Lyapunov Constant Research Articles

The linear term of the Poincaré map at singularities of planar vector fields

The aim of this work is to give conditions on the parameters of a family of analytic planar vector fields with a fixed Newton diagram and a monodromic singularity in order to guarantee that the coefficient (generalized first Lyapunov constant) of the linear part of the Poincaré map has an explicit formula that depends only on the Newton diagram. As far as we know, all the monodromic classes appearing in the literature satisfy that formula.

Electrochemical Noise Analysis Using Experimental Chaos Theory, Power Spectral Density and Hilbert–Huang Transform in Anodized Aluminum Alloys in Tartaric–Phosphoric–Sulfuric Acid Solutions

Aluminum and its alloys find widespread applications across diverse industries such as the automotive, construction, and aeronautics industries. When these alloys come into contact with ambient air, an Al2O3 thin oxide layer is naturally formed, typically measuring 2 to 4 nm and exhibiting remarkable hardness and protective qualities, rendering the alloys corrosion-resistant in specific atmospheric and chemical environments. This study aimed to characterize the electrochemical behaviors of anodized AA2024 and AA7075 alloys within a complex three-component electrolyte composed of tartaric–phosphoric–sulfuric acid (TPSA) solutions. The anodized specimens were subsequently exposed to 3.5 wt.% NaCl solution at room temperature, and their electrochemical performances were meticulously evaluated using an electrochemical noise (EN) analysis in accordance with ASTM G-199, respectively. In the EN, three methods of data analysis were used: the time domain analysis (chaos analysis: application of Lyapunov exponent and dimension correlation), the frequency domain analysis (power spectral density, PSD), and the time–frequency domains analysis (Hilbert–Huang transform, HHT). Scanning electron microscopy (SEM) was used to observe the morphologies of the anodized surfaces. The results indicated that the AA2024-0, AA2024-1, and AA2024-2 alloys and the AA7075-2 and AA7075-3 samples exhibited mixed corrosion according to the Lyapunov constant, with a notable inclination towards localized corrosion when analyzed using the PSD and HHT methods. The surface was not homogenous, and the corrosion process was predominately localized in specific zones.

Seismic Response of Two-way Asymmetric building with Semi-Active Stiffness Damper Under Bi-Directional Excitations

This article presents a seismic response of a two-way asymmetric building installed with a semi-active stiffness damper (SASD). The seismic response analysis of asymmetric structure is studied in terms of peak responses subjected to bi-directional excitations. The responses are obtained by numerically solving the governing equation of motion using state-space method. The effect of stiffness ratio and lyapunov constant on peak response which includes lateral, torsional displacement, and acceleration. The responses of asymmetric controlled and uncontrolled systems are simulated through MATLAB. To study the effectiveness of dampers, the controlled response of the asymmetric system is compared with the corresponding uncontrolled response. The parametric study indicated that the semi-active stiffness damper is quite effective in reducing various responses. Furthermore, on the optimum value of stiffness ratio, effectiveness of semi-active stiffness damper is reduced for lateral and torsional acceleration.

Bifurcation of limit cycles from a parabolic–parabolic type critical point in a class of planar piecewise smooth quadratic systems

In this paper we study small amplitude limit cycles bifurcated from a parabolic–parabolic (PP) type critical point of planar piecewise smooth quadratic systems having exactly one switching line given by the x-axis. Besides the non-smoothness, more difficulties arise when the critical point has a parabolic contact. For such kind of critical point, to make sure that the return map is analytic, one has to use the generalized polar coordinates instead of the classic polar coordinates to compute the corresponding Lyapunov constants. Consequently, much more complicated computations are involved. By the recent results of Novaes and Silva, the index of the first nonzero Lyapunov constant for a PP type critical point is always an even number 2ℓ+2 with ℓ≥1. We call the corresponding weak focus (0,0) is of order ℓ. We obtain eight center conditions and six conditions under which (0,0) is a weak focus of order 6. Furthermore, we prove that at least seven limit cycles can bifurcate from (0,0).

On the stability of the zero solution of differential equation of the second order under critical case

Differential equation of the form x¨+x2sgnx = Y (t, x, x˙), is considered, where the right-hand side is a small periodic perturbation of t, a sufficiently differentiable function in the origin neighborhood with variables x, x˙. It is assumed that X perturbation is of order smallness not lower than the fifth, if x is assigned the second order, and x˙ is assigned the third order. Periodic functions are introduced that are the solution of the equation above with zero righthand side. Since differentiability of the quadratic part is bounded, then differentiability of the introduced functions is also bounded. These functions are used to transfer from the initial equation to the system of equations in coordinates similar to polar. This system with the help of the polynomial replacement is reduced to a system with Lyapunov constants. Replacement coefficients are found by the partial fraction decomposition. The conclusion about the nature of stability of the zero solution is made by a sign of the first non zero constant. Due to the bounded differentiability of the introduced functions, the degree of the polynomial replacement must be limited. The system of differential equations for the replacement coefficients is solved recursively. The number of found Lyapunov constants is also bounded. This article considers the case when all found constants are zero. To study this problem there used the method of isolating the main part of the introduced functions and their combinations as a result of the expantion of the latter in the Fourier series. The remainder of the series is assumed to be sufficiently small and it is shown that its presence can be neglected. The transition to the main parts instead of functions allows to compensate for the lack of differentiability of the introduced functions. Considering such systems, polynomial replacement can be again used and the Lyapunov constant for each main part can be found. It is shown that the sign of the constant for any main part is preserved. Sufficient conditions for stability and instability are indicated.

