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We introduce a class of multi-scale systems with discrete time, motivated by the problem of inviscid limit in fluid dynamics in the presence of small-scale noise. These systems are infinite-dimensional and defined on a scale-invariant space-time lattice. We propose a qualitative theory describing the vanishing regularization (inviscid) limit as an attractor of the renormalization group operator acting in the space of flow maps or respective probability kernels. If the attractor is a nontrivial probability kernel, we say that the inviscid limit is spontaneously stochastic: it defines a stochastic (Markov) process solving deterministic equations with deterministic initial and boundary conditions. The results are illustrated with solvable models: symbolic systems leading to digital turbulence and systems of expanding interacting phases. [ABSTRACT FROM AUTHOR]

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the quasi-Lagrangian description, making the developed theory applicable to the Euler and Navierâ€"Stokes systems. [ABSTRACT FROM AUTHOR]

We expose a hidden scaling symmetry of the Navier--Stokes equations in the limit of vanishing viscosity, which stems from dynamical space--time rescaling around suitably defined Lagrangian scaling centres. At a dynamical level, the hidden symmetry projects solutions which differ up to Galilean invariance and global temporal scaling onto the same representative flow. At a statistical level, this projection repairs the scale invariance, which is broken by intermittency in the original formulation. Following previous work by the first author, we here postulate and substantiate with numerics that hidden symmetry statistically holds in the inertial interval of fully developed turbulence. We show that this symmetry accounts for the scale-invariance of a certain class of observables, in particular, the Kolmogorov multipliers. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'. [ABSTRACT FROM AUTHOR]

We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence. [ABSTRACT FROM AUTHOR]

Open problems in fluid dynamics, such as the existence of finite-time singularities (blowup), explanation of intermittency in developed turbulence, etc, are related to multi-scale structure and symmetries of underlying equations of motion. Significantly simplified equations of motion, called toy-models, are traditionally employed in the analysis of such complex systems. In these models, equations are modified preserving just a part of the structure believed to be important. Here we propose a different approach for constructing simplified models, in which instead of simplifying equations one introduces a simplified configuration space: velocity fields are defined on multi-dimensional logarithmic lattices with proper algebraic operations and calculus. Then, the equations of motion retain their exact original form and, therefore, naturally maintain most scaling properties, symmetries and invariants of the original systems. Classification of such models reveals a fascinating relation with renowned mathematical constants such as the golden mean and the plastic number. Using both rigorous and numerical analysis, we describe various properties of solutions in these models, from the basic concepts of existence and uniqueness to the blowup development and turbulent dynamics. In particular, we observe strong robustness of the chaotic blowup scenario in the three-dimensional incompressible Euler equations, as well as the Fourier mode statistics of developed turbulence resembling the full three-dimensional Navier–Stokes system. [ABSTRACT FROM AUTHOR]

We consider a class of ordinary differential equations featuring a non-Lipschitz singularity at the origin. Solutions exist globally and are unique up until the first time they hit the origin. After 'blowup', infinitely many solutions may exist. To study continuation, we introduce physically motivated regularizations: they consist of smoothing the vector field in a ν-ball. We show that the limit ν → 0 can be understood using a certain autonomous dynamical system obtained by a solution-dependent renormalization. This procedure maps the pre-blowup dynamics to the solution ending at infinitely large renormalized time. The asymptotic behavior near blowup is described by an attractor. The post-blowup dynamics is mapped to a different renormalized solution starting infinitely far in the past and, consequently, it is associated with another attractor. The regularization establishes a relation between these two different 'lives' of the renormalized system and generically selects a restricted family of solutions, not depending on the regularization. [ABSTRACT FROM AUTHOR]

