Nonsmooth Perturbations Research Articles

Limit Cycles Bifurcating from Planar Polynomial Quasi-Homogeneous Centers of Weight-Degree 3 with Nonsmooth Perturbations

In this paper, we estimate the number of limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3, when they are perturbed inside the class of piecewise smooth polynomial systems of degree n. By using the first-order averaging method, we give the upper and lower bounds on the maximal number of limit cycles which can bifurcate from the period annuluses. Our results indicate that the nonsmooth systems can have more limit cycles than the smooth ones.

On the number of limit cycles in piecewise planar quadratic differential systems

We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si’s in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.

A class of discontinuous systems exhibit perturbed period doubling bifurcation

<p>This research considers discontinuous dynamical systems, which have related vector fields that shift over a discontinuity surface. These systems appear in a variety of applications, including ecology, medicine, neuroscience, and nonsmooth mechanics. The purpose of this paper is to develop a perturbation technique that measures the effect of a nonsmooth perturbation on the period doubling bifurcation of an unperturbed system. The unperturbed system is assumed to be close to a period doubling orbit, such that when the bifurcation parameter varies, the response changes from a period one to a period two limit cycle. The generalized determination of the Poincaré map associated with perturbed systems subjected to nonsmooth transitions is derived. The main techniques used in the proof of the results are normal forms and Melnikov functions, which are defined in two zones. Various examples are presented to show that non-smoothness is responsible for period doubling. To illustrate the interesting period doubling phenomenon that emerges from an existing flat periodic orbit via the non-smooth perturbation, a simple and novel discontinuous system is provided. An additional example is provided to show the emergence of a perturbed period doubling orbit near an unperturbed one.</p>

Chaotic dynamics of an extended Duffing-van der Pol system with a non-smooth perturbation and parametric excitation

Abstract Chaotic dynamics of a fifth-order extended Duffing-van der Pol system with a non-smooth periodic perturbation and parametric excitation are investigated both analytically and numerically in this paper. With the Fourier series, the system is expanded to the equivalent smooth system. The Melnikov perturbation method is used to derive the horseshoe chaos condition in the cases of homoclinic and heteroclinic intersections. The chaotic features for different system parameters are investigated in detail. The monotonic variation of the coefficients of parametric excitation and non-smooth periodic disturbance is found. With numerical methods, we validate the analytical results obtained by Melnikov’s method. The impact of initial conditions is carefully analyzed by basins of attraction and the effect of non-smooth periodic disturbance on the basin of attraction is also investigated. Besides, the effect of different parameters on the bifurcation pathway into chaotic attractors is examined.

Stochastic bifurcation and chaos analysis for a class of ships rolling motion under non-smooth perturbation and random excitation

The actions of wind and wave have a great influence on the ship rolling motion, which may cause chaotic situation even capsizing. Stochastic bifurcation and chaos for a class of ships rolling motion in longitudinal waves under non-smooth perturbation and random excitation, i.e., the effects of wind and wave, is studied by analytical and numerical methods. The stationary probability density function (SPDF) for rolling motion is obtained through the stochastic averaging method as well as stochastic phenomenological (P) bifurcation is analysed by the qualitative change of it. The parameter conditions for chaos are strictly derived through Melnikov method and random Melnikov method, respectively. Chaotic regions and characteristics of systems parameters are illustrated and discussed in detail. In addition, taking the maritime patrol ship as an example, the analytical results as well as the effects of parameter excitation and noise on chaos are verified and analysed by relevant numerical simulations. These results demonstrate that changing parameter excitation amplitude or noise intensity can induce or suppress chaos.

On the number of limit cycles of a pendulum-like equation with two switching lines

This paper is devoted to study the limit cycle bifurcations of a pendulum equation x˙=y,y˙=−sinx under non-smooth perturbations of polynomials of cosx, sinx and y of degree n with switching lines x=0 and y=0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by expressing the corresponding first order Melnikov functions as several generating functions, some of which are complete elliptic integrals of the first and second kind.

Limit Cycles in Two Kinds of Quadratic Reversible Systems with Non-smooth Perturbations

Limit Cycles in Two Kinds of Quadratic Reversible Systems with Non-smooth Perturbations

Limit Cycles of a Kind of Nonsmooth Differential Systems with Two Switching Lines

This paper is concerned with the number of limit cycles of nonsmooth differential systems [Formula: see text] under nonsmooth perturbations of polynomials of degree at most [Formula: see text], where [Formula: see text]. We first obtain the detailed expansion of the first Melnikov function by computing its generators for [Formula: see text]. Then by using the expansion, we give the upper bounds for the number of limit cycles bifurcating from each period annulus for two cases: [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text].

LIMIT CYCLE BIFURCATIONS IN A CLASS OF PIECEWISE SMOOTH DIFFERENTIAL SYSTEMS UNDER NON-SMOOTH PERTURBATIONS

<abstract> This paper deals with the problem of limit cycles of a class of piecewise smooth integrable differential systems with switching line $ x = 0 $. The generating functions of the associated first order Melnikov function satisfy two different Picard-Fuchs equations. By using the property of Chebyshev space, we obtain an upper bound for the number of limit cycles bifurcating from the period annulus under non-smooth perturbations of polynomials of degree $ n $. Finally, we present a concrete example to illustrate the theoretical result. </abstract>

Stability in shape optimization with second variation

We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the Hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.

