Nonlinear Invariants Research Articles

Chaos in nonlinear dynamics of air bearing slider in contact

This paper characterizes the nonlinear dynamic properties of air-bearing slider based on numerical simulation. The FFT spectrum of slider responses is shown to exhibit primary resonance frequency of air-bearing modes, harmonic frequencies, and narrowband and broadband frequencies. The chaos dynamical invariants of the response including largest Lyapunov exponents and K-entropy are examined and evaluated to characterize the nonlinear dynamics properties of the system response. It is found that system response exhibit chaos features under certain conditions. The broadband features of slider response are correlated with the chaotic properties characterized by the specific nonlinear invariants. The analysis provides an insight into the nonlinear dynamic properties of the air-bearing dynamics and control.

Excitation of flow instabilities due to nonlinear scale invariance

A novel route to instabilities and turbulence in fluid and plasma flows is presented in kinetic Vlasov-Maxwell model. New kind of flow instabilities is shown to arise due to the availability of new kinetic energy sources which are absent in conventional treatments. The present approach is based on a scale invariant nonlinear analytic formalism developed to address irregular motions on a chaotic attractor or in turbulence in a more coherent manner. We have studied two specific applications of this turbulence generating mechanism. The warm plasma Langmuir wave dispersion relation is shown to become unstable in the presence of these multifractal measures. In the second application, these multifractal measures are shown to induce naturally non-Gaussian, i.e., a stretched, Gaussian distribution and anomalous transport for tracer particles from the turbulent advection-diffusion transport equation in a Vlasov plasma flow.

On the applicability of nonlinear time series methods for partial discharge analysis

In this work, the application of nonlinear invariants and phase space methods for Partial Discharge (PD) analysis are discussed and potential pitfalls are identified. Unsupervised statistical inference techniques based on the use of surrogate data sets are proposed and employed for the purpose of testing the applicability of nonlinear algorithms and methods. The Generalized Hurst Exponent and Lempel-Ziv Complexity are used for finding the location of the system under test on the spectrum between determinism and stochasticity. The surrogate generation method is shown to produce phase-resolved relations with high correspondence with the original sets and the nonlinear invariants employed are found to have strong classification abilities at discerning between surrogates and original point series.

Characterization of local complex structures in a recurrence plot to improve nonlinear dynamic discriminant analysis

Structures in recurrence plots (RPs), preserving the rich information of nonlinear invariants and trajectory characteristics, have been increasingly analyzed in dynamic discrimination studies. The conventional analysis of RPs is mainly focused on quantifying the overall diagonal and vertical line structures through a method, called recurrence quantification analysis (RQA). This study extensively explores the information in RPs by quantifying local complex RP structures. To do this, an approach was developed to analyze the combination of three major RQA variables: determinism, laminarity, and recurrence rate (DLR) in a metawindow moving over a RP. It was then evaluated in two experiments discriminating (1) ideal nonlinear dynamic series emulated from the Lorenz system with different control parameters and (2) data sets of human heart rate regulations with normal sinus rhythms (n = 18) and congestive heart failure (n = 29). Finally, the DLR was compared with seven major RQA variables in terms of discriminatory power, measured by standardized mean difference (DSMD). In the two experiments, DLR resulted in the highest discriminatory power with DSMD = 2.53 and 0.98, respectively, which were 7.41 and 2.09 times the best performance from RQA. The study also revealed that the optimal RP structures for the discriminations were neither typical diagonal structures nor vertical structures. These findings indicate that local complex RP structures contain some rich information unexploited by RQA. Therefore, future research to extensively analyze complex RP structures would potentially improve the effectiveness of the RP analysis in dynamic discrimination studies.

Immersion and invariance adaptive control of a class of nonlinear systems with unknown control direction

The goal of this article is to extend the adaptive control problem of parametric strict feedback form nonlinear systems, using immersion and invariance to the case of unknown, possibly, time-varying control direction. The idea is to immerse a target system in ℝ n−1, which is stabilised through the design of virtual controllers, into an extended system in ℝ n+p . The designed controller takes advantage of the well-known Nussbaum functions to deal with the unknown sign of input multiplier and is designed through the manifold dynamics belonging to ℝ p+1. The effectiveness of the proposed method is shown through a simulation example and is also compared to the classical adaptive backstepping approach with an unknown control direction.

