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In this paper, we present an example of piecewise linear systems with infinitely many crossing limit cycles defined in two zones separated by a piecewise linear curve with countable corners. Then we prove that under piecewise linear perturbations, the perturbed system can have infinitely many limit cycles, or exactly l limit cycles for any given nonnegative integer l. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin (0 , 0) , which are the positive x - and y -axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The bifurcations of periodic solutions and the generation of chaos in a ring of three unidirectionally coupled linear elements with a single reverse coupling through a piecewise linear function are considered. A discontinuous and a continuous piecewise linear function are employed for the reverse coupling. A chaotic attractor is generated immediately through a Hopf-like boundary equilibrium bifurcation of a focus in both cases. A chaotic attractor is also generated directly through a grazing bifurcation in the case of the discontinuous function, which is replaced with a cascade of period-doubling bifurcations in the case of the continuous function. A chaotic oscillation with the same form is also observed in an experiment on an analog circuit constructed with operational amplifiers. In a smooth version of the system, a ring of three unidirectionally coupled sigmoid neurons with a reverse coupling, the Hopf-like boundary equilibrium bifurcation is replaced with a period-doubling cascade following after the Hopf bifurcation. [ABSTRACT FROM AUTHOR]

Bifurcations and chaos of a three-dimensional single-loop feedback system with a discontinuous piecewise linear feedback function are examined. Chaotic attractors are generated at the same time of the destabilization of foci accompanied with grazing. Multiple periodic solutions are connected with homoclinic orbits based at a pseudo saddle-focus, which satisfies the condition of Shil'nikov chaos formally. The generation of chaotic oscillations is shown in a circuit experiment on a linear ring oscillator with a comparator. The homoclinic bifurcations and chaos are also shown in a ring neural network with a nonmonotonic neuron. [ABSTRACT FROM AUTHOR]

The bifurcations and chaos of a ring of three unidirectionally coupled neuron-like elements are examined as a minimal chaotic neural network. The output function of one neuron is nonmonotonic and piecewise constant while those of the other two neurons are linear. Two kinds of nonmonotonic output functions are considered and it is shown that periodic solutions undergo grazing bifurcations owing to discontinuity in the nonmonotonic functions. Chaotic attractors are created directly through a grazing bifurcation and homoclinic orbits based at pseudo steady states are generated. It is shown that homoclinic/heteroclinic orbits satisfying the condition of Shil'nikov chaos are caused by overshoot in the nonmonotonic functions. [ABSTRACT FROM AUTHOR]

Chaotic oscillations induced by single rectification in networks of linear neuron-like elements are examined on three prototype models: one nonautonomous system and two autonomous systems. The first is a system of coupled neurons with periodic input; the second is a system of three coupled neurons with six couplings; the third is a ring of four unidirectionally coupled neurons with one reverse coupling. In each system, the output function of one neuron is ramp and that of the others is linear. Each system is piecewise linear and the phase space is separated into two domains by a single border. Steady states, periodic solutions and homoclinic orbits are derived rigorously and their stability is evaluated with the eigenvalues of the Jacobian matrices. The bifurcation analysis of the three systems shows that chaotic attractors could be generated through cascades of period-doubling bifurcations of periodic solutions. [ABSTRACT FROM AUTHOR]

For a class of three-dimensional piecewise linear systems with an admissible saddle-focus, the existence of three kinds of homoclinic loops is shown. Moreover, the birth and number of the periodic orbits induced by homoclinic bifurcation are investigated, and various sufficient conditions are obtained to guarantee the appearance of only one periodic orbit, finitely many periodic orbits or countably infinitely many periodic orbits. Furthermore, the stability of these newborn periodic orbits is analyzed in detail and some conclusions are made about them to be periodic saddle orbits or periodic sinks. Finally, some examples are given. [ABSTRACT FROM AUTHOR]

