Piecewise Linear Systems Research Articles

Vibration Suppression of Piecewise-Linear Stiffness Nes System Under Random Excitation

Nonlinear energy sinks (NESs) are critically important for structural vibration suppression. They can absorb vibrational energy across a broad frequency spectrum, possess strong robustness, and have a relatively small mass. This study addresses the vibration suppression in a piecewise linear stiffness NES system under random excitation. Initially, a theoretical model of the piecewise linear stiffness NES system is developed. The piecewise linear stiffness function is approximated using Legendre polynomial approximation. Following this, the steady-state Fokker–Planck–Kolmogorov (FPK) equation of the system is formulated via the Generalized Harmonic Function Method. The FPK equation is solved using the fourth-order central difference method, and the effectiveness of this approach is validated by comparing the FDM results with numerical simulations. Lastly, the influence of varying system parameters on the stability of the piecewise linear stiffness NES system is analyzed.

Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

Four Limit Cycles of Three-Dimensional Discontinuous Piecewise Differential Systems Having a Sphere as Switching Manifold

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in [Formula: see text]. In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in [Formula: see text], where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles.

Saddle–node canard cycles in slow–fast planar piecewise linear differential systems

By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The obtained results are comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Dumortier and Roussarie in Mem. Am. Math. Soc. 1996, and Krupa and Szmolyan in J. Differ. Equ. 2001. In addition, some novel slow–fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle–node bifurcation of canard limit cycles, one involving headless canard cycles, and the other involving canard cycles with head. This configuration also occurs in smooth systems with N-shaped fast nullcline. However, it has not been previously reported in the Van der Pol system. Our results provide justification for this observation.

Design of Calibration System for Multi-Channel Thermostatic Metal Bath

Abstract The use of the thermostatic metal bath is becoming more and more extensive and the requirements for its precision and reliability are also increasing. To meet the needs of the metal bath calibration, a 12-channel thermostatic metal bath temperature field calibration system based on a four-wire PT100 has been designed. The system design includes a front-end temperature measurement component, which consists of a four-wire PT100 and a thermostatic block, and a signal processing component, which consists of a bidirectional constant current source excitation unit, a signal conditioning unit and a high-precision acquisition unit. The STM32f407 is used as the main control chip, and the analog channel selector is used for 12-channel selection. The constant current source is used for signal excitation, the proportional method is used to measure the PT100 resistance, and an acquisition circuit with a high-precision 32-bit ADS1263 analog-to-digital converter is used to amplify, filter and convert the analog signal. After piecewise linear fitting and system calibration, the temperature measurement accuracy can reach 0.4 mK, which meets the calibration requirements of the thermostatic metal bath.

Unveiling the principles of stochastic resonance and complex potential functions for bearing fault diagnosis

This paper introduces a novel and complex Unsaturated Piecewise Linear Quad-Stable Stochastic Resonance System (UPLQSR) to address the issue of output saturation in the Classical Quad-Stable Stochastic Resonance (CQSR) system. By linearizing the structure of the potential function, the constraints imposed by high-order terms are effectively eliminated, allowing Brownian particles to move more freely. Numerical simulations demonstrate that UPLQSR achieves significantly higher output signal amplitudes compared to CQSR, highlighting its remarkable signal amplification capability. By analyzing the structure of the potential function, the relationship between the height of the potential barrier and the aggressiveness of the particles jumping is determined. Utilizing adiabatic approximation theory, the paper derives the Steady-state Probability Density (SPD), Mean First Passage Time (MFPT), and Spectral Amplification (SA), revealing the specific process of the particle jumping, as well as the influence of the parameters on the performance of the UPLQSR. After optimizing parameters using the Adaptive Genetic Algorithm (GA), UPLQSR is applied to the early fault diagnosis of various bearing models under Gaussian white noise, demonstrating superior fault detection capabilities. In summary, this study pioneered the theory of non-saturation of quad-stable systems, optimized the weak signal detection technique, and provided a more accurate means of signal identification and analysis, highlighting its great value of application in engineering.

