamical prediction of changed v1-flow at energies available at the CERN Large Hadron Collider

Substantial collective flow is observed in collisions between lead nuclei at Large Hadron Collider (LHC) as evidenced by the azimuthal correlations in the transverse momentum distributions of the produced particles. Our calculations indicate that the global v1-flow, which at RHIC peaked at negative rapidities (named third flow component or antiflow), now at LHC is going to turn toward forward rapidities (to the same side and direction as the projectile residue). Potentially this can provide a sensitive barometer to estimate the pressure and transport properties of the quark-gluon plasma. Our calculations also take into account the initial state center-of-mass rapidity fluctuations, and demonstrate that these are crucial for v1 simulations. In order to better study the transverse momentum flow dependence we suggest a new"symmetrized" vS1(pt) function, and we also propose a new method to disentangle global v1 flow from the contribution generated by the random fluctuations in the initial state. This will enhance the possibilities of studying the collective Global v1 flow both at the STAR Beam Energy Scan program and at LHC.

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which selfsimilarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N21 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features

Color ad spectral image assessment using novel quality and fidelity techniques

The ongoing development of the digital media has brought a new set of challenges with it. As images containing more than three wavelength bands, often called spectral images, are becoming a more integral part of everyday life, problems in the quality of the RGB reproduction from the spectral images have turned into an important area of research. The notion of image quality is often thought to comprise two distinctive areas – image quality itself and image fidelity, both dealing with similar questions, image quality being the degree of excellence of the image, and image fidelity the measure of the match of the image under study to the original. In this thesis, both image fidelity and image quality are considered, with an emphasis on the influence of color and spectral image features on both. There are very few works dedicated to the quality and fidelity of spectral images. Several novel image fidelity measures were developed in this study, which include kernel similarity measures and 3D-SSIM (structural similarity index). The kernel measures incorporate the polynomial, Gaussian radial basis function (RBF) and sigmoid kernels. The 3D-SSIM is an extension of a traditional gray-scale SSIM measure developed to incorporate spectral data. The novel image quality model presented in this study is based on the assumption that the statistical parameters of the spectra of an image influence the overall appearance. The spectral image quality model comprises three parameters of quality: colorfulness, vividness and naturalness. The quality prediction is done by modeling the preference function expressed in JNDs (just noticeable difference). Both image fidelity measures and the image quality model have proven to be effective in the respective experiments.

Phase effects on synchronization by dynamical relaying in delay-coupled systems

Synchronization in an array of mutually coupled systems with a finite time delay in coupling is studied using the Josephson junction as a model system. The sum of the transverse Lyapunov exponents is evaluated as a function of the parameters by linearizing the equation about the synchronization manifold. The dependence of synchronization on damping parameter, coupling constant, and time delay is studied numerically. The change in the dynamics of the system due to time delay and phase difference between the applied fields is studied. The case where a small frequency detuning between the applied fields is also discussed.

Closure of the Monte Carlo dynamical equation in the spherical Sherrington-Kirkpatrick model

We study the analytical solution of the Monte Carlo dynamics in the spherical Sherrington-Kirkpatrick model using the technique of the generating function. Explicit solutions for one-time observables (like the energy) and two-time observables (like the correlation and response function) are obtained. We show that the crucial quantity which governs the dynamics is the acceptance rate. At zero temperature, an adiabatic approximation reveals that the relaxational behavior of the model corresponds to that of a single harmonic oscillator with an effective renormalized mass.

Neurofuzzy design and model construction of nonlinear dynamical processes from data

A common problem in many data based modelling algorithms such as associative memory networks is the problem of the curse of dimensionality. In this paper, a new two-stage neurofuzzy system design and construction algorithm (NeuDeC) for nonlinear dynamical processes is introduced to effectively tackle this problem. A new simple preprocessing method is initially derived and applied to reduce the rule base, followed by a fine model detection process based on the reduced rule set by using forward orthogonal least squares model structure detection. In both stages, new A-optimality experimental design-based criteria we used. In the preprocessing stage, a lower bound of the A-optimality design criterion is derived and applied as a subset selection metric, but in the later stage, the A-optimality design criterion is incorporated into a new composite cost function that minimises model prediction error as well as penalises the model parameter variance. The utilisation of NeuDeC leads to unbiased model parameters with low parameter variance and the additional benefit of a parsimonious model structure. Numerical examples are included to demonstrate the effectiveness of this new modelling approach for high dimensional inputs.

The control of dynamical systems by neural networks

In this paper the use of neural networks for the control of dynamical systems is considered. Both identification and feedback control aspects are discussed as well as the types of system for which neural networks can provide a useful technique. Multi-layer Perceptron and Radial Basis function neural network types are looked at, with an emphasis on the latter. It is shown how basis function centre selection is a critical part of the implementation process and that multivariate clustering algorithms can be an extremely useful tool for finding centres.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

The effect of the equivalent-weights particle filter on dynamical balance in a primitive equation model

The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed. The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed. This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

Prediction of dynamical, nonlinear, and unstable process behavior

Process scheduling techniques consider the current load situation to allocate computing resources. Those techniques make approximations such as the average of communication, processing, and memory access to improve the process scheduling, although processes may present different behaviors during their whole execution. They may start with high communication requirements and later just processing. By discovering how processes behave over time, we believe it is possible to improve the resource allocation. This has motivated this paper which adopts chaos theory concepts and nonlinear prediction techniques in order to model and predict process behavior. Results confirm the radial basis function technique which presents good predictions and also low processing demands show what is essential in a real distributed environment.