Nonlinear Autonomous Equations Research Articles

Uniqueness and Liouville type results for radial solutions of some classes of k -Hessian equations

We establish a uniqueness theorem and a Liouville type result for positive radial solutions of some classes of nonlinear autonomous equation with the k -Hessian operator. We also give some interesting qualitative properties of solutions. We provide an approach, based upon a Pohozaev-type identity, that unifies all our results.

Synchronizing Stability Analysis and Region of Attraction Estimation of Grid-Feeding VSCs Using Sum-of-Squares Programming

Grid-synchronising Stability (GSS) is an emerging issue that exists in grid-feeding voltage-source converters (VSCs). Its occurrence is primarily related to the nonlinear dynamics of a type of vastly applied synchronization units - Phase-locked Loops (PLL). Dynamic characterisation and modelling for the GSS analysis can be achieved by using a simplified system model, which is a second-order and autonomous nonlinear equation; however, with the presence of an indefinite damping term. As revealed and demonstrated in this work, this indefinite damping effect can result in an inaccurate Region-of-Attraction (ROA) estimation of the traditional Equal Area Principle (EAP)-based method. To overcome this issue and achieve a valid ROA estimation, this paper adopts the sum-of-squares (SOS) programming technique, which is a numeric optimisation method with SOS relaxations. The development and implementation of the SOS program for ROA estimation are presented. Numerical case studies and time-domain verifications demonstrate that this method is valid for GSS analysis, and an almost precise estimation is achieved in the first quadrant. This evidence makes the SOS method a promising tool for GSS analysis because the GSS problem is most concerned with the stability within the first swing cycle, i.e., in the first quadrant.

Admissibility in the Strong and Weak Senses

For a dynamics with continuous time determined by a nonautonomous linear equation, we characterize completely the notion of an exponential dichotomy with respect to a family of norms in terms of an admissibility property. Moreover, we consider both strong and weak admissibility properties, which are expressed, respectively, in terms of classical and mild solutions. As a nontrivial application, we establish the robustness of the notions of an exponential dichotomy with respect to a family of norms and of a nonuniform exponential dichotomy in a very simple manner.

Exponential Dichotomies and Fredholm Operators of Dynamic Equations on Time Scales

For time-varying non-regressive linear dynamic equations on a time scale with bounded graininess, we introduce the concept of the associative operator with linear systems on time scales. The purpose of this research is the characterizations of the exponential dichotomy obtained in terms of Fredholm property of that associative operator. Particularly, we use Perron’s method, which was generalized on time scales by J. Zhang, M. Fan, H. Zhu in [1], to show that if the associative operator is semi-Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both T + and T-. Moreover, we also give the converse result that the linear systems have an exponential dichotomy on both T + andT- then the associative operator is Fredholm on T.

The Behavior of Solutions to a Special Abel Equation of the Second Kind near a Nodal Singular Point

The propagation of a diffusion–reaction plane traveling wave (for example, a flame front), the charge distribution inside a heavy atom in the Thomas–Fermi model, and some other models in natural sciences lead to bounded solutions of a certain autonomous nonlinear second-order ordinary differential equation reducible to an Abel equation of the second kind. In this study, a sufficient condition is obtained under which all solutions to a special second-kind Abel equation that pass through a nodal singular point of the equation can be represented by a convergent power series (in terms of fractional powers of the variable) in a neighborhood of this point. Under this condition, new parametric representations of bounded solutions to the corresponding autonomous nonlinear equation are derived. These representations are efficient for numerical implementation.

Non-modal stability analysis of miscible viscous fingering with non-monotonic viscosity profiles

A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavourable (when a less viscous fluid displaces a high viscous fluid) and with favourable (when a more viscous fluid displaces a less viscous fluid) endpoint viscosities. A linear stability analysis yields instabilities for such viscosity variations. Using the optimal perturbation structure, we are able to show that an initially unconditional stable state becomes unstable corresponding to the most unstable initial disturbance. In addition, we also show that to understand the spatio-temporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration $c_{m}$. For the favourable endpoint viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favourable endpoint viscosity contrast. Further, the usefulness of the non-modal analysis demonstrating the physical mechanism of the quadruple structure of the perturbations from the optimal concentration disturbances is discussed. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favourable and unfavourable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities than the corresponding Arrhenius type relationship.