Inherited Real Risk of Type 2 Diabetes Mellitus: bedside diagnosis, pathophysiology and primary prevention.

OPINION article Front. Endocrinol., 26 February 2013Sec. Clinical Diabetes https://doi.org/10.3389/fendo.2013.00017

Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods

Aiming at automatic verification and analysis techniques for hybrid discrete-continuous systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODE-LP, as a state-of-the-art interval solver for ODEs, and with bracketing systems, which exploit monotonicity properties allowing to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply the combined iSAT-ODE solver to the analysis of a variety of non-linear hybrid systems by solving predicative encodings of reachability properties and of an inductive stability argument, and evaluate the impact of the different enclosure methods, decision heuristics and their combination. Our experiments include classic benchmarks from the literature, as well as a newly-designed conveyor belt system that combines hybrid behavior of parallel components, a slip-stick friction model with non-linear dynamics and flow invariants and several dimensions of parameterization. In the paper, we also present and evaluate an extension of VNODE-LP tailored to its use as a deduction mechanism within iSAT-ODE, to allow fast re-evaluations of enclosures over arbitrary subranges of the analyzed time span.

Scale Selection Properties of Generalized Scale-Space Interest Point Detectors

AbstractScale-invariant interest points have found several highly successful applications in computer vision, in particular for image-based matching and recognition.This paper presents a theoretical analysis of the scale selection properties of a generalized framework for detecting interest points from scale-space features presented in Lindeberg (Int. J. Comput. Vis. 2010, under revision) and comprising:an enriched set of differential interest operators at a fixed scale including the Laplacian operator, the determinant of the Hessian, the new Hessian feature strength measures I and II and the rescaled level curve curvature operator, as well asan enriched set of scale selection mechanisms including scale selection based on local extrema over scale, complementary post-smoothing after the computation of non-linear differential invariants and scale selection based on weighted averaging of scale values along feature trajectories over scale.It is shown how the selected scales of different linear and non-linear interest point detectors can be analyzed for Gaussian blob models. Specifically it is shown that for a rotationally symmetric Gaussian blob model, the scale estimates obtained by weighted scale selection will be similar to the scale estimates obtained from local extrema over scale of scale normalized derivatives for each one of the pure second-order operators. In this respect, no scale compensation is needed between the two types of scale selection approaches. When using post-smoothing, the scale estimates may, however, be different between different types of interest point operators, and it is shown how relative calibration factors can be derived to enable comparable scale estimates for each purely second-order operator and for different amounts of self-similar post-smoothing.A theoretical analysis of the sensitivity to affine image deformations is presented, and it is shown that the scale estimates obtained from the determinant of the Hessian operator are affine covariant for an anisotropic Gaussian blob model. Among the other purely second-order operators, the Hessian feature strength measure I has the lowest sensitivity to non-uniform scaling transformations, followed by the Laplacian operator and the Hessian feature strength measure II. The predictions from this theoretical analysis agree with experimental results of the repeatability properties of the different interest point detectors under affine and perspective transformations of real image data. A number of less complete results are derived for the level curve curvature operator.

Nonlinear invariants for linear loops and eigenpolynomials of linear operators

The concept of an eigenpolynomial of a linear operator is introduced, an algorithm for the construction of eigenpolynomials is formulated, and a relationship between eigenpolynomials and polynomial invariants of linear loops of programs is established. The main result obtained is the construction of sets of L-invariants for loops of operators whose Jordan forms contain nontrivial blocks.

Application and Research of Fuzzy Control Simulation in Twin Screw Extruder

The quality of rubber machinery equipment does not only depend on its machinofacture, but also depend on its automation control level. Twin screw extruder is the important auxiliary engine of internal mixer. Due to the intermittent working of internal mixer, the speed control of twin screw extruder has nonlinear and time invariance problems. According to twin screw extruder system's characteristics, this paper proposed adopting fuzzy control to improve its system with the view of advancing twin screw extruder's production efficiency and service life. It also built twin screw extruder's traditional screw speed control system and the system adopted fuzzy control. It can be seen from the comparison that the modified twin screw extruder is more advantageous.