In this work, we present a systematic numerical study of the post-blowup dynamics of singular solutions of the 1D focusing critical NLS equation in the framework of a nonlinear damped perturbation. This study shows that initially, the post-blowup is described by the adiabatic approximation, in which the collapsing core approaches a universal profile and the solution width is governed by a system of ODEs (a reduced system). After that, a non-adiabatic regime is observed soon after the maximum of the solution, in which our direct numerical simulations show a clear deviation from the dynamics based on the reduced system. Our study suggests that such a non-adiabatic regime is caused by the increasing influx of mass into the collapsing core of the solution, which is not considered in the derivation of the reduced system. Also, adiabatic theoretical predictions related to the wave maximum and wave dissipation are compared with our numerical simulations. At the end of the post-collapse, the numerical simulations reveal a dominant quasi-linear regime caused by the rapid defocusing process. The collapsing core approaches the universal profile, after removing some oscillations resulting from interference with the tail. Finally, our numerical study suggests that in the limit of vanishing dissipation and, a free-space domain, the critical mass is radiated to infinity instantly at collapse time. • Evidence of no dissipation of mass in the limit of the vanishing damping term. • Adiabatic approximation is not valid in the neighborhood of the maximum amplitude. • The universal profile is valid in the non-adiabatic stage. • The last instants of the post-blowup dynamics are described by a quasi-linear regime. • The critical mass is radiated to infinity instantly after the collapse time. [ABSTRACT FROM AUTHOR]

We analyze the phenomenon of spontaneous stochasticity in fluid dynamics formulated as the nonuniqueness of solutions resulting from viscosity at infinitesimal scales acting through intermediate scales on large scales of the flow. We study the finite-time onset of spontaneous stochasticity in a real version of the Gledzer-Ohkitani-Yamada shell model of turbulence. This model allows high-accuracy numerical simulations for a wide range of scales (up to 10 orders of magnitude) and demonstrates nonchaotic dynamics but leads to an infinite number of solutions in the vanishing viscosity limit after the blowup time. We provide the numerical and theoretical description of the system dynamics at all stages. This includes the asymptotic analysis before and after the blowup leading to universal (periodic and quasi-periodic) renormalized solutions, followed by nonunique stationary states at large times. [ABSTRACT FROM AUTHOR]

Volatile oil recovery through air injection is a promising method for highly heterogeneous low permeability reservoirs. Thermal effects due to oxidation reaction, vaporization/condensation and flue gas drive are the most important mechanisms for light oil recovery by medium pressure air injection. To gather evidence for this claim, we performed ramped temperature experiments with consolidated cores filled with hexadecane injecting either air or nitrogen at different injection rates at medium pressures. The experiments show that oxygen is removed from the injected air through physical and chemical sorption by the hydrocarbon at low temperatures. Most of the bonded oxygen is released later at higher temperatures and reacts to form carbon oxides. The amount of oil burned in the air injection process relative to the amount of oil recovered increased from 2% at 10 bar to 18% at 30 bar, and again decreased to 5% at 45 bar and 3% at 70 bar. This trend was predicted theoretically in an earlier study of the medium temperature oxidation process. [ABSTRACT FROM AUTHOR]

In this paper we study the combined effects of bearing and rotor asymmetry on the stability of the classical Laval rotor using analytical techniques. This setting intrinsically features equations of motion with periodic coefficients. We obtain closed form approximations for the stability boundaries which give insights in the interaction of different effects which are elsewhere mostly considered in isolated form. Using a classical two degree of freedom model of a rotor we investigate dissipative and follower force type of forcing in combination with internal and external damping in the presence of both rotating and stationary asymmetries. Since in many technical applications the resulting forces are small compared to the elastic restoring forces of the corresponding symmetric system, the former are treated as perturbations. The results are benchmarked with numerical simulations indicating the range of validity of the approximations. Our model is designed to illustrate the most generic effects originating from the coupling of rotor and bearing asymmetries, which cannot be captured in models with constant coefficients. [ABSTRACT FROM AUTHOR]

We formulate a mathematical model for one dimensional flow with chemical reactions resulting from injection of air into a porous medium initially filled with gas, water and volatile oil. Our goal is to investigate the effect of water and steam on oil recovery, and we do so for a medium pressure air injection process. We show that, when the boiling point of the volatile oil is below or slightly above the boiling point of water, the hot steam region moves upstream of the medium temperature oxidation (MTO) wave (where oil vaporization and combustion occur), while the volatile oil and steam condense at the same location; it leads to considerable improvement of oil recovery by the MTO wave. Remarkably, the recovery curves (recovery fraction vs. time) depend weakly on the initial water and light oil saturations. If the volatile oil boiling point is much higher than the boiling point of water, the steam region moves downstream of the MTO wave. In this case the water effect on recovery is weaker and becomes negative for high water saturations. Numerical calculations suggest the existence of an oil boiling point at which a bifurcation occurs that separates solutions with the steam region upstream or downstream of the combustion zone. [ABSTRACT FROM AUTHOR]