Time-series-based nonlinear system identification of strongly nonlinear attachments

This work introduces a new nonlinear system identification procedure for identifying the dynamics of an otherwise linear mechanical system augmented by strongly nonlinear local attachments directly from measured transient response data. The method employs the proper orthogonal decomposition to extract energy-dependent proper orthogonal modes from the measured time series. Then, using known linear properties, the characteristic frequencies of the system are estimated by applying the Rayleigh quotient, and an estimated frequency-energy plot (FEP) is created by plotting the characteristic frequencies as functions of the mechanical energy in the system. The estimated FEP directly reveals the presence of strongly nonlinear modal interactions, in the form of non-smooth perturbations (spikes) that result from transient resonance captures between different harmonic components of the measured data. The nonlinearity is identified by plotting the estimated characteristic frequencies as functions of a defined characteristic displacement and fitting a frequency equation based on the model of the nonlinearity. The method is demonstrated computationally and experimentally using the response of a cantilevered model airplane wing with a strongly nonlinear attachment connected to its tip.

On uniqueness of recovering the convolution integro-differential operator from the spectrum of its non-smooth one-dimensional perturbation

We consider a one-dimensional perturbation of the convolution integro-differential operator of arbitrary order on a finite interval. The inverse problem of recovering the convolution component from the spectrum is studied, provided that the perturbation term is known a priori. We prove the uniqueness theorem for this inverse problem and establish the minimal requirements on the perturbation term guarantying the informativity of the spectrum.

Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system

We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.

Direct detection of nonlinear modal interactions from time series measurements

Abstract This work describes a new method for the detection and characterization of strongly nonlinear modal interactions that occur in dynamical systems with an underlying linear structure and smooth, local nonlinearities directly from transient response data. The method relies on proper orthogonal decomposition (POD) to extract energy-dependent proper orthogonal mode shapes, which are used as trial vectors in the Rayleigh quotient to estimate the system’s frequency content at varying energy levels. By plotting the estimated frequencies as functions of energy, an approximate frequency-energy plot (FEP) is created, which reveals the rich and interesting nonlinear dynamical interactions directly from the measured time series. The method is first applied to the simulated response of a cantilever beam with local, smooth nonlinearity and then to the experimentally measured response of a comparable cantilever beam. In both applications, the approximate FEP reveals the presence of nonsmooth perturbations in the frequency estimates emanating from the curves that connect different nonlinear normal modes (NNMs) of the system. Using wavelet-bounded empirical mode decomposition and slow-flow analysis, the nonsmooth perturbations are shown to be the result of internal resonances between two or more interacting NNMs. Ultimately, the method provides significant insight into the nonlinear physics governing dynamical systems with local, smooth nonlinearities while remaining conceptually and computationally simple compared to traditional methods.

Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones

This work is devoted to study the existence of periodic solutions for a class of ε-family of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers.

Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder

Detect the birth of limit cycles in non-smooth vector fields is a very important matter into the recent theory of dynamical systems and applied sciences. The goal of this paper is to study the bifurcation of limit cycles from a continuum of periodic orbits filling up a two-dimensional isochronous cylinder of a vector field in R3. The approach involves the regularization process of non-smooth vector fields and a method based in the Malkin bifurcation function for C0 perturbations. The results provide sufficient conditions in order to obtain limit cycles emerging from the cylinder through smooth and non-smooth perturbations of it. To the best of our knowledge they also illustrate the implementation by the first time of a new method based in the Malkin bifurcation function. In addition, some points concerning the number of limit cycles bifurcating from non-smooth perturbations compared with smooth ones are studied. In summary the results yield a better knowledge about limit cycles in non-smooth vector fields in R3 and explicit a manner to obtain them by performing non-smooth perturbations in codimension one Euclidean manifolds.

Pointwise Description for the Linearized Fokker–Planck–Boltzmann Model

In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker–Planck–Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543–1608, 2004) for Boltzmann equation, but the Fokker–Planck term in this paper creates some technical difficulties.

On Topological Derivatives for Contact Problems in Elasticity

In this article, a general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses domain decomposition combined with specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with a rigid foundation. The contact model allows a small interpenetration of the bodies in the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced at an arbitrary point of the elastic body. For the asymptotic analysis, we use a nonoverlapping domain decomposition technique and the associated Steklov---Poincare pseudodifferential operator. The differentiability of the energy with respect to the nonsmooth perturbation is established, and the topological derivative is presented in the closed form.

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m − 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.

Topological method for analysis of periodic canards

Singularly perturbed systems of ordinary differential equations are studied. A method for analysis of canard-type trajectories in such systems based on the topological degree theory is suggested. The method does not require smoothness of the right-hand side of the system. A result on the existence of periodic canards in systems with non-smooth perturbations is obtained. The trajectories located in this way are not necessarily Lyapunov stable, and appropriate control algorithms are required to stabilize them, e.g., feedback control.