Cosserat point element (CPE) for finite deformation of orthotropic elastic materials

An eight node brick Cosserat point element (CPE) has been developed for the numerical solution of three-dimensional problems of hyperelastic nonlinear orthotropic elastic materials. In the Cosserat approach, a strain energy function for the CPE is proposed which satisfies restrictions due to a nonlinear form of the patch test. Part of the strain energy of the CPE is characterized by a three-dimensional strain energy function that depends on physically based nonlinear orthotropic invariants. Special attention has been focused on developing closed form expressions for constitutive coefficients in another part of the strain energy that characterizes the response to inhomogeneous deformations appropriate for orthotropic material response. A number of example problems are presented which demonstrate that the CPE is a robust user friendly element for finite deformations of orthotropic elastic materials, which does not exhibit unphysical locking or hourglassing for thin structures or nearly incompressible materials.

Low-Rank Kalman Filtering for Efficient State Estimation of Subsurface Advective Contaminant Transport Models

AbstractAccurate knowledge of the movement of contaminants in porous media is essential to track their trajectory and later extract them from the aquifer. A two-dimensional flow model is implemented and then applied on a linear contaminant transport model in the same porous medium. Because of different sources of uncertainties, this coupled model might not be able to accurately track the contaminant state. Incorporating observations through the process of data assimilation can guide the model toward the true trajectory of the system. The Kalman filter (KF), or its nonlinear invariants, can be used to tackle this problem. To overcome the prohibitive computational cost of the KF, the singular evolutive Kalman filter (SEKF) and the singular fixed Kalman filter (SFKF) are used, which are variants of the KF operating with low-rank covariance matrices. Experimental results suggest that under perfect and imperfect model setups, the low-rank filters can provide estimates as accurate as the full KF but at much low...

Quantum Biophysical Semeiotics

This paper provides a brief biography on Sergio Stagnaro, in addition to his innovative research with Quantum Biophysical Semeiotics (QBS), which is an extension of classical semeiotics, an original medical science which purports to interpret the body’s signals for diagnostic purposes. The key of this new discipline is the awareness that human bodies are a continuum of biological systems whose dynamics follow the laws of deterministic chaos, which are able to be measured by means of nonlinear statistical invariants. Furthermore, there is the recent discovery that energy information and communication between DNA and bio-systems are strictly connected with quantum behavior. An integrated quantum-chaotic approach is evidenced as an epistemological base for a new approach to modern sciences. A full overview of theoretical perspectives behind QBS is overviewed, as well as scientific evidence for its promising use to prevent and/or change the course of various diseases.

A quantitative Arrow theorem

Arrow’s Impossibility theorem states that any constitution which satisfies independence of irrelevant alternatives (IIA) and unanimity and is not a dictator has to be non-transitive. In this paper we study quantitative versions of Arrow theorem. Consider n voters who vote independently at random, each following the uniform distribution over the six rankings of three alternatives. Arrow’s theorem implies that any constitution which satisfies IIA and unanimity and is not a dictator has a probability of at least 6−n for a non-transitive outcome. When n is large, 6−n is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every $${\epsilon > 0}$$ , there exists a $${\delta = \delta(\epsilon) > 0}$$ , which depends on $${\epsilon}$$ only, such that for all n, and all constitutions on three alternatives, if the constitution satisfies: Then the probability of a non-transitive outcome is at least δ. Our results generalize to any number k ≥ 3 of alternatives and to other distributions over the alternatives. We further derive a quantitative characterization of all social choice functions satisfying the IIA condition whose outcome is transitive with probability at least 1 − δ. Our results provide a quantitative statement of Arrow theorem and its generalizations and strengthen results of Kalai and Keller who proved quantitative Arrow theorems for k = 3 and for balanced constitutions only, i.e., for constitutions which satisfy for every pair of alternatives a, b, that the probability that the constitution ranks a above b is exactly 1/2. The main novel technical ingredient of our proof is the use of inverse-hypercontractivity to show that if the outcome is transitive with high probability then there are no two different voters who are pivotal with for two different pairwise preferences with non-negligible probability. Another important ingredient of the proof is the application of non-linear invariance to lower bound the probability of a paradox for constitutions where all voters have small probability for being pivotal.