Energy harvesters based on a piecewise linear system make it possible to extract more energy from environment vibrations. The amplitude truncation effect is a phenomenon of an energy harvester with a piecewise linear system. It shows a wider energy harvesting bandwidth and a lower voltage response than the purely linear counterpart. This work investigates the influence of spring length on the amplitude truncation effect and power density of electromagnetic energy harvesters. We first establish an analytical model of the piecewise linear system. Then, we apply the averaging method to explore the voltage responses in the frequency domain and compare the influence of various parameters. Later, we conduct a finite element simulation to analyze the voltage response at a constant frequency. The results obtained by the finite element simulation and averaging method match well with the experimental measurements. The experimental results demonstrate that the amplitude truncation effect is more visible at the shorter spring. Spring length plays a key role in the power density of an electromagnetic energy harvester. The case of 40 mm yields a maximum average power of 45.13 mW. The case of 30 mm displays a maximum power density of 2.35 mW/cm 3.

This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first provide a negative answer via a counterexample to a conjecture on the global and finite convergence of the Newton iteration for symmetric and positive definite matrices. Additionally, we discuss some surprising features of the semi-smooth Newton iteration in low dimensions and its behavior in higher dimensions. Secondly, we present two iterative schemes inspired by the classical Jacobi and Gauss-Seidel methods for linear systems of equations for finding a solution to the problem. We study sufficient conditions for the convergence of both proposed procedures, which are also sufficient for the existence and uniqueness of solutions to the problem. Lastly, we perform some computational experiments designed to illustrate the behavior (in terms of CPU time) of the proposed iterations versus the semi-smooth Newton method for dense and sparse large-scale problems. Moreover, we included the numerical solution of a discretization of the Boussinesq PDE modeling a two-dimensional flow in a homogeneous phreatic aquifer. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary of the attraction basin of a class of piecewise linear systems arising from anti-stable linear systems with saturated linear state feedback. In three-dimensional cases, for this class of systems, we prove that the equilibrium points other than the origin lie on the boundary of the attraction basin of the origin. This gives strong evidence that the boundary of the attraction basin is homeomorphic to a sphere. Some examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we show that any switching hypersurface of n -dimensional continuous piecewise linear systems is an (n − 1) -dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The existence of homoclinic orbits is discussed analytically for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, that is, under the same parameter setting, homoclinic orbits and chaos appear simultaneously in the system. In addition, homoclinic chaos can be suppressed to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

Let A be a real n × n matrix and z , b ∈ R n . The piecewise linear equation system z - A | z | = b is called an absolute value equation. In this note we consider two solvers for uniquely solvable instances of the latter problem, one direct, one semi-iterative. We slightly extend the existing correctness, resp. convergence, results for the latter algorithms and provide numerical tests. [ABSTRACT FROM AUTHOR]

Consider a piecewise linear dynamical system in ℝ 3 , divided by a switching plane, with the Teixeira Singularity (TS). Under the antiparallelism hypothesis, it is known that the system undergoes the so-called pseudo-Hopf bifurcation (pH). In this paper, we consider a particular position of the antiparallel vectors to yield a two-parametric unfolding of the TS. The bifurcation analysis gives us, besides the curve of pH bifurcation points, a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin bifurcation, since it exhibits the exact same behavior as the Bautin bifurcation for smooth dynamical systems. [ABSTRACT FROM AUTHOR]

First, this paper provides a new proof for the expression of the generalized first order Melnikov function on piecewise smooth differential systems with multiply straight lines. Then, by using the Melnikov function, we consider the limit cycle bifurcation problem of a 3-piecewise near Hamiltonian system with two switching lines, obtaining 2n + 3[n+1/2] limit cycles near the double generalized homoclinic loop. [ABSTRACT FROM AUTHOR]

The study of strange nonchaotic attractors (SNAs) has been mainly restricted to quasiperiodically forced systems. At present, SNAs have also been uncovered in several periodically forced smooth systems with noise. In this work, we consider a periodically forced nonsmooth system and find that SNAs are created by a small amount of noise. SNAs can be generated in different periodic windows with weak noise perturbation. If the parameter is varied further from the chaotic range, a larger noise intensity is required to induce SNAs. Besides, noise-induced SNAs can be generated by the periodic attractors near the boundary crisis. In addition, with the increasing noise intensity, the intermittency between SNAs and periodic attractors can be induced by transient chaos. The characteristics of SNAs are analyzed by the Lyapunov exponent, power spectrum, singular continuous spectrum, spectral distribution functions, and finite time Lyapunov exponent. [ABSTRACT FROM AUTHOR]