Stability and Bifurcation Analysis in a Nonlinear Harvested Predator–Prey Model with Simplified Holling Type IV Functional Response

In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

On the stability of the equilibrium under periodic perturbations of an oscillator with a power-law restoring force with a rational exponent

Abstract Small time-periodic perturbations of the oscillator where p and q are odd numbers, p > q , are considered. The stability of the equilibrium x = 0 is investigated. The problem is distinguished by the fact that the frequency of unperturbed oscillations is an infinitesimal function of the amplitude. It is shown that in the case of a general equilibrium, for fixed value of q , the Lyapunov constant for values of p that are equal modulo 4 q is calculated by the same algorithms, i.e., the problem reduces to a consideration of a finite number (equal to 2 q − 2 if q > 1, and equal to 2 if q = 1) of values of p . An estimate, depending on q , of the number of terms of the transformation required for the calculation of the Lyapunov constant for values of p that are equal modulo 4 q is given. Particular cases are considered.

Center problem and the bifurcation of limit cycles for a cubic polynomial system

In this study, we consider the center problem and the bifurcation of limit cycles for a cubic system that lies in a symmetrical vector field about the origin. By analyzing and calculating the focal values (or the Lyapunov constant), we obtain the conditions where two equilibrium points, (1,1) and (−1,−1), become a pair of simultaneous centers. Moreover, six limit cycles, including three stable limit cycles, can bifurcate from (1,1) under a specific condition. From the symmetric quality, (−1,−1) can also bifurcate into six limit cycles by simultaneous Hopf bifurcation, which is an interesting result.

Period Doubling Bifurcation and Center Manifold Reduction in a Time-periodic and Time-delayed Model of Machining

A closed-form calculation is presented for the analysis of the period-doubling bifurcation in the time-periodic delay-differential equation model of interrupted machining processes such as milling where the nonlinearity is essentially nonsymmetric. We prove the subcritical sense of this period-doubling bifurcation and approximate the emerging period-two oscillations by the Lyapunov—Perron method for computing the center manifold and by calculating the Poincaré—Lyapunov constant of the bifurcation analytically at certain characteristic parameter values. The existence of the unstable period-two oscillations around the stable stationary cutting is confirmed using a numerical continuation algorithm developed for time-periodic delay-differential equations.

Tesselation of a triangle by repeated barycentric subdivision

Under iterated barycentric subdivision of a triangle, most triangles become flat in the sense that the largest angle tends to $\pi$. By analyzing a random walk on $SL_2(\mathbb{R})$ we give asymptotics with explicit constants for the number of flat triangles and the degree of flatness at a given stage of subdivision. In particular, we prove analytical bounds for the upper Lyapunov constant of the walk.

A Maple program for computing a terms of a center manifold, and element of bifurcations for a class of retarded functional differential equations with Hopf singularity

In this paper, we present explicit formulas for computing the coefficients of a center manifold for the Hopf singularity in autonomous retarded functional differential equations. As consequence, an algorithm for computing Lyapunov constant and normal forms up to an arbitrary order is derived. The explicit formulas have been implemented using the computer algebra system Maple. The paper ends with some examples in order to show the applicability of the methodology and the convenience of the computer software.

The focus-centre problem for a type of degenerate system

We consider differential systems in the plane defined by the sum of two homogeneous vector fields. We assume that the origin is a degenerate singular point for these differential systems. We characterize when the singular point is of focus-centre type in a generic case. The problem of its local stability is also considered. We compute the first generalized Lyapunov constant when some non-degeneracy conditions are assumed.

Instability in the critical case of pair of pure imaginary roots for a class of systems with aftereffect

The stability of a system described by Volterra integrodifferential equations is investigated in the critical case when the characteristic equation has a pair of pure imaginery roots. Conditions for instability, analogous to the well-known conditions from the theory of differential equations [1], are derived. (A similar result was established previously in [2] for integrodifferential equations of simpler structure with integral kernels of exponential-polynomial type). For the proof, several manipulations are used to simplify the original equation and, in particular, to reduce the linearized equation to the form of a differential equation with constant diagonal matrix. (An analogous approach was used to analyse instability for Volterra integrodifferential equations in the critical case of zero root in [3, 4]). As an example, the sign of the Lyapunov constant in the problem of the rotational motion of a rigid body with viscoelastic supports is calculated.