Describing Function and Event Based Analysis of a General Class of Piecewise Feedback Control Systems

The analysis of nonlinear systems can be greatly improved using computer software. In this work the Sridhar nonlinearity is described, and a new parameterization is given that allows to systematically obtain a large number of the classical piecewise nonlinearities as particular cases. The calculation of limit cycles using the describing function (DF) approach for the case of autonomous nonlinear control systems and an event-based simulation method are presented and compared. Finally, a user-friendly, graphically oriented interactive tool based on Sysquake is presented. In this way the students can easily assimilate some concepts of introductory courses in nonlinear control such as the behavior of piecewise linear systems, stable and unstable limit cycles, event-based simulation and the DF method.

Two-parametric unfoldings for planar invisible double-tangency singularities.

We consider the family of planar discontinuous piecewise linear systems in two regions with a straight line as boundary and one tangency point in each region. First, we recall a seven-parametric canonical form for such a family. Next, we select two parameters as bifurcation parameters. One of them represents the distance between the tangency points. It is known that this bifurcation parameter unfolds the called pseudo-Hopf bifurcation. The coefficient that determines the stability of the crossing limit cycle that emerges from this bifurcation mechanism, called the first Lyapunov coefficient, is considered our second bifurcation parameter. Finally, with these two bifurcation parameters, we give a two-parametric unfolding for the invisible fold-fold and focus-fold singularities. The bifurcation diagram for each unfolding consists of two curves of bifurcation points: a curve of pseudo-Hopf bifurcation points and a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin (pB) bifurcation because of the dynamical behavior that is same as the Bautin bifurcation for smooth dynamical systems.

The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line

We study discontinuous piecewise linear differential systems formed by linear centers and/or linear Hamiltonian saddles and separated by a nonregular straight line. There are two classes of limit cycles: the ones that intersect the separation line at two points and the ones that intersect the separation line in four points, named limit cycles of type [Formula: see text] and limit cycles of type [Formula: see text], respectively. We prove that the maximum numbers of limit cycles of types [Formula: see text] and [Formula: see text] are two and one, respectively. We show that all these upper bounds are reached providing explicit examples.

Vibration Suppression Study of Parallel Combined Damping Nonlinear Energy Sink on Piecewise Linear Systems

The Nonlinear Energy Sink (NES) is a passive control device capable of achieving targeted energy transfer and effectively suppressing system vibrations. This paper conducts dynamic modeling and vibration suppression analysis of parallel combined damping nonlinear energy sinks in piecewise linear systems. Initially, the system's amplitude-frequency response curve is obtained using the harmonic balance method, validated by the Runge-Kutta method for accuracy. Subsequently, an analysis is performed on the vibration reduction capabilities of the parallel combined damping nonlinear energy sink and the influence of parameters on its vibration suppression effectiveness. Finally, the study investigates changes in the main system's maximum response amplitude with varying damping under different stiffness conditions and determines optimal parameter values based on trend analysis. Research findings indicate that the vibration reduction performance of the parallel combined damping nonlinear energy sink surpasses that of a single connected damping nonlinear energy sink. Furthermore, after parameter optimization, the main system achieves superior vibration reduction effects.

Limit Cycles of a Planar Piecewise Linear System with an Improper Node

This paper is concerned with the number of limit cycles for a planar piecewise linear (PWL) system with two zones separated by a straight line. Assume that one of the subsystems of the PWL system has an improper node. The number of limit cycles for saddle-improper node type, focus-improper node type and center-improper node type (the focus or the center is a virtual or boundary equilibrium) are studied. First, we introduce displacement functions and study the number of zeros of displacement functions for different types. Then, we give the parameter regions where the exact number of limit cycles is one or two (at least two) for different types.