The forced vibration of a single-degree-of-freedom piecewise linear system containing fractional time-delay feedback was investigated. The approximate analytical solution of the system was obtained by employing an averaging method. A frequency response equation containing time delay was obtained by studying a steady-state solution. The stability conditions of the steady-state solution, the amplitude–frequency results, and the numerical solutions of the system under different time-delay parameters were compared. Comparison results indicated a favorable goodness of fit between the two parameters and revealed the correctness of the analytical solution. The effects of the time-delay and fractional parameters, piecewise stiffness, and piecewise gap on the principal resonance and bifurcation of the system were emphasized. Results showed that fractional time delay occurring in the form of equivalent linear dampness and stiffness under periodic variations in the system and influenced the vibration characteristic of the system. Moreover, piecewise stiffness and gap induced the nonlinear characteristic of the system under certain parameters. [ABSTRACT FROM AUTHOR]

We reconsider the multiplier–accelerator model of business cycles, first introduced by Samuelson and then modified by many authors. The original simple model, besides damped oscillations, also leads to divergent oscillations. To avoid this, we introduce two different types of governmental expenditures leading a two-dimensional continuous piecewise linear map that can generate sustained oscillations (attracting cycles). The map is defined by three different linear functions in three different partitions of the phase plane, and this peculiarity influences the overall dynamics of the system. We show that, similar to the classical Samuelson model, there is a unique feasible equilibrium as well as converging oscillations. However, close to the center bifurcation value the attracting equilibrium coexists with attracting cycles of different periods, which lose stability via a center bifurcation simultaneously with the equilibrium. Moreover, we show that attracting cycles of particular type also exist when the equilibrium becomes an unstable focus. For several families of attracting cycles, by introducing the symbolic representation, we obtain boundaries of the related periodicity regions, associated with border collision bifurcations. [ABSTRACT FROM AUTHOR]

In this article, we study a rich and complex phenomena of planar piecewise linear systems having two domains of which the separation curves are not straight. We show that for positive integers n and m and non-negative integers $n_1,n_2,\ldots,n_m$, there exist two types of piecewise linear systems: one has a period annulus possessing exactly n critical periods; the other has m period annuli possessing exactly $n_1,n_2,\dots,n_m$ critical periods. Moreover, an algebraic curve can be chosen as the separation line in system of the first type.

In (Euzébio et al., 2016 [10] ; Chen and Tang, 2020 [8]), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F (x) − y , y ˙ = g (x) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincaré disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve. [ABSTRACT FROM AUTHOR]

In this article, a novel mass and momentum conservative semi‐implicit method is presented for the numerical solution of the incompressible free‐surface Navier–Stokes equations. This method can be seen as an extension of the semi‐implicit mass‐conservative scheme presented by Casulli. The domain is covered by the fluid, by potential solid obstacles, and by the surrounding void via a scalar volume fraction function for each phase, according to the so‐called diffuse interface approach. The semi‐implicit finite volume discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure. The nonlinearity on the diagonal of the system stems from the nonlinear definition of the volume, while the remaining linear part of the pressure system is symmetric and at least positive semi‐definite. Hence, the pressure can be efficiently obtained with the family of nested Newton‐type techniques recently introduced and analyzed by Brugnano and Casulli. The time step size is only limited by the flow speed and eventually by the velocity of moving rigid obstacles contained in the computational domain, and not by the gravity wave speed. Therefore, the method is efficient also for low Froude number flows. Moreover the scheme is formulated to be locally and globally conservative: for this reason it fits well in the presence of shock waves, too. In the special case of only one grid cell in vertical direction, the proposed scheme automatically reduces to a mass and momentum conservative discretization of the shallow water equations. The proposed method is first validated against the exact solution of a set of one‐dimensional Riemann problems for inviscid flows. Then, some computational results are shown for non‐hydrostatic flow problems and for a simple fluid‐structure interaction problem. [ABSTRACT FROM AUTHOR]

