Piecewise Linear Oscillator Research Articles

Synchronous behavior in directed networks of heterogeneous piecewise linear oscillators

In this paper, we investigate the onset of synchronous behavior on strongly connected directed networks of heterogeneous piecewise linear (PWL) oscillators. Each oscillator is composed of a linear part, which is assumed to be identical for all oscillators, and a commutation function, which is different for each oscillator, creating in this way the heterogeneity in the whole network. Due to this heterogeneity, the oscillators do not achieve perfect or complete synchronization but instead, practical synchronization emerges. The stability properties of the practical synchronous solution in the network are analyzed using standard Lyapunov theory for systems with nonvanishing perturbations and the concept of disagreement vectors. Our results provide an ultimate bound on the norm of the disagreement vector and also a maximal bound on the synchronization error, such that a relationship between the disagreement bound and practical synchronization is established. Additionally, we provide numerical simulations to illustrate the theoretical results.

Model reduction of a periodically forced slow–fast continuous piecewise linear system

AbstractIn this paper, singular perturbation theory is exploited to obtain a reduced-order model of a slow–fast piecewise linear 2-DOF oscillator subjected to harmonic excitation. The nonsmooth nonlinearity of piecewise linear nature is studied in the case of bilinear damping as well as with bilinear stiffness characteristics. We propose a continuous matching of the locally invariant slow manifolds obtained in each subregion of the state space, which yields a reduced-order model of the same nature as the full dynamics. The frequency-response curves obtained from the full system and the reduced-order models suggest that the proposed reduction method can capture nonlinear behaviors such as super- and subharmonic resonances.

Double grazing bifurcation route in a quasiperiodically driven piecewise linear oscillator.

Considering a piecewise linear oscillator with quasiperiodic excitation, we uncover the route of double grazing bifurcation of quasiperiodic torus to strange nonchaotic attractors (i.e., SNAs). The maximum displacement for double grazing bifurcation of the quasiperiodic torus can be obtained analytically. After double grazing of quasiperiodic orbits, the smooth quasiperiodic torus wrinkles increasingly with the continuous change of the parameter. Subsequently, the whole quasiperiodic torus loses the smoothness by becoming everywhere non-differentiable, which indicates the birth of SNAs. The Lyapunov exponent is adopted to verify the nonchaotic property of the SNA. The strange property of SNAs can be characterized by the phase sensitivity, the power spectrum, the singular continuous spectrum, and the fractal structure. Our detailed analysis shows that the SNAs induced by double grazing may exist in a short parameter interval between 1 T quasiperiodic orbit and 2 T quasiperiodic orbit or between 1 T quasiperiodic orbit and 4 T quasiperiodic orbit or between 1 T quasiperiodic orbit and chaotic motion. Noteworthy, SNAs may also exist in a large parameter interval after double grazing, which does not lead to any quasiperiodic or chaotic orbits.

Semi-analytical method to study piecewise linear oscillators

This article proposes a semi-analytical method to investigate the dynamics and bifurcation scenarios of piecewise linear oscillators. The method is based on a mapping technique with a matrix structure that allows easy and rapid construction of any periodic orbit. When validated against direct numerical integration simulations, a good correlation and an accurate prediction of bifurcation phenomena were shown. The method is applied to analyse the nonlinear dynamic responses and bifurcations scenarios causes by changes of stiffness and viscous damping. A set of minimum conditions that the system must meet to present period doubling bifurcations and sub-harmonic orbits was given.

Time-scale synchronisation of oscillatory responses can lead to non-monotonous R-tipping

Rate-induced tipping (R-tipping) describes the fact that, for multistable dynamic systems, an abrupt transition can take place not only because of the forcing magnitude, but also because of the forcing rate. In the present work, we demonstrate through the case study of a piecewise-linear oscillator (PLO), that increasing the rate of forcing can make the system tip in some cases but might also prevent it from tipping in others. This counterintuitive effect is further called non-monotonous R-tipping (NMRT) and has already been observed in recent studies. We show that, in the present case, the reason for NMRT is the peak synchronisation of oscillatory responses operating on different time scales. We further illustrate that NMRT can be observed even in the presence of additive white noise of intermediate amplitude. Finally, NMRT is also observed on a van-der-Pol oscillator with an unstable limit cycle, suggesting that this effect is not limited to systems with a discontinuous right-hand side such as the PLO. This insight might be highly valuable, as the current research on tipping elements is shifting from an equilibrium to a dynamic perspective while using models of increasing complexity, in which NMRT might be observed but hard to understand.

