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9-fold/9.3-fold for the healthy mucosa and 46.6-fold/24.2-fold for the pathological mucosa, while from direct calculations, those ratios were 19.7-fold/10.1-fold for the healthy mucosa and 33.1-fold/17.3-fold for the pathological mucosa. The higher values obtained in this study indicate a higher blood content in the pathological samples used to measure the diffuse reflectance spectra. In light of such accuracy and sensibility to the presence of hidden absorbers, with a different accumulation between healthy and pathological tissues, good perspectives become available to develop minimally invasive spectroscopy methods for in vivo early detection and monitoring of colorectal cancer.The amplitude-dependent frequency of the oscillations, termed nonisochronicity, is one of the essential characteristics of nonlinear oscillators. In this paper, the dynamics of the Rössler oscillator in the presence of nonisochronicity is examined. In particular, we explore the appearance of a new fixed point and the emergence of a coexisting limit-cycle and quasiperiodic attractors. We also describe the sequence of bifurcations leading to synchronized, desynchronized attractors and oscillation death states in the coupled Rössler oscillators as a function of the strength of nonisochronicity and coupling parameters. Furthermore, we characterize the multistability of the coexisting attractors by plotting the basins of attraction. Our results open up the possibilities of understanding the emergence of coexisting attractors and into a qualitative change of the collective states in coupled nonlinear oscillators in the presence of nonisochronicity.We investigate, by direct numerical simulations and for certain parametric regimes, the dynamics of the damped and forced nonlinear Schrödinger (NLS) equation in the presence of a time-periodic forcing. It is thus revealed that the wave number of a plane-wave initial condition dictates the number of emerged Peregrine-type rogue waves at the early stages of modulation instability. The formation of these events gives rise to the same number of transient "triangular" spatiotemporal patterns, each of which is reminiscent of the one emerging in the dynamics of the integrable NLS in its semiclassical limit, when supplemented with vanishing initial conditions. We find that the L2-norm of the spatial derivative and the L4-norm detect the appearance of rogue waves as local extrema in their evolution. The impact of the various parameters and noisy perturbations of the initial condition in affecting the above behavior is also discussed. The long-time behavior, in the parametric regimes where the extreme wave events are observable, is explained in terms of the global attractor possessed by the system and the asymptotic orbital stability of spatially uniform continuous wave solutions.We investigate the synchronization of coupled electrochemical bursting oscillators using the electrodissolution of iron in sulfuric acid. The dynamics of a single oscillator consisted of slow chaotic oscillations interrupted by a burst of fast spiking, generating a multiple time-scale dynamical system. A wavelet analysis first decomposed the time series data from each oscillator into a fast and a slow component, and the corresponding phases were also obtained. The phase synchronization of the fast and slow dynamics was analyzed as a function of electrical coupling imposed by an external coupling resistance. For two oscillators, a progressive transition was observed With increasing coupling strength, first, the fast bursting intervals overlapped, which was followed by synchronization of the fast spiking, and finally, the slow chaotic oscillations synchronized. With a population of globally coupled 25 oscillators, the coupling eliminated the fast dynamics, and only the synchronization of the slow dynamics can be observed. The results demonstrated the complexities of synchronization with bursting oscillations that could be useful in other systems with multiple time-scale dynamics, in particular, in neuronal networks.We present an approach to construct structure-preserving emulators for Hamiltonian flow maps and Poincaré maps based directly on orbit data. Intended applications are in moderate-dimensional systems, in particular, long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. selleck screening library The method is based on multi-output Gaussian process (GP) regression on scattered training data. To obtain long-term stability, the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation, we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both are related to symplectic Euler methods in terms of numerical integration but fulfill a complementary purpose. The developed methods are first tested on the pendulum and the Hénon-Heiles system and results compared to spectral regression of the flow map with orthogonal polynomials. Chaotic behavior is studied on the standard map. Finally, the application to magnetic field line tracing in a perturbed tokamak configuration is demonstrated. As an additional feature, in the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. For implicit GP methods, we demonstrate regression performance comparable to spectral bases and artificial neural networks for symplectic flow maps, applicability to Poincaré maps, and correct representation of chaotic diffusion as well as a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.We characterize a stochastic dynamical system