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In this paper, we consider a perturbation of planar general piecewise Hamiltonian systems with nonregular separation line. A general explicit expression of the first and second order Melnikov functions is presented for such piecewise Hamiltonian systems. In addition, the maximum number of limit cycles for piecewise linear differential system is an important topic that many researchers are concerned with. Different from considering a piecewise linear perturbation of a linear center in the previous research, we apply our explicit expression to estimate the number of limit cycles bifurcated from a class of piecewise linear Hamiltonian systems. More specifically, the unperturbed system we study is piecewise smooth and constructed by a linear center and a constant differential system, which is difficult to be considered by a polar coordinate transformation. And we obtain the upper bounds to be 4 and 5 for the limit cycles bifurcated from the periodic orbits by using the first and second order Melnikov functions, respectively. Moreover the upper bounds are sharp. [ABSTRACT FROM AUTHOR]

Recurrent neural networks use sequential or time series data. These algorithms can be used in temporal or ordinal tasks, such as speech recognition, image captioning, and forecasting, among many others. This study introduces a novel approach to the real-time identification of chaotic trajectories using a Recurrent Wavelet First-Order Artificial Neural Network (RWFONN) in conjunction with a Field-Programmable Analog Array (FPAA). The FPAA physically implements the chaotic dynamics of a third-order Jerky equation system with reconfigurable analog electronic components through its voltage output. The trajectory acquisition of the voltage signals is carried out by a dSPACE 1104 board connected to MATLAB & Simulink and Control Desk, which serves as real-time identification data for the RWFONN. The first-order artificial network implements a Morlet wavelet activation function for the online training. It has the particularity to work without the need for multiple epoch training, validation, or dividing the data into batches. Instead, it validates the process by the filtered and mean square errors, affirming the RWFONN's precision in identifying and reproducing the chaotic behavior of the FPAA-modeled system. This breakthrough in real-time identification holds promise for applications demanding dynamic and unpredictable behaviors, such as random number generators, neural controllers, and cryptographic systems. [Display omitted] • Real-time implementation of analog chaotic trajectories by a RWFONN. • The continuous-time system is implemented in an FPAA. • An analysis of error filtering of the trajectories and neural training by Morlet wavelet activation function. • Identification of the RWFONN output and the states of the analog circuit presents a low mean square error. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative is studied. First, a mathematical model of the single-degree-of-freedom system is established, and the approximate steady-state solution associated with the amplitude–frequency equation is obtained based on the averaging method. Then, the amplitude–frequency response equations are used for stability analysis, and the stability condition is founded. To validate the correctness and precision, the approximate solutions determined by the analytical method are compared with the solutions based on the numerical integration method. It is found that the approximate solutions and the numerical solutions are in excellent agreement. Finally, the effects of system parameters, such as fractional-order coefficient, order, clearance, and piecewise stiffness, on the complex dynamical behaviors of the piecewise linear single-degree-of-freedom oscillator are studied. The results show that the system parameters not only influence resonance amplitude and resonance frequency but also affect the size of the unstable region. [ABSTRACT FROM AUTHOR]

The dynamics of a two-degrees-of-freedom (pitch–plunge) aeroelastic system is investigated. The aerodynamic force is modeled as a piecewise linear function of the effective angle of attack. Conditions for admissible (existing) and virtual equilibria are determined. The stability and bifurcations of equilibria are analyzed. We find saddle-node, border collision and rapid bifurcations. The analysis shows that the pitch–plunge model with a simple piecewise linear approximation of the aerodynamic force can reproduce the transition from divergence to the complex aeroelastic phenomenon of stall flutter. A linear tuned vibration absorber is applied to increase stall flutter wind speed and eliminate limit cycle oscillations. The effect of the absorber parameters on the stability of equilibria is investigated using the Liénard–Chipart criterion. We find that with the vibration absorber the onset of the rapid bifurcation can be shifted to higher wind speed or the oscillations can be eliminated altogether. [ABSTRACT FROM AUTHOR]