Dynamics of excited piecewise linear oscillators

Dynamics of excited piecewise linear oscillators

Birth of strange nonchaotic attractors in a piecewise linear oscillator.

Nonsmooth systems are widely encountered in engineering fields. They have abundant dynamical phenomena, including some results on the complex dynamics in such systems under quasiperiodically forced excitations. In this work, we consider a quasiperiodically forced piecewise linear oscillator and show that strange nonchaotic attractors (SNAs) do exist in such nonsmooth systems. The generation and evolution mechanisms of SNAs are discussed. The torus-doubling, fractal, bubbling, and intermittency routes to SNAs are identified. The strange properties of SNAs are characterized with the aid of the phase sensitivity function, singular continuous spectrum, rational frequency approximation, and the path of the partial Fourier sum of state variables in a complex plane. The nonchaotic properties of SNAs are verified by the methods of maximum Lyapunov exponent and power spectrum.

Recurrent Motions in a Piecewise Linear Oscillator

We study the oscillator ddot{x} + n^2 x + h(x) = p(t), where h is a piecewise linear saturation function and p is a continuous 2pi -periodic forcing. It is shown that there is recurrence if and only if p satisfies the Lazer–Leach condition. This condition relates the n-th Fourier coefficient of p(t) with the maximum of h and was first introduced to characterize the existence of periodic solutions.

A 43 nW, 32 kHz, ±4.2 ppm Piecewise Linear Temperature-Compensated Crystal Oscillator With ΔΣ-Modulated Load Capacitance

This article describes an ultralow-power (ULP) temperature-compensated crystal oscillator (TCXO) with a pulsed-injection XO driver for IoT applications. Temperature compensation is achieved by changing the load capacitance ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${C}_{L}$ </tex-math></inline-formula> ) between two values using a delta-sigma modulator ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta \Sigma \text{M}$ </tex-math></inline-formula> ). The complex modulation profile across temperature is approximated as piecewise linear elements that is selected by a coarse temperature sensor. As a result, the power and area of fine-grain look-up tables (LUTs) or a polynomial engine used in prior works can be avoided. The proposed pulsed-injection XO driver that directly replenishes the energy of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${C}_{L}$ </tex-math></inline-formula> sustains the XO oscillation for the two different <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${C}_{L}$ </tex-math></inline-formula> states. Implemented in 40-nm CMOS, the proposed 32.768-kHz TCXO achieves an accuracy of ±4.2 ppm from −20 °C to 85 °C with just three-point trimming and an Allan deviation floor of 34 ppb while consuming 43 nW, which is an approximate 8 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> improvement over the recent state-of-the-art TCXOs.

Computation of periodic orbits for piecewise linear oscillator by Harmonic Balance Methods

In this paper, Harmonic Balance based methods, namely Incremental Harmonic Balance Method and the method of Harmonic Balance with Alternating Frequency and Time traditionally used to compute periodic orbits of smooth nonlinear dynamical systems, are employed to investigate the dynamics of a non-smooth system, specifically a piecewise linear oscillator with a play. The Incremental Harmonic Balance Method was used to compute the period one orbits, including those exhibiting grazing and large impacts. The method of Harmonic Balance with Alternating Frequency and Time was implemented to calculate more complex orbits and multi stability. A good agreement between obtained approximate solutions and numerically calculated responses indicates robustness of the implemented HBMs, which should allow to effectively study the global dynamics of non-smooth systems.