Global analysis on a continuous planar piecewise linear differential system with three zones

This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincare disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/83/abstr.html

Emergent Behaviors in Coupled Multi-scroll Oscillators in Network with Subnetworks

This paper presents the emergence of two collective behaviors in interconnected networks. Specifically, the nodes in these networks belong to a particular class of piece-wise linear systems. The global topology of the network is designed in the form of connected subnetworks, which do not necessarily share the same structure and coupling strength. In particular, it is considered that there are two levels of connection, the internal level is related to the connection between the nodes of each subnetwork; while the external level is related to connections between subnetworks. In this configuration, the internal level is considered to provide lower bounds on the coupling strength to ensure internal synchronization of subnetworks. The external level has a relevant value in the type of collective behavior that can be achieved, for which, we determine conditions in the coupling scheme, to achieve partial or complete cluster synchronization, preserving the internal synchronization of each cluster. The analysis of the emergence of stable collective behavior is presented by using Lyapunov functions of the different coupling. The theoretical results are validated by numerical simulations.

Phase Portraits of a Class of Continuous Piecewise Linear Differential Systems

AbstractThe phase portraits of the planar linear differential systems are very well known. This is not the case for the phase portraits of the planar continuous piecewise linear differential systems. In this paper we classify the phase portraits of the class of planar continuous piecewise linear differential systems of the form $$\begin{aligned} {\dot{x}}= a|x|+by+c,\qquad {\dot{y}}= \alpha |x|+\beta y+\gamma , \end{aligned}$$ x ˙ = a | x | + b y + c , y ˙ = α | x | + β y + γ , in the Poincaré disc when $$a\beta -b\alpha \ne 0$$ a β - b α ≠ 0 , and prove the existence and uniqueness of limit cycles. Note that on the straight line $$x=0$$ x = 0 these differential systems are only continuous.

Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

Dynamic Investigations of an Endogenous Business Cycle Model with Heterogeneous Agents

AbstractWe introduce agents’ heterogeneity into a model of endogenous business cycles, in which agents can invest either in ‘good’ projects that contribute to future capital formation, or in ‘bad’ projects without that property. The resulting map involves three distinct regimes, two of which we linearize. Using theoretical results on piecewise linear systems and on border collision bifurcations, we are able to provide a thourough analysis of the dynamics.

Bifurcation Analysis of a Planar Piecewise Linear System of Focus–Focus Type

This paper is devoted to discussing the sliding bifurcation phenomena of a planar piecewise linear system with two zones separated by a straight line. We consider those systems for which in each zone there is precisely one real focus. By introducing Poincaré map and defining proper displacement function, some results on the existence of crossing periodic orbits, sliding periodic orbits, sliding-crossing periodic orbits and sliding homoclinic connections are obtained. Moreover, we establish some conditions for the simultaneous occurrence of crossing and sliding periodic orbits and show a concrete example.

Periodic trajectories in planar discontinuous piecewise linear systems with only centers and with a nonregular switching line

In this paper periodic trajectories of dynamical systems presenting discontinuities are studied. The considered model consists of two distinct linear differential systems, each one containing a single equilibrium point of centre type. Each system is defined on disjoint regions of the plane, the separation line is a union of two half-straight lines contained on the coordinate axes. The obtained differential system is non-smooth, so we apply Filippov’s theory to study the transitions from one dynamical system to another. The combination of the two linear plus the Filippov system acting on the separation line generates a nonlinear regime observed by the presence of limit cycles, sliding and tangential periodic trajectories as well as the coexistence of such objects. In theorem we establish the location, stability and hyperbolicity of limit cycles for certain classes of the considered model. In theorem we perform the global analysis of a representative model through bifurcation theory to analyse the birth of limit cycles, sliding periodic trajectories and tangential ones. We also provide some results addressing the coexistence of periodic trajectories and two potential physical interpretations of the model considered in the paper, one addressing nonlinear oscillations and the other considering slow-fast systems of neuron models. The main techniques employed to obtain the results are first integrals, Poincaré half return maps, and elements of bifurcation theory.

Multi-scroll Attractors and Multi-stability Generated Via Fractional-order Switching Linear Systems

This work presents a method for fractional order dynamical systems design that exhibits multi-scrolls by means of a fractional-order switching piecewise linear system. This method is based on the equilibrium points positioning and the region sizes in which the system switches. Numerical results of the proposed method are also presented and the multi-stability phenomenon is exhibited.

On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one

On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one