Class Of Nonlinear Oscillators Research Articles

On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

Multiple timescale effects can be reflected bursting oscillations in many classical nonlinear oscillators. In this work, we are concerned about the bursting oscillations induced by two timescale effects in the damped Helmholtz-Rayleigh-Duffing oscillator (written as DHRDO for short) excited by slow-changing parametrical and external forcings. By using trigonometric function variation and authenticating the slow excitations as a slowly varying state variable, the time-varying DHRDO can be rewritten as a new time-invariant system. Then, the critical conditions of some typical bifurcations are presented by bifurcation theory. With the help of bifurcation analyses, six bursting patterns, i.e., ‘Hopf/Hopf-Hopf/Hopf’ bursting, ‘fold/Homoclinic-Hopf/Hopf’ bursting, ‘fold/Homoclinic/Hopf’ bursting, ‘Hopf/fold/Homoclinic/Hopf’ bursting, ‘Hopf/Homoclinic/Homoclinic/Hopf’ bursting and ‘Hopf/Homoclinic/Hopf-Hopf/Homoclinic/Hopf’ bursting, are explored by the slow/fast decomposition method and the other techniques. Our findings provide different forms of the excited state oscillation modes as well as the bursting patterns. In addition, we use the numerical simulation to prove the correctness of the theoretical analyses.

Quantum quasi-Lie systems: properties and applications

A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a t-dependent vector field that is equivalent to a t-dependent family of vector fields within a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised in the literature to deal with t-dependent Schrödinger equations determined by a particular class of t-dependent Hamiltonian operators, the quantum Lie systems, and other systems of differential equations through the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and quantum Lie systems to cope with t-dependent Schrödinger equations associated with the here-called quantum quasi-Lie systems. To illustrate our methods, we propose and study a quantum analogue of the classical nonlinear oscillator searched by Perelomov, and we analyse a quantum one-dimensional fluid in a trapping potential along with quantum t-dependent Smorodinsky–Winternitz oscillators.

Quantum asymptotic phases reveal signatures of quantum synchronization

Synchronization of quantum nonlinear oscillators has attracted much attention recently. To characterize the quantum oscillatory dynamics, we recently proposed a fully quantum-mechanical definition of the asymptotic phase, which is a key quantity in the synchronization analysis of classical nonlinear oscillators (Kato and Nakao 2022 Chaos 32 063133). In this work, we further extend this theory and introduce multiple asymptotic phases using the eigenoperators of the adjoint Liouville superoperator of the quantum nonlinear oscillator associated with different fundamental frequencies. We analyze a quantum van der Pol oscillator with Kerr effect in the strong quantum regime and show that the system has several different fundamental frequencies. By introducing order parameters and power spectra in terms of the associated quantum asymptotic phases, we reveal that phase locking of the system with a harmonic drive at several different frequencies, an explicit quantum signature observed only in the strong quantum regime, can be interpreted as synchronization on a torus rather than a simple limit cycle.

A class of nonlinear oscillators with non-autonomous first integrals and algebraic limit cycles

In this paper, we present a class of autonomous nonlinear oscillators with non-autonomous first integral. We prove explicitly the existence of a global sink which is, under some conditions, an algebraic limit cycle. For that class, we draw the possible phase portraits in the Poincaré disk.

Nonstationary stochastic response determination of nonlinear oscillators endowed with fractional derivatives

Fractional calculus has been broadly used in diverse engineering applications. In this regard, vibrations of fractional oscillators subject to stochastic loads have attracted considerable attention. This paper proposes a semi-analytical approach to determine the nonstationary response statistics of nonlinear oscillators endowed with fractional derivatives. Specifically, the fractional derivative term is represented by introducing a discretization scheme and associated first-order differential equations. This leads to an augmented dimension dynamic system. Further, in conjunction with the statistical linearization technique, the evolution of the system response is captured by a set of coupled ordinary differential equations with time-dependent coefficients. Furthermore, solving the associated Lyapunov equation for the randomly excited dynamic system yields the nonstationary statistics of the oscillator response. The reliability of the proposed method is demonstrated by Monte Carlo simulations pertaining to classical nonlinear oscillators.

Counterdiabatic driving in the classical β-Fermi-Pasta-Ulam-Tsingou chain.

Shortcuts to adiabaticity (STAs) have been used to make rapid changes to a system while eliminating or minimizing excitations in the system's state. In quantum systems, these shortcuts allow us to minimize inefficiencies and heating in experiments and quantum computing protocols, but the theory of STAs can also be generalized to classical systems. We focus on one such STA, approximate counterdiabatic (ACD) driving, and numerically compare its performance in two classical systems: a quartic anharmonic oscillator and the β Fermi-Pasta-Ulam-Tsingou lattice. In particular, we modify an existing variational technique to optimize the approximate driving and then develop classical figuresof merit to quantify the performance of the driving. We find that relatively simple forms for the ACD driving can dramatically suppress excitations regardless of system size. ACD driving in classical nonlinear oscillators could have many applications, from minimizing heating in bosonic gases to finding optimal local dressing protocols in interacting field theories.

Scrambling and quantum feedback in a nanomechanical system

The question of how swiftly entanglement spreads over a system has attracted vital interest. In this regard, the out-of-time ordered correlator (OTOC) is a quantitative measure of the entanglement spreading process. Particular interest concerns the propagation of quantum correlations in the lattice systems, {\it e.g.}, spin chains. In a seminal paper D. A. Roberts, D. Stanford and L. Susskind, J. High Energy Phys. 03, 051, (2015) the concept of the OTOC's radius was introduced. The radius of the OTOC defines the front line reached by the spread of entanglement. Beyond this radius operators commute. In the present work, we propose a model of two nanomechanical systems coupled with two Nitrogen-vacancy (NV) center spins. Oscillators are coupled to each other directly while NV spins are not. Therefore, the correlation between the NV spins may arise only through the quantum feedback exerted from the first NV spin to the first oscillator and transferred from the first oscillator to the second oscillator via the direct coupling. Thus nonzero OTOC between NV spins quantifies the strength of the quantum feedback. We show that NV spins cannot exert quantum feedback on classical nonlinear oscillators. We also discuss the inherently quantum case with a linear quantum harmonic oscillator indirectly coupling the two spins and verify that in the classical limit of the oscillator, the OTOC vanishes.

Oscillatory Circuits With a Real Non-Volatile Stanford Memristor Model

Stanford memristor model is a widely used model that accurately characterizes real non-volatile metal-oxide resistive random access memory (RRAM) devices with bipolar switching characteristics. The paper studies for the first time the dynamics and bifurcations in a class of nonlinear oscillators with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">real non-volatile memristor devices</i> obeying Stanford model. This is in contrast with papers in the literature considering oscillators with ideal, abstract, or artificial memristor models, that are unable to describe physical memristors implemented in nanotechnology. One main new idea in the paper is to use the memristor as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">programmable nonlinear resistor</i> . Namely, two principal modes of operation are considered. 1) Analogue transient phase: the oscillator is designed so that in the transient oscillations the voltage on the memristor is below threshold, hence the main memristor state variable, i.e., the gap of the insulating material, is almost constant and the memristor behaves as a static nonlinear resistor. 2) Programming phase: the nonlinear characteristic of the memristor, which depends on the gap, can be changed via the application of voltages above threshold. The paper studies nonlinear oscillations in the transient phase for a fixed gap as well as the bifurcations phenomena displayed when the gap is varied. The paper also discusses the differences between the approach in the paper and those to design other memristor oscillators with non-volatile memristors.

Investigating the Solutions of Two Classical Nonlinear Oscillators by the AG Method

Investigating the Solutions of Two Classical Nonlinear Oscillators by the AG Method

Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

&lt;p style='text-indent:20px;'&gt;The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.&lt;/p&gt;

Chaotic modes of a non-linear fractional oscillator

A non-linear fractional oscillator is a generalization of a classical non-linear oscillator in consideration of the hereditary or the memory effect. The memory effect is a property of a dynamic system in which its current state depends on a finite number of its previous states. Therefore, a non-linear fractional oscillator can be mathematically described using integro-differential equations with difference kernels or fractional order derivatives. In this paper, a fractional non-linear oscillator has been investigated to identify chaotic oscillatory modes. The quantitative measure of chaotic regimes is the largest (maximal) Lyapunov exponents. For calculating the maximal Lyapunov exponents, the Wolff algorithm based on the Gram-Schmidt orthogonalization procedure has been selected, using both numerical solution for an initial fractional dynamical system by using variational equations. The Wolff algorithm also makes it possible to plot the spectrum of Lyapunov exponents as a function of control parameters for the initial dynamical system. It has been shown that some spectra of Lyapunov exponents contain positive values indicating the existence of chaotic modes, which are also confirmed by the corresponding phase trajectories.

Sachdev-Ye-Kitaev superconductivity: Quantum Kuramoto and generalized Richardson models

Sachdev-Ye-Kitaev (SYK) model has emerged as a new paradigm of the non-Fermi-liquid behavior. Here we investigate a possibility of having a superconducting off-diagonal long-range order (ODLRO) and a pseudogap phase within the SYK framework. We found that ODLRO may be established in spin-1/2 version of the model with the time-reversal invariance and an extra attractive interaction. If the latter is taken as the on-site negative-$U$ Hubbard term, it leads to the pseudogap phase at $U<U_c$ dominated by quantum fluctuations of local phases. These fluctuations are described by a quantum version of the Kuramoto model, traditionally employed to illustrate synchronization of classical non-linear oscillators. In the opposite limit of large $U$, the SYK+Hubbard model is approaching a certain generalization of the integrable Richardson model. We present exact diagonalization studies, along with analytic solutions of the aforementioned limiting cases. We also discuss possible holographic interpretations of the model, ODLRO and the pseudogap.

Heat flux in one-dimensional systems.

Understanding heat transport in one-dimensional systems remains a major challenge in theoretical physics, both from the quantum as well as from the classical point of view. In fact, steady states of one-dimensional systems are commonly characterized by macroscopic inhomogeneities, and by long-range correlations, as well as large fluctuations that are typically absent in standard three-dimensional thermodynamic systems. These effects violate locality-material properties in the bulk may be strongly affected by the boundaries, leading to anomalous energy transport-and they make more problematic the interpretation of mechanical microscopic quantities in terms of thermodynamic observables. Here, we revisit the problem of heat conduction in chains of classical nonlinear oscillators, following a Lagrangian and a Eulerian approach. The Eulerian definition of the flux is composed of a convective and a conductive component. The former component tends to prevail at large temperatures where the system behavior is increasingly gaslike. Finally, we find that the convective component tends to be negative in the presence of a negative pressure.

First Integrals and Integrating Factors of Second-Order Autonomous Systems

We present a new approach to the construction of first integrals for second-order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first-order equations and their Lie point symmetry and integrating factors of second-order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first-order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.

A new representation for the nonlinear classical oscillator

We use the q-exponential function defined by Tsallis to make a new representation of the classical nonlinear oscillator in which the solution of the correspondent differential equation is mapped in another nonlinear differential equation with the coordinate x(t) deformed in a new coordinate x q (t). The solution is given in the form of a sum of two complex q-exponential functions defined in the Tsallis theory. We obtain the correspondent nonlinear equation to this solution and derive some of its properties.

Modified Parker-Sochacki method for solving nonlinear oscillators

ABSTRACTA new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.

Coherent response of a stochastic nonlinear oscillator to a driving force: analytical characterization of the spectral signatures

We study the dynamics of a classical nonlinear oscillator subject to noise and driven by a sinusoidal force. In particular, we give an analytical identification of the mechanisms responsible for the supernarrow peaks observed recently in the spectrum of a mechanical realization of the system. Our approach, based on the application of averaging techniques, simulates standard detection schemes used in practice. The spectral peaks, detected in a range of parameters corresponding to the existence of two attractors in the deterministic system, are traced to characteristics already present in the linearized stochastic equations. It is found that, for specific variations of the parameters, the characteristic frequencies near the attractors converge on the driving frequency and, as a consequence, the widths of the peaks in the spectrum are significantly reduced. The implications of the study to the control of the observed coherent response of the system are discussed.

Coherent energy transport in classical nonlinear oscillators: An analogy with the Josephson effect.

By means of a simple theoretical model and numerical simulations, we demonstrate the presence of persistent energy currents in a lattice of classical nonlinear oscillators with uniform temperature and chemical potential. In analogy with the well-known Josephson effect, the currents are proportional to the sine of the phase differences between the oscillators. Our results elucidate general aspects of nonequilibrium thermodynamics and point towards a way to practically control transport phenomena in a large class of systems. We apply the model to describe the phase-controlled spin-wave current in a bilayer nanopillar.

Design of nonlinear isochronous oscillators

This study provides two theorems defining two classes of conservative nonlinear oscillators that have some characteristics of the linear harmonic oscillators, one of which is an amplitude-independent period. These theorems also define their response in terms of initial conditions. The first class of these nonlinear oscillators is also characterized by a constant amplitude, while the amplitude of the second one decreases in time. A few illustrative examples are also presented.

Exact formula for the high harmonic response of a strongly driven two-level system

In this paper we investigate the polarization response for a two-level system under the influence of a strong pump field. We use a strong-field perturbation theory to derive a simple formula for the full spectrum. We use the formula to show that the spectrum has a plateau and a cutoff and is in this way analogous to the well-known high harmonic generation (HHG) spectra generated by noble gases in a strong laser field. Using our formula we derive a simple relation describing the cutoff energy for the two-level system and show that it is similar to the noble gas case but is different in an important way in that it does not depend on the pump wavelength. This is in contrast to the noble gas case where the cutoff depends quadratically on the pump wavelength. Our strong-field perturbation theory can with small modifications be applied to the investigation of the HHG-like spectra generated by strongly driven classical nonlinear oscillator systems.