Phase space reconstruction of chaotic dynamical system based on wavelet decomposition

In view of the disadvantages of the traditional phase space reconstruction method, this paper presents the method of phase space reconstruction based on the wavelet decomposition and indicates that the wavelet decomposition of chaotic dynamical system is essentially a projection of chaotic attractor on the axes of space opened by the wavelet filter vectors, which corresponds to the time-delayed embedding method of phase space reconstruction proposed by Packard and Takens. The experimental results show that, the structure of dynamical trajectory of chaotic system on the wavelet space is much similar to the original system, and the nonlinear invariants such as correlation dimension, Lyapunov exponent and Kolmogorov entropy are still reserved. It demonstrates that wavelet decomposition is effective for characterizing chaotic dynamical system.

Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

Strong global solvability is difficult to prove for high-dimensional hydrodynamic systems because of the complex interplay between nonlinearity and scale invariance. We define the Ladyzhenskaya-Lions exponent α(L)(n)=(2+n)/4 for Navier-Stokes equations with dissipation -(-Δ)(α) in R(n), for all n≥2. We review the proof of strong global solvability when α≥α(L)(n), given smooth initial data. If the corresponding Euler equations for n>2 were to allow uncontrolled growth of the enstrophy (1/2)∥∇u∥(L²)(2), then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for α<α(L)(n). The energy is critical under scale transformations only for α=α(L)(n).

On duality symmetries of supergravity invariants

The role of duality symmetries in the construction of counterterms for maximal supergravity theories is discussed in a field-theoretic context from different points of view. These are: dimensional reduction, the question of whether appropriate superspace measures exist and information about non-linear invariants that can be gleaned from linearised ones. The former allows us to prove that F-term counterterms cannot be E 7(7)-invariant in D = 4, N = 8 supergravity or E 6(6)-invariant in D = 5 maximal supergravity. This is confirmed by the two other methods which can also be applied to D = 4 theories with fewer supersymmetries and allow us to prove that N = 6 supergravity is finite at three and four loops and that N = 5 supergravity is three-loop finite.

Prediction of microdrill breakage using rough sets

AbstractThis study attempts to correlate the nonlinear invariants' with the changing conditions of a drilling process through a series of condition monitoring experiments on small diameter (1 mm) drill bits. Run-to-failure tests are performed on these drill bits, and vibration data are consecutively gathered at equal time intervals. Nonlinear invariants, such as the Kolmogorov entropy and correlation dimension, and statistical parameters are calculated based on the corresponding conditions of the drill bits. By intervariations of these values between two successive measurements, a drop–rise table is created. Any variation that is within a certain threshold (±20% of the measurements in this case) is assumed to be constant. Any fluctuation above or below is assumed to be either a rise or a drop. The reduct and conflict tables then help eliminate incongruous and redundant data by the use of rough sets (RSs). Inconsistent data, which by definition is the boundary region, are classified through certainty and coverage factors. By handling inconsistencies and redundancies, 11 rules are extracted from 39 experiments, representing the underlying rules. Then 22 new experiments are used to check the validity of the rule space. The RS decision frame performs best at predicting no failure cases. It is believed that RSs are superior in dealing with real-life data over fuzzy set logic in that actual measured data are never as consistent as here and may dominate the monitoring of the manufacturing processes as it becomes more widespread.

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques

We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.

Noise stability of functions with low influences: Invariance and optimality

In this paper, we study functions with low influences on product probability spaces. The analysis of Boolean functions f {-1, 1}/sup n/ /spl rarr/ {-1, 1} with low influences has become a central problem in discrete Fourier analysis. is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly smoothed; this extension is essential for our applications to noise stability -type problems. In particular; as applications of the invariance principle we prove two conjectures: the Majority Is Stablest conjecture [29] from theoretical computer science, which was the original motivation for this work, and the It Ain't Till It's Over conjecture [27] from social choice theory. The Majority Is Stablest conjecture and its generalizations proven here, in conjunction with the Unique Games Conjecture and its variants, imply a number of (optimal) inapproximability results for graph problems.