Symplectic integrators for second-order linear non-autonomous equations

Two families of symplectic methods specially designed for second-order time-dependent linear systems are presented. Both are obtained from the Magnus expansion of the corresponding first-order equation, but otherwise they differ in significant aspects. The first family is addressed to problems with low to moderate dimension, whereas the second is more appropriate when the dimension is large, in particular when the system corresponds to a linear wave equation previously discretised in space. Several numerical experiments illustrate the main features of the new schemes.

Robustness of exponential behavior in Banach spaces

We consider a generalization of the notion of exponential dichotomy in which the exponential behaviors on \(\mathbb {R}^+\) and \(\mathbb {R}^-\) need not agree at the origin, although they still satisfy a certain compatibility condition. A nontrivial example is the change from stable to unstable behavior in a given direction, such as in a saddle-node bifurcation. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations. We emphasize that we consider arbitrary evolution families in Banach spaces. This includes any differentiable evolution family obtained from a nonautonomous linear equation \(x^{\prime } =A(t) x\) possibly with A(t) unbounded, although in general we do not require the evolution families to be differentiable.

A non-autonomous Conley index

In this papier, a homotopy index (Conley index) which can be applied to non-autonomous differential equations is defined. It is proved that the index is well defined, and several theorems concerning its basic properties are established. The second part of this paper is concerned with the application of this index to (non-autonomous) ordinary differential equations as well as (non-autonomous) semilinear parabolic equations. Finally, several existence results for bounded solutions of asymptotically linear non-autonomous equations are proved. We also consider the existence of recurrent or Poisson stable solutions.

The Modelling of Crater Wear in Cutting with TiN Coated High Speed Steel Tool

The flow zone of the chip in contact with the tool reaches a high temperature in cutting. According to chip hardening experiments α-γ transformation may occur in steel, so the tool is in contact with a high-temperature γ phase at high pressure. The microscopic examination of worn surfaces showed that the degradation of the tool is the result of adhesive/abrasive and thermally activated processes, therefore both friction length and temperature must be taken into consideration in the modelling of crater wear. Wear rate can be described by a non-linear autonomous equation. TiN coating, which increases tool life in high speed steel, changes and slows down the wear of the tool. The activation energy of wear can be calculated from the constants of the wear equation determined by cutting experiments. The deoxidation products to be found in the workpiece in cutting may form a protective layer on the TiN layer that blocks or slows down wear.

Non-zero Lyapunov exponents and Lyapunov's coefficient of regularity for flows

We obtain sharp lower and upper bounds for the Lyapunov coefficient of regularity defined by the solutions of a non-autonomous linear equation. We also show that the dynamics can always be reduced to one defined by an upper triangular coefficient matrix with the property that the canonical basis is normal. The bounds are then used to show in a simple manner that the coefficient of regularity is related to the notion of non-uniform hyperbolicity, more precisely to the non-uniform part of a non-uniform exponential contraction or dichotomy. As an application, we show that for almost all trajectories of a measure-preserving flow with negative Lyapunov exponents, the non-uniformity can be made arbitrarily small.

On Conservation Forms and Invariant Solutions for Classical Mechanics Problems of Liénard Type

In this study we apply partial Noether andλ-symmetry approaches to a second-order nonlinear autonomous equation of the formy′′+fyy′+g(y)=0, called Liénard equation corresponding to some important problems in classical mechanics field with respect tof(y)andg(y)functions. As a first approach we utilize partial Lagrangians and partial Noether operators to obtain conserved forms of Liénard equation. Then, as a second approach, based on theλ-symmetry method, we analyzeλ-symmetries for the case thatλ-function is in the form ofλ(x,y,y′)=λ1(x,y)y′+λ2(x,y). Finally, a classification problem for the conservation forms and invariant solutions are considered.

Strong and weak [formula omitted]-admissibility

We consider the notion of a strong exponential dichotomy with respect to a family of norms for a dynamics with continuous time and we characterize it in terms of the admissibility of the pair (Lp,Lq) for p,q∈[1,∞) with p⩾q. Moreover, we consider both strong and weak admissibility, in the sense that the solutions are, respectively, classical solutions of a nonautonomous linear equation or mild solutions of an integral equation obtained from perturbing an evolution family. We then use the characterization of the notion of a strong exponential dichotomy to establish its robustness in a very simple manner.