with tempered stable noise, by examining its probability density evolution. This probability density function satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition principle that the probability measure-valued solution to this nonlocal Fokker-Planck equation is equivalent to the martingale solution composed with the inverse stochastic flow. This result together with a Schauder estimate leads to the existence and uniqueness of strong solution for the nonlocal Fokker-Planck equation. Second, we devise a convergent finite difference method to simulate the probability density function by solving the nonlocal Fokker-Planck equation. Finally, we apply our aforementioned theoretical and numerical results to a nonlinear filtering system by simulating a nonlocal Zakai equation.We consider the problem of data-assisted forecasting of chaotic dynamical systems when the available data are in the form of noisy partial measurements of the past and present state of the dynamical system. Recently, there have been several promising data-driven approaches to forecasting of chaotic dynamical systems using machine learning. Particularly promising among these are hybrid approaches that combine machine learning with a knowledge-based model, where a machine-learning technique is used to correct the imperfections in the knowledge-based model. Such imperfections may be due to incomplete understanding and/or limited resolution of the physical processes in the underlying dynamical system, e.g., the atmosphere or the ocean. Previously proposed data-driven forecasting approaches tend to require, for training, measurements of all the variables that are intended to be forecast. We describe a way to relax this assumption by combining data assimilation with machine learning. We demonstrate this technique using the Ensemble Transform Kalman Filter to assimilate synthetic data for the three-variable Lorenz 1963 system and for the Kuramoto-Sivashinsky system, simulating a model error in each case by a misspecified parameter value. We show that by using partial measurements of the state of the dynamical system, we can train a machine-learning model to improve predictions made by an imperfect knowledge-based model.We develop an information-theoretic framework to quantify information upper bound for the probability distributions of the solutions to the McKean-Vlasov stochastic differential equations. More precisely, we derive the information upper bound in terms of Kullback-Leibler divergence, which characterizes the entropy of the probability distributions of the solutions to McKean-Vlasov stochastic differential equations relative to the joint distributions of mean-field particle systems. The order of information upper bound is also figured out.In this research paper, a novel approach in dengue modeling with the asymptomatic carrier and reinfection via the fractional derivative is suggested to deeply interrogate the comprehensive transmission phenomena of dengue infection. The proposed system of dengue infection is represented in the Liouville-Caputo fractional framework and investigated for basic properties, that is, uniqueness, positivity, and boundedness of the solution. We used the next-generation technique in order to determine the basic reproduction number R0 for the suggested model of dengue infection; moreover, we conduct a sensitivity test of R0 through a partial rank correlation coefficient technique to know the contribution of input factors on the output of R0. We have shown that the infection-free equilibrium of dengue dynamics is globally asymptomatically stable for R0 less then 1 and unstable in other circumstances. The system of dengue infection is then structured in the Atangana-Baleanu framework to represent the dynamics of dengue with the non-singular and non-local kernel. The existence and uniqueness of the solution of the Atangana-Baleanu fractional system are interrogated through fixed-point theory. Finally, we present a novel numerical technique for the solution of our fractional-order system in the Atangana-Baleanu framework. We obtain numerical results for different values of fractional-order ϑ and input factors to highlight the consequences of fractional-order ϑ and input parameters on the system. On the basis of our analysis, we predict the most critical parameters in the system for the elimination of dengue infection.Koopman mode decomposition and tensor component analysis [also known as CANDECOMP (canonical decomposition)/PARAFAC (parallel factorization)] are two popular approaches of decomposing high dimensional datasets into modes that capture the most relevant features and/or dynamics. Despite their similar goal, the two methods are largely used by different scientific communities and are formulated in distinct mathematical languages. We examine the two together and show that, under certain conditions on the data, the theoretical decomposition given by the tensor component analysis is the same as that given by Koopman mode decomposition. This provides a "bridge" with which the two communities should be able to more effectively communicate. Our work provides new possibilities for algorithmic approaches to Koopman mode decomposition and tensor component analysis and offers a principled way in which to compare the two methods. Additionally, it builds upon a growing body of work showing that dynamical systems theory and Koopman operator theory, in particular, can be useful for problems that have historically made use of optimization theory.