Harmonic solutions for a class of non-autonomous piecewise linear oscillators

The study deals with the existence and uniqueness conditions of harmonic solutions of a class of piecewise linear oscillators with a general periodic excitation. Based on the relationship between natural frequency and driving frequency, we divide the discussion of the harmonic solutions into resonant case and non-resonant case. For resonant case, the existence conditions dependent on periodic excitation and clearance of harmonic solutions are given based on the Poincaré-Bohl fixed point theorem. Moreover, we give a necessary and sufficient condition of bounded solutions, where harmonic solutions, n-subharmonic solutions, and quasi-periodic solutions can occur simultaneously. For non-resonant case, the existence of harmonic solutions are analyzed based on contract mapping. The uniqueness of harmonic solutions for resonant and non-resonant cases are proven by variational method and reduction to absurdity. Numerical examples are presented to illustrate the theoretical results.

Method for controlling vibration by exploiting piecewise-linear nonlinearity in energy harvesters.

Vibration energy is becoming a significant alternative solution for energy generation. Recently, a great deal of research has been conducted on how to harvest energy from vibration sources ranging from ocean waves to human motion to microsystems. In this paper, a theoretical model of a piecewise-linear (PWL) nonlinear vibration harvester that has potential applications in variety of fields is proposed and numerically investigated. This new technique enables automatic frequency tunability in the energy harvester by controlling the gap size in the PWL oscillator so that it is able to adapt to changes in excitations. To optimize the performance of the proposed system, a control method combining the response prediction, signal measurement and gap adjustment mechanism is proposed in this paper. This new energy harvester not only overcomes the limitation of traditional linear energy harvesters that can only provide the maximum power generation efficiency over a narrow frequency range but also improves the performance of current nonlinear energy harvesters that are not as efficient as linear energy harvesters at resonance. The proposed system is demonstrated in several case studies to illustrate its effectiveness for a number of different excitations.

Distribution tail structure and extreme value analysis of constrained piecewise linear oscillators

A single-degree-of-freedom random oscillator with a piecewise linear restoring force (experiencing softening after a certain point value of the response, called a “knuckle” point) is studied with the goal of understanding the structure of the distribution tail of its response or (local) maximum. A theoretical analysis is carried out by two approaches: first, by focusing on the maximum and response after crossing the “knuckle” point, where explicit calculations can be performed assuming standard distributions for the derivative at the crossing, and second, by considering the white noise random external excitation, where the stationary distribution of the response is readily available from the literature. Both approaches reveal the structure of the distribution tails where a Gaussian core is followed by a heavier tail, possibly having a power-law form, which ultimately turns into a tail having a finite upper bound (referred to as a light tail). The extent of the light tail region is also investigated, and shown to be the result of the conditioning for the system not to reach the unstable equilibrium. The study is motivated by applications to ship motions, where the considered random oscillator serves as a prototypical model for ship roll motion in beam seas, and estimation of the probabilities of these motions exceeding large critical angles is of interest. Standard statistical methodology for such estimation is based on the peaks-over-threshold approach, for which several lessons are drawn from the analysis of the tail structure of the considered random oscillator.

Forced Response Analysis of 2-DOF Piecewise-Linear Oscillator

Forced Response Analysis of 2-DOF Piecewise-Linear Oscillator

A multiple focus-center-cycle bifurcation in 4D discontinuous piecewise linear memristor oscillators

The dynamical richness of 4D memristor oscillators has been recently studied in several works, showing different regimes, from stable oscillations to chaos. Typically, only numerical simulations have been reported and so there is a lack of mathematical results. We focus our analysis in the existence of multiple stable oscillations in the 4D piecewise linear version of the canonical circuit proposed by Itoh and Chua (Int J Bifurc Chaos 18(11):3183–3206, 2008). This oscillator is modeled by a discontinuous piecewise linear dynamical system. By adding one parameter that stratifies the 4D dynamics, it is shown that the dynamics in each stratum is topologically equivalent to a 3D continuous piecewise linear dynamical system. Some previous results on bifurcations in such reduced system allow to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits.

Nonlinear Resonances of Chains of Thin Elastic Beams With Intermittent Contact

This paper deals with the forced response analysis of chains of thin elastic beams that are subject to periodic external loading and frictionless intermittent contact between the beams. Our study shows that the beams show nonlinear resonances whose frequencies are the same as the linear resonant frequencies if all the beams have the same stiffness. Furthermore, it is also shown that small gaps between the beams and small deviation or mistuning in the stiffness of each beam can cause drastic changes in the nonlinear resonant frequencies of the system dynamics. The system is modeled as a semidiscrete system of piecewise-linear oscillators with multiple degrees-of-freedom (DOF) that are subject to unilateral constraints, which is derived from a finite element discretization of the beams. The resulting equations of motions are solved by a second-order numerical integration scheme, and steady-state solutions are sought for various driving frequencies. Results of parametric studies with respect to the gaps between the beams and the number of beams are presented to discuss how these parameters affect the resonant behavior of the system.

The analytical method of studying subharmonic periodic orbits for planar piecewise-smooth systems with two switching manifolds

Piecewise linear approach is one of important methods to study nonlinear dynamics of oscillators with complex nonlinear restoring forces, so piecewise-smooth systems with multiple switching manifolds sometimes are ideal mathematical models to analyze the nonlinear dynamics of nonlinear oscillators. In this paper, inspired by the work presented by Cao et al. (Philos Trans R Soc A 366(1865):635–652, 2008) and Granados et al. (SIAM J Appl Dyn Syst 11(3):801–830, 2012), we want to formulate an analytical method for studying subharmonic periodic orbits for piecewise-smooth systems with two switching manifolds. We will define what are subharmonic orbits for this class of piecewise-smooth systems and develop a Melnikov-type analytical method to detect the existence of subharmonic orbits. In order to obtain this objective, we assume that the plane is divided into three zones by the two switching manifolds which may not be symmetric, and the dynamics in each zone is decided by a smooth system. Furthermore, we suppose that the unperturbed system is a piecewise-defined Hamiltonian system and possesses a pair of piecewise-smooth homoclinic orbits crossing the two switching manifolds transversally exactly twice and connecting origin to itself. The region outside the pair of homoclinic orbits is fully covered by periodic orbits crossing respectively the two switching manifolds transversally exactly twice. Finally, we want to study the persistence of those continuum of periodic orbits outside the pair of homoclinic orbits when a small time-periodic perturbation is considered. By choosing an appropriate Poincare section, constructing a Poincare map and applying perturbation techniques, we obtain the Melnikov function of subharmonic orbits for the class of planar piecewise-smooth systems and employ it to study the existence of subharmonic periodic motions for a piecewise-linear oscillator. Numerical simulations are also shown the effectiveness of the analytical method to detect the parameters and initial conditions for the existence of subharmonic orbits in piecewise-smooth oscillators.

Clusters in nonsmooth oscillator networks.

For coupled oscillator networks with Laplacian coupling, the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory, this approach has recently been extended to treat more general cluster states. However, the MSF and its generalizations require the determination of a set of Floquet multipliers from variational equations obtained by linearization around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by, the framework is often explored using numerical techniques. Here, we show that further insight into network dynamics can be obtained by focusing on piecewise linear (PWL) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of saltation matrices. By augmenting the variational approach for studying smooth dynamical systems with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration, we analyze an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled network built from planar PWL nodes, including a reduction of the popular Morris-Lecar neuron model. We use these examples to emphasize that the stability of network cluster states can depend as much on the choice of single node dynamics as it does on the form of network structural connectivity. Importantly, the procedure that we present here, for understanding cluster synchronization in networks, is valid for a wide variety of systems in biology, physics, and engineering that can be described by PWL oscillators.

On Discontinuous Piecewise Linear Models for Memristor Oscillators

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh &amp; Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello &amp; Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold

In this paper, we extend the classical Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth system subjected to a time-periodic perturbation. In this class, we suppose there exists a switching manifold with a more general form such that the plane is divided into two zones, and the dynamics in each zone is governed by a smooth system. Furthermore, we assume that the unperturbed system is a general planar piecewise-smooth system with non-zero trace and possesses a piecewise-smooth homoclinic orbit transversally crossing the switching manifold. We also define a reset map to describe the instantaneous impact rule on the switching manifold when a trajectory arrives at the switching manifold. Through a series of geometrical analysis and perturbation techniques, we obtain a Melnikov-type function to measure the separation of the unstable manifold and stable manifold under the effect of the time-periodic perturbations and the reset map. Finally, we use the presented Melnikov function to study global bifurcations and chaotic dynamics for a concrete planar piecewise-linear oscillator.