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of the mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949, Onsager noticed that the scaling properties in the inerţial range imply nondifferentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup . indeed represents the driving mechanism of the inerţial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inerţial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations [ABSTRACT FROM AUTHOR]

Abstract: The paper presents new results on instability of general rotating systems with small axial asymmetry and damping. Rotating systems with arbitrary finite degrees of freedom are considered. Instability conditions are derived for nonresonant and resonant rotation speeds characterized by simple and double eigenvalues. As an application, the stability problem of a rotating elastic shaft with small internal and external damping forces and asymmetric imperfections is studied. The numerical example shows validity and accuracy of the derived analytical formulae for the instability regions. [Copyright &y& Elsevier]

We show that multiscaling properties of developed turbulence in shell models, which lead to anomalous scaling exponents in the inertial range, are determined exclusively by instanton dynamics. Instantons represent correlated extreme events localized in space-time, whose structure is described by self-similar statistics with a single universal scaling exponent. We show that anomalous scaling exponents appear due to the process of instanton creation. A simplified model of instanton creation is suggested, which adequately describes this anomaly. [ABSTRACT FROM AUTHOR]

We consider self-similar solutions describing intermittent bursts in shell models of turbulence and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows. It is shown that the renormahzed wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows. [ABSTRACT FROM AUTHOR]

There is a renewed interest in using combustion for the recovery of medium viscosity oil. We consider the combustion process when air is injected into the porous medium containing some fuel and inert gas. Commonly the reaction rate is negligible at low temperatures, hence the possibility of oxygen breakthrough. In this case, the oxygen gets in contact with the fuel in the downstream zone leading to slow reaction. We focus on the case when the reaction is active for all temperatures, but heat losses are negligible. For a combustion model that includes heat and mass balance equations, we develop a method for calculating the wave profile in the form of an asymptotic expansion and derive its zero- and first-order approximations. This wave profile appears to be different from wave profiles analyzed in other papers, where only the reaction at the highest temperatures was taken into account. The combustion wave has a long decaying tail. This tail is hard to observe in the laboratory because heat losses must be very small for the long tail to form. Numerical simulations were performed in order to validate our asymptotic formulae. [ABSTRACT FROM AUTHOR]

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler's paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form. [ABSTRACT FROM AUTHOR]

Abstract: The classical Chelomei''s problem of stabilization of a statically unstable elastic column by axial harmonic vibration is reconsidered. The excitation frequency is assumed to be arbitrary. Two types of boundary conditions of the column, simply supported and clamped–hinged ends, are considered. Both cases are studied analytically with check by numerical analysis. For undamped columns with the axial force close to the critical stability value, a finite number of triangular stabilization zones appears. This implies that stabilization is attained at medium frequency range (compared to the second eigenfrequency of the unexcited column). The high-frequency stabilization is impossible, as the stabilization regions vanish with the increasing excitation frequency. With addition of internal damping, a continuous stabilization region appears for small excitation amplitudes, which starts at medium frequencies and extends to high-frequency range. The influence of external damping on stabilization regions is shown to be small. [Copyright &y& Elsevier]

A linear vibrational system with multiple degrees of freedom subjected to parametric excitation is considered. It is assumed that the system is statically unstable but close to a critical point, the excitation amplitude and damping are small, and the excitation frequency is arbitrary. A new stabilization condition is derived in terms of integrals depending on eigenfrequencies and modes of the undisturbed conservative system and the symmetric excitation matrix. As a special case, an approximation for high-frequency excitation is deduced from this condition. Influence of damping on stabilization region is shown to be very small. Two examples for systems with one and two degrees of freedom are presented. It is shown that stabilization of statically unstable systems is possible for low, medium and high excitation frequencies. [Copyright &y& Elsevier]

Small amplitude shocks involving a state with complex characteristic speeds arise in mixed-type systems of two or more conservation laws. We study such shocks in detail in the generic case, when they appear near the codimension-1 elliptic boundary. Then we classify all exceptional codimension-2 states on smooth parts of the elliptic boundary. Asymptotic formulae describing shock curves near regular and exceptional states are derived. The type of singularity at the exceptional point depends on the second and third derivatives of the flux function. The main application is understanding the structure of small amplitude Riemann solutions where one of the initial states lies in the elliptic region. [ABSTRACT FROM AUTHOR]