Stability of nonautonomous equations and Lyapunov functions

We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting nonuniform exponential contractions in terms of strict Lyapunov functions. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniformexponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works,the persistence of the asymptotic stability under sufficiently smalllinear and nonlinear perturbations.

Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type

We prove the existence of time-periodic, small amplitude solutions of autonomous quasi-linear or fully nonlinear completely resonant pseudo-PDEs of Benjamin–Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero.At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions).The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order.The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have non-constant coefficients.The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash–Moser iteration in Sobolev spaces, and Lyapunov–Schmidt decomposition.

Stability Analysis of Normalized Frequency Adaptive Comb Filter

A normalized frequency adaptive comb filter composed of a number of normalized frequency estimators in parallel is proposed to track unknown frequency and unknown amplitude of each component of an almost periodic signal. The filter consists of two coupled nonlinear differential equations respectively for updating frequencies and estimating state variables. A nonlinear autonomous equation for frequency estimation is deduced after average method is applied to the almost periodic dynamic system resulting from system decoupling by slow integral manifold. Three kinds of local stability of the autonomous system:the exponential stability of isolated equilibrium point,the semistability on center manifold,and the robustness under unknown periodic disturbance are proved. The convergence and the boundedness of amplitude estimation and signal tracking as well as the effect of parameters on the transient and steady-state performance of frequency and amplitude are investigated. Each sinusoidal component and its amplitude are accurately tracked on the condition of given intervals instead of the values of frequencies. The response speed is independent of the component amplitude. Simulation results reveal the validity of the proposed algorithm.

Persistence of bounded solutions to degenerate Sobolev-Galpern equations

In this paper the persistence of bounded solutions to degenerate evolution equations of Sobolev-Galpern type is discussed. In order to define the evolution operator well, we study the existence and uniqueness of solutions to its linear form. On this basis we discuss exponential dichotomies of the evolution operator and give the Fredholm alternative result for bounded solutions of nonhomogeneous linear degenerate equations. This result enables us to give a condition for the persistence of bounded solutions of a general degenerate nonlinear autonomous equation under a nonautonomous perturbation.

Lyapunov functions versus exponential contractions

For nonautonomous linear equations in a Banach space admitting a nonuniform version of exponential contraction, we give an optimal characterization of the exponential behavior in terms of strict Lyapunov sequences. In particular, we construct explicitly strict Lyapunov sequences for each exponential contraction. We also consider the particular case of quadratic Lyapunov functions, and we use the corresponding characterization of the exponential behavior in terms of these functions to show that the stability of an exponential contraction persists under sufficiently small perturbations.

Optimal regularity of robustness for parameterized perturbations

For exponential dichotomies defined by nonautonomous linear equations, we show that sufficiently small C 1 -parameterized perturbations originate a family of exponential dichotomies of class C 1 in the parameter. We consider the general case of nonuniform exponential dichotomies, and also the general case of arbitrary growth rates of the form e λ ρ ( t ) where ρ is an arbitrary function. This includes the usual exponential behavior as a very special case when ρ ( t ) = t .

Lyapunov functions for trichotomies with growth rates

We consider linear equations x ′ = A ( t ) x that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates e c ρ ( t ) determined by a function ρ ( t ) . For example, the usual exponential behavior with ρ ( t ) = t is included as a very special case, and when ρ ( t ) = log t we obtain a polynomial behavior. We emphasize that we also consider the general case of nonuniform exponential behavior, which corresponds to the existence of what we call a ρ- nonuniform exponential trichotomy. This is known to occur in a large class of nonautonomous linear equations. Our main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting a ρ-nonuniform exponential trichotomy. This includes criteria for the existence of a ρ-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions we show that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomies in terms of strict Lyapunov functions, we establish the robustness of ρ-nonuniform exponential trichotomies under sufficiently small linear perturbations. We emphasize that in comparison with former works, our proof of the robustness is much simpler even when ρ ( t ) = t .