A novel semi-active vibration control concept with a serial-switch-stiffness-system was previously presented in our work. Differing from conventional vibration control systems, this system does not dissipate but converts vibration energy as potential energy stored in springs and then reacts against external disturbance. As a piecewise linear system, whether or not energy conversion limit happens is an interesting nonlinear dynamic issue related to the systems steady state response. This paper formulates this issue in depth using the approach called equivalence in control. The systems control force represented by the converted vibration energy is approximately decomposed into two portions. One is responsible for low-frequency free response and the other for high-frequency switching response. An equivalent linear system suffering from a decomposed high-frequency switching force is obtained instead of the original switched system. The steady state response of the disturbed system can be delivered through linear superposition as executed in a linear system. Energy conversion limit occurring in the system under a harmonic disturbance is numerically shown by means of fast Fourier transformation. Analytical formulation and numerical simulation for open- and closed-loop control of the system are further carried out, respectively. The results give that the proposed approach is capable of solving the stead state response of the switched system accurately, and meanwhile, energy conversion limit occurs in the vibration control system indeed. Experimental discussion is also executed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of finite number of Jordan curves can be realized by a discontinuous piecewise linear system with two zones separated by a continuous curve. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type "focus‐focus" and "focus‐center." Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case "focus‐center," we use a pure geometric approach. In the case "focus‐focus," we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two‐dimensional system describing the qualitative behavior of a two‐dimensional excitable membrane model. [ABSTRACT FROM AUTHOR]

In this paper, we consider planar piecewise linear differential systems with a line of discontinuity sharing a linear part. We study not only the number of crossing limit cycles, but also the number of sliding ones, and the coexistence of two configurations of limit cycles. In particular, we proved that both numbers of crossing limit cycles and sliding ones are at most 2 , but the total number of limit cycles is at most 3. Finally, by complete analysis on the number of limit cycles, we show some bifurcations which exist in generic Filippov systems, revealing also two nongeneric bifurcations. [ABSTRACT FROM AUTHOR]

Abstract We study a planar piecewise linear differential system with three zones and asymmetry x ̇ = F (x) − y , y ̇ = g (x) , where x g (x) > 0 for x ≠ 0 and the graph of F (x) is a U-shaped curve. The system was researched in Llibre et al. (2015) for its limit cycles. However, the number and hyperbolicity of limit cycles were still unclear in some parameter regions. We give the exact number of limit cycles and obtain the hyperbolicity of limit cycles in these parameter regions, where the maximum number is two. Highlights • The exact number of limit cycles of a piecewise linear system with three zones and asymmetry. • The hyperbolicity of limit cycles of the piecewise linear system. • The development and generalization of Coppel's method in the piecewise linear system. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic behaviour of a family of piecewise linear structures, namely the vibration of beams on block-and-tackle suspension system is analysed. The regularity of the vibration modes in one of the linear states induces non-harmonic, yet periodic free vibration modes. The periodicity constraint of the continuous structure is formulated using modal analysis in the regular state. The required number of modes in the finite modal analysis is specified so that the numerical damping caused by the omitted modes does not change the periodic or non-periodic nature of the free vibration of the continuous structure. It is shown, that the application of five excess passive modes allows to draw conclusions about the behaviour of the continuous structure. The periodic behaviour depends on the number and position of the suspension points and the number of the active vibration modes. Analysis of the limits of the periodic behaviour reveals that suspension points close to the middle of the beam, or first few active vibration modes result in periodic vibration of the nonlinear system. [ABSTRACT FROM AUTHOR]

We continue the study of the structural stability and the bifurcations of planar continuous bimodal linear dynamical systems (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we complete the study when one of the subsystems is a saddle, leading to a 3D bifurcation diagram where a large catalogue of bifurcations appears: four surfaces of codimension-1 bifurcations; two sequences of surfaces of additional codimension-1 bifurcations; two lines of codimension-2 bifurcations; and one codimension-3 bifurcation. [ABSTRACT FROM AUTHOR]