Bifurcations of equilibria at bimodal branching points in potential systems are investigated. General formulas describing postbuckling paths and conditions for their stability are derived in terms of the original potential energy. Formulas describing unfolding of bimodal branching points due to a change of system parameters are given. A full list of possible cases for postbuckling paths, their stability, and unfolding depending on three system coefficients is presented. In order to calculate these coefficients, one needs the derivatives of the potential energy and eigenvectors of the linearized problem taken at the bifurcation point. The presented theory is illustrated by a mechanical example on stability and postbuckling behavior of an articulated elastic column having four degrees of freedom and depending on three problem parameters (stiffness coefficients at the hinges). For some of the bimodal critical points, numerical results are obtained illustrating influence of parameters on postbuckling paths, their stability, and unfolding. A surprising phenomenon that a symmetric bimodal column loaded by an axial force can buckle with a stable asymmetric mode is recognized. An example with a constrained sum of the stiffnesses of the articulated column shows that the maximum critical load (optimal design) is attained at the bimodal point. [ABSTRACT FROM AUTHOR]

Abstract: A linear oscillatory system with a positive definite mass matrix M, a positive definite stiffness matrix K and a positive semi-definite damping matrix D can be asymptotically stable. In this case the damping D of the system is called pervasive. The condition for the damping to be pervasive was given in a paper by Moran in 1970. Moran''s result has been reinvented in a recent paper from 2004 motivated by a long discussion on “intuitive” examples of marginal and asymptotic stability of systems with incomplete damping. The aim of the present note is to draw attention of the readers to Moran''s result. We reveal the importance of Moran''s criterion for suppression of oscillations by incomplete damping in systems with multiple eigenfrequencies. Finally, we provide extensions to the case of indefinite damping. [Copyright &y& Elsevier]

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study S i, j dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of S i, j shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of S i, j shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing S i, j shocks and classical waves are described. As a physical application, all types of S i, j shocks with i> j are detected and studied in a family of models for multi-phase flow in porous media. [ABSTRACT FROM AUTHOR]

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Linear multi-input dynamical systems smoothly depending on parameters are considered. A set of parameter values corresponding to uncontrollable systems (an uncontrollability set) is studied. The typical (generic) structure of the uncontrollability set is described. A constructive method of perturbation analysis of the uncontrollability set is developed. Formulae of first-order approximations for the uncontrollability set and generalized eigenvalues (uncontrollable modes) are derived and used for numerical construction of the uncontrollability set. The method is based on the versal deformation theory for matrix pairs under feedback equivalence. As an example, the uncontrollability set is found for a three-parameter two-degree-of-freedom mechanical system. [ABSTRACT FROM AUTHOR]

Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29–43] and Galin [Uspekhi Mat. Nauk, 27 (1972),pp. 241–242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found. [ABSTRACT FROM AUTHOR]

This paper deals with the study of generic singularities of a boundary of the stability domain in a parameter space for systems governed by autonomous linear differential equations y = Ay or x( m) + a1x( m-1) + ··· + amx = 0. It is assumed that elements of the matrix A and coefficients of the differential equation of mth order smoothly depend on one, two, or three real parameters. A constructive approach allowing the geometry of singularities (orientation in space, magnitudes of angles, etc.) to be determined with the use of tangent cones to the stability domain is suggested. The approach allows the geometry of singularities to be described using only first derivatives of the coefficients ai of the differential equation or first derivatives of the elements of the matrix A with respect to problem parameters with its eigenvectors and associated vectors calculated at the singular points of the boundary. Two methods of study of singularities are suggested. It is shown that they are constructive and can be applied to investigate more complicated singularities for multiparameter families of matrices or polynomials. Two physical examples are presented and discussed in detail. [ABSTRACT FROM AUTHOR]

The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem. [ABSTRACT FROM AUTHOR]

Almost twenty years ago, light was slowed down to less than 10-7 of its vacuum speed in a cloud of ultracold atoms of sodium. Upon a sudden turn-off of the coupling laser, a slow light pulse can be imprinted on cold atoms such that it can be read out and converted into a photon again. In this process, the light is stopped by absorbing it and storing its shape within the atomic ensemble. Alternatively, the light can be stopped at the band edge in photonic-crystal waveguides, where the group speed vanishes. Here, we extend the phenomenon of stopped light to the new field of parity-time (PT) symmetric systems. We show that zero group speed in PT symmetric optical waveguides can be achieved if the system is prepared at an exceptional point, where two optical modes coalesce. This effect can be tuned for optical pulses in a wide range of frequencies and bandwidths, as we demonstrate in a system of coupled waveguides with gain and loss. [ABSTRACT FROM AUTHOR]

The butterfly effect is today commonly identified with the sensitive dependence of deterministic chaotic systems upon initial conditions. However, this is only one facet of the notion of unpredictability pioneered by Lorenz, who actually predicted that multiscale fluid flows could spontaneously lose their deterministic nature and become intrinsically random. This effect, which is radically different from chaos, have remained out of reach for detailed physical observations. Here we show that this scenario is inherent to the elementary Kelvin–Helmholtz hydrodynamical instability of an initially singular shear layer. We moreover provide evidence that the resulting macroscopic flow displays universal statistical properties that are triggered by, but independent of specific physical properties at micro-scales. This spontaneous stochasticity is interpreted as an Eulerian counterpart to Richardson's relative dispersion of Lagrangian particles, giving substance to the intrinsic nature of randomness in turbulence. Intrinsic randomness, also known as spontaneous stochasticity, has been suggested to limit the finite-time predictability of multiscale chaotic dynamics, but an explicit demonstration of its effect is still lacking. Here, the authors use numerical simulations to show the existence of spontaneous stochasticity in the discontinuous shear layer leading to the Kelvin-Helmholtz instability. [ABSTRACT FROM AUTHOR]

We discuss a theoretical framework to define an optimal subgrid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three-dimensional Navier-Stokes turbulence. We also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers. We show numerically that such low-order closures work well, reproducing all known properties of the large-scale dynamics including anomalous scaling. We found small but systematic discrepancies only for a range of scales close to the subgrid threshold, which do not tend to disappear by increasing the order of the approximation. We speculate that the lack of convergence might be due to a structural instability, at least for the evolution of very fast degrees of freedom at small scales. Connections with similar problems for large eddy simulations of the three-dimensional Navier-Stokes equations are also discussed. [ABSTRACT FROM AUTHOR]

The article focuses on the study conducted by Oene Bottema regarding the two-degree-of-freedom system related to the 4th order polynomial coefficients. It mentions that the study was associated with Ziegler's paradox n multiple degrees-of-freedom systems which needs the singularity theory done by V.I. Arnold in 1970. Result shows that it offers a final step to its quality understanding as a singularity, where it will remain for the simple specific cases without its understanding.

The article discusses the aspect of the study regarding the paradoxes in the dissipation-induced destabilization or who opened Whitney's umbrella conducted by Oene Bottema. It mentions that the study with damping was compared with the domain of marginal stability and asymptotic stability depending on its characteristic polynomials. It states that he resolved it through the singularity theory on the boundary of two-degrees-of freedom non-conservative systems including the Ziegler's pendulum.

In this paper we consider the combined effect of rotating and non-rotating asymmetries on a rotor under non-conservative loading, motivated by technical applications in which breaking of symmetries may be used to avoid self-excited vibrations. We consider a rotating disk with two translatory degrees of freedom describing the displacements of the center of the disk with respect to the axis of rotation and two rotational degrees of freedom describing the tilting angles with respect to the axis of rotation. It is shown that breaking of symmetries of the rotating and non-rotating restoring terms has opposite effects for the stability behavior of translatory and rotational motions. Thus, if neither the translatory nor rotational degrees of freedom are dominant, it will be difficult to find robust designs against self-excited vibrations using the approach of breaking symmetries. [ABSTRACT FROM AUTHOR]

Following the exact decomposition in eigenstates of helicity for the Navier-Stokes equations in Fourier space [F. Waleffe, Phys. Fluids A 4, 350 (1992)], we introduce a modified version of helical shell models for turbulence with nonlocal triadic interactions. By using both an analytical argument and numerical simulation, we show that there exists a class of models, with a specific helical structure, that exhibits a statistically stable inverse energy cascade, in close analogy with that predicted for the Navier-Stokes equations restricted to the same helical interactions. We further support the idea that turbulent energy transfer is the result of a strong entanglement among triads possessing different transfer properties. [ABSTRACT FROM AUTHOR]