Piecewise Smooth Research Articles

Hopf Cyclicity and Center Conditions for Some Piecewise Smooth Cubic Differential Systems

In this paper, we consider piecewise smooth differential systems with subsystems having cubic reversible isochronous centers at the origin, where the separation line is a straight line passing through the origin. By computing Lyapunov constants, we obtain Hopf cyclicity at the origin with necessary and sufficient conditions for the origin to be a center in this type of piecewise smooth differential systems.

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status

Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status

Bifurcation of Limit Cycles for a Kind of Piecewise Smooth Differential Systems with an Elementary Center of Focus-Focus Type

Bifurcation of Limit Cycles for a Kind of Piecewise Smooth Differential Systems with an Elementary Center of Focus-Focus Type

Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

1D Piecewise Smooth Map: Exploring a Model of Investment Dynamics under Financial Frictions with Three Types of Investment Projects

1D Piecewise Smooth Map: Exploring a Model of Investment Dynamics under Financial Frictions with Three Types of Investment Projects

Integrable 2D and 3D piecewise smooth vector fields with chaotic behavior and preserving energy or not

In this paper we exhibit 2D and 3D piecewise smooth vector fields that have, at the same time, first integrals and chaotic behavior. The fact of existing first integrals means that the trajectories are tangent to foliations. Also, in some of these examples the trajectories live in an unique level of the first integrals, and so the system preserves energy, and in other cases the trajectories visit distinct levels of the first integrals and the system does not preserves the energy.

Planar quartic–quadratic fold–fold singularity of Filippov systems and its bifurcation

In this paper it is exhibited the bifurcation scenario concerning a typical singularity of planar piecewise smooth vector fields in two zones. This singularity is characterized by a quadratic contact of one vector field and a quartic contact of another vector field at the same point of the switching manifold. By means of a three parameter perturbation, we observe the presence of bifurcations like: Hopf Bifurcation, Sliding Hopf Bifurcation, the birth and annihilation of chaotic scenarios, among others. The two and three dimensional bifurcation diagrams are included.

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at (0,0). Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system.

Total Generalized Variation Regularized Piecewise Smooth Mumford-Shah Point Cloud Surface Segmentation

Total Generalized Variation Regularized Piecewise Smooth Mumford-Shah Point Cloud Surface Segmentation

Continuous Solutions of Linear Functional Equations on Piecewise Smooth Curves in Mathematical Models of Boundary Value Problems with a Shift

Continuous Solutions of Linear Functional Equations on Piecewise Smooth Curves in Mathematical Models of Boundary Value Problems with a Shift

3-dimensional piecewise linear and quadratic vector fields with invariant spheres

We consider the class X of 3 -dimensional piecewise smooth vector fields that admit a first integral which leaves invariant any sphere centered at the origin. In this class, we prove that a linear vector field does not admit isolated invariant cones. Moreover, we provide the existence of at least ten 1 -parameter families of crossing closed trajectories for quadratic vector fields in X .

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O(h2+(Δt)2). Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.

An Interface Formulation of the Laplace–Beltrami Problem on Piecewise Smooth Surfaces

An Interface Formulation of the Laplace–Beltrami Problem on Piecewise Smooth Surfaces

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS.

Nonlinear 2D C1 Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions

The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers’ Equation

Burgers’ equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical technique for the numerical solution of Burgers’ equation. The space derivative is discretized using cubic B-splines with new approximations for the second order. Typical finite differences are used to estimate the time derivative. Additionally, the scheme undergoes a stability study to ensure minimal error accumulation, and its convergence is investigated. The primary advantage of this scheme is that it generates an approximate solution as a smooth piecewise continuous function, enabling approximation at any point within the domain. The scheme is subjected to a numerical study, and the obtained results are compared to those previously reported in the literature to demonstrate the effectiveness of the proposed approach. Overall, this study aims to contribute to the development of efficient and accurate numerical methods for solving Burgers’ equation. The spline-based approach presented herein has the potential to advance our understanding of complex fluid flow phenomena and facilitate more reliable predictions in a range of practical applications.

Global Dynamics of Planar Piecewise Linear Inelastic systems having straight lines as switching manifolds

In this paper we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider.

ASPID: An asymmetric pinching damaged hysteresis model for timber structures

This paper presents a new high-fidelity phenomenological-based hysteretic model called ASPID that includes asymmetry, pinching, and strength/stiffness degradation for timber members. To capture asymmetry, one force–displacement envelope and ten physical-based hysteretic parameters are used for each loading direction. Pinching is added with an unloading–reloading monotone smooth piecewise function composed of two quadratic Bézier polynomials and one linear segment. Strength degradation at target displacement is taken using a novel fatigue law based on the maximum displacement and hysteretic dissipated energy. Exponential stiffness degradation of pinching and reloading phases is adopted in terms of their maximum displacement, whereas irreversible damage is captured using two threshold displacements independent of each loading direction. The update force algorithm is included for their computational implementation, and a Python version can be downloaded for free. The model is validated using six experimental benchmark tests of mass and lightweight timber connections and assemblies with symmetric/asymmetric behavior, where an optimized parameter identification process is considered. Moreover, the model test responses are compared with three well-known hysteretic models: SAWS, Pinching4, and DowelType. The ASPID model’s test results show an error less than 7.6% for the capacity and 2.9% for the cumulative dissipated energy as well as fits with high precision the force and dissipated energy history, getting a Normalized Root Mean Square (NRMS) error less than 4.3% and 2.8%, a Normalized Mean Absolute (NMA) error less than 2.7% and 1.5%, and a coefficient of determination R2 over 94.88% and 99.09%, respectively. Finally, comparing the four hysteretic models, the ASPID model gives the highest R2 values in all tests and has the smallest NRMS and NMA errors regarding force history and the smallest NMA errors for the dissipated energy history.