Limit Cycles In Perturbations Research Articles

Optimal phase-based control of strongly perturbed limit cycle oscillators using phase reduction techniques.

Phase reduction is a well-established technique for analysis and control of weakly perturbed limit cycle oscillators. However, its accuracy is diminished in a strongly perturbed setting where information about the amplitude dynamics must also be considered. In this paper, we consider phase-based control of general limit cycle oscillators in both weakly and strongly perturbed regimes. For use at the strongly perturbed end of the continuum, we propose a strategy for optimal phase control of general limit cycle oscillators that uses an adaptive phase-amplitude reduced order model in conjunction with dynamic programming. This strategy can accommodate large magnitude inputs at the expense of requiring additional dimensions in the reduced order equations, thereby increasing the computational complexity. We apply this strategy to two biologically motivated prototype problems and provide direct comparisons to two related phase-based control algorithms. In situations where other commonly used strategies fail due to the application of large magnitude inputs, the adaptive phase-amplitude reduction provides a viable reduced order model while still yielding a computationally tractable control problem. These results highlight the need for discernment in reduced order model selection for limit cycle oscillators to balance the trade-off between accuracy and dimensionality.

Limit cycles in piecewise smooth perturbations of a class of cubic differential systems

In this paper, we study the bifurcation of limit cycles from a class of cubic integrable non-Hamiltonian systems under arbitrarily small piecewise smooth perturbations of degree n . By using the averaging theory and complex method, the lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems are given at first order in ε . It is also shown that in this case, the maximum number of limit cycles produced by piecewise smooth perturbations is almost twice the upper bound of the maximum number of limit cycles produced by smooth perturbations for the considered systems.

An Adaptive Phase-Amplitude Reduction Framework without $\mathcal{O}(\epsilon)$ Constraints on Inputs

Phase reduction is a well-established technique used to analyze the timing of oscillations in response to weak external inputs. In the preceding decades, a wide variety of results have been obtained for weakly perturbed oscillators that place restrictive limits on the magnitude of the inputs or on the magnitude of the time derivatives of the inputs. By contrast, no general reduction techniques currently exist to analyze oscillatory dynamics in response to arbitrary, large magnitude inputs and comparatively very little is understood about these strongly perturbed limit cycle oscillators. In this work, the theory of isostable reduction is leveraged to develop an adaptive phase-amplitude transformation that does not place any restrictions on the allowable input. Additionally, provided some of the Floquet multipliers of the underlying periodic orbits are near-zero, the proposed method yields a reduction in dimension comparable to that of other phase-amplitude reduction frameworks. Numerical illustrations show that the proposed method accurately reflects synchronization and entrainment of coupled oscillators in regimes where a variety of other phase-amplitude reductions fail.

Optimisation of chaotically perturbed acoustic limit cycles

In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure. The objective of manufacturers is to design stable thermoacoustic configurations. In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation. In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down. First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time-averaged cost functionals exist and can be robustly calculated by a shadowing method. Third, we calculate the sensitivities of the time-averaged acoustic energy and Rayleigh index to small changes to the heat-source intensity and time delay. By embedding the sensitivities into a gradient-update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification.

Phase-amplitude reduction far beyond the weakly perturbed paradigm.

While phase reduction is a well-established technique for the analysis of perturbed limit cycle oscillators, practical application requires perturbations to be sufficiently weak thereby limiting its utility in many situations. Here, a general strategy is developed for constructing a set of phase-amplitude reduced equations that is valid to arbitrary orders of accuracy in the amplitude coordinates. This reduction framework can be used to investigate the behavior of oscillatory dynamical systems far beyond the weakly perturbed paradigm. Additionally, a patchwork phase-amplitude reduction method is suggested that is useful when exceedingly large magnitude perturbations are considered. This patchwork method incorporates the high-accuracy phase-amplitude reductions of multiple nearby periodic orbits that result from modifications to nominal parameters. The proposed method of high-accuracy phase-amplitude reduction can be readily implemented numerically and examples are provided where reductions are computed up to fourteenth order accuracy.

Random fluctuations around a stable limit cycle in a stochastic system with parametric forcing.

Many real populations exhibit stochastic behaviour that appears to have some periodicity. In terms of populations, these time series can occur as limit cycles that arise through seasonal variation of parameters such as, e.g., disease transmission rate. The general mathematical context is that of a stochastic differential system with periodic parametric forcing whose solution is a stochastically perturbed limit cycle. Earlier work identified the power spectral density (PSD) features of these fluctuations by computation of the autocorrelation function of the stochastic process and its transform. Here, we present an alternative analysis which shows that the structure of the fluctuations around the limit cycle is analogous to that of fluctuations about a fixed point. Furthermore, we show that these fluctuations can be expressed, approximately, as a factorization which reveals the combined frequencies of the limit cycle and the stochastic perturbation. This result, based on a new limit theorem near a Hopf point, yields an understanding of the previously found features of the PSD. Further insights are obtained from the corresponding stochastic equations for phase and amplitude.

Augmented Phase Reduction of (Not So) Weakly Perturbed Coupled Oscillators

While phase reduction is a tremendously useful tool for understanding the dynamics of weakly perturbed limit cycle oscillators, its assumptions break down as perturbations become larger, limiting i...

Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular separation line

We study Poincaré bifurcation for a planar piecewise near-Hamiltonian system with two regions separated by a non-regular separation line, which is formed by two rays starting at the origin and such that the angle between them is α ∈ (0, π). The unperturbed system is a piecewise linear system having a periodic annulus between the origin and a homoclinic loop around the origin for all α ∈ (0, π). We give an estimation of the maximal number of the limit cycles which bifurcate from the periodic annulus mentioned above under n-th degree polynomial perturbations. Compared with the results in [13], where a planar piecewise linear Hamiltonian system with a straight separation line was perturbed by n-th degree polynomials, one more limit cycle is found. Moreover, based on our Lemma 2.5 we improve the upper bounds on the maximal number of zeros of the first order Melnikov functions derived in [19].

An Operational Definition of Phase Characterizes the Transient Response of Perturbed Limit Cycle Oscillators

Phase reduction is an essential tool used in the study of perturbed limit cycle oscillators. Usually, phase is defined with respect to some asymptotic limit, often making phase reduction's ability ...

The Maximal Number of Limit Cycles in Perturbations of Piecewise Linear Hamiltonian Systems with Two Saddles

We study the maximal number of limit cycles of piecewise linear near-Hamiltonian systems, whose unperturbed system has two saddles. Using the method developed in [Xiong, 2015], we prove that two limit cycles can be bifurcated from the periodic orbits.

Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method

In this article we study limit cycles in piecewise smooth perturbations of a linear center. In this setting it is common to adapt classical results for smooth systems, like Melnikov functions, to non-smooth ones. However, there is little justification for this procedure in the literature. By using the regularization method we give a theoretical proof that supports the use of Melnikov functions directly from the original non-smooth problem.

Phase reduction theory for hybrid nonlinear oscillators.

Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable. On the basis of the generalized theory, we analyze injection locking of hybrid limit-cycle oscillators by periodic forcing and reveal their characteristic synchronization properties, such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition. We also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion.

Study of limit cycles in piecewise smooth perturbations of Hamiltonian centers via regularization method

Study of limit cycles in piecewise smooth perturbations of Hamiltonian centers via regularization method

The number of limit cycles in perturbations of polynomial systems with multiple circles of critical points

This paper investigates the problem for limit cycle bifurcations of system x˙=yF(x,y)+εp(x,y), y˙=−xF(x,y)+εq(x,y), where F(x,y) consists of multiple circles and p(x,y),q(x,y) are polynomials of degree n. The upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order Melnikov function. Furthermore, Hopf bifurcation for a cubic system of this type is discussed.

On the number of limit cycles in small perturbations of a piecewise linear Hamiltonian system with a heteroclinic loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n (n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [n+1/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in e. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

Phase Reduction Method for Strongly Perturbed Limit Cycle Oscillators

The phase reduction method for limit cycle oscillators subjected to weak perturbations has significantly contributed to theoretical investigations of rhythmic phenomena. We here propose a generalized phase reduction method that is also applicable to strongly perturbed limit cycle oscillators. The fundamental assumption of our method is that the perturbations can be decomposed into a slowly varying component as compared to the amplitude relaxation time and remaining weak fluctuations. Under this assumption, we introduce a generalized phase parameterized by the slowly varying component and derive a closed equation for the generalized phase describing the oscillator dynamics. The proposed method enables us to explore a broader class of rhythmic phenomena, in which the shape and frequency of the oscillation may vary largely because of the perturbations. We illustrate our method by analyzing the synchronization dynamics of limit cycle oscillators driven by strong periodic signals. It is shown that the proposed method accurately predicts the synchronization properties of the oscillators, while the conventional method does not.

On the Limit Cycles Bifurcating from a Quadratic Reversible Center of Genus One

The paper is concerned with the bifurcation of limit cycles in perturbations of a quadratic reversible system with a center of genus one. By studying the properties of the auxiliary curve and centroid curve defined by the Abelian integrals, we have proved that under small quadratic perturbations, at most two limit cycles arise from the period annulus surrounding the quadratic reversible center, and the bound is sharp. This partially verifies Conjecture 1 given in Gautier et al. (Discrete Contin Dyn Syst 25:511–535, 2009).

Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors

Lean premix gas turbine combustors are prone to high amplitude pressure oscillations driven by non-linear thermoacoustic coupling. These pulsations are unwanted because they can affect the lifetime of the combustor parts. The standard strategy to get rid of these oscillations is to implement acoustic damping devices. Knowing the deterministic components characterizing the acoustic-flame coupling, the linear growth rates in particular, is necessary to properly design the dampers. However, the time scale associated with the variations of the engine operating conditions is much larger than the one of the acoustic pressure amplitude dynamics. Therefore, the linearly unstable system cannot be observed during the exponential amplitude growth of one of the acoustic eigenmodes and it is not possible to directly determine the linear growth rates. They can only be estimated from signals recorded when the system is operating on limit cycling states. Fortunately, these states are driven by a strong stochastic forcing produced by the highly turbulent reactive flow. It is shown in this article that the deterministic quantities can be extracted from the noise perturbed limit cycles data by making use of the stochastic differential equations describing the combustion instabilities. A straightforward experimental set-up allowing to reproduce the main features of the thermoacoustic coupling observed in gas turbines is used to validate the proposed identification methods. In a second step, these latter methods are applied to engine data.

Phase reduction of weakly perturbed limit cycle oscillations in time-delay systems

The phase reduction method is applied to a general class of weakly perturbed time-delay systems exhibiting periodic oscillations. The adjoint equation with an appropriate initial condition for the infinitesimal phase response curve of a time-delay system is derived. The method is demonstrated numerically for the Mackey–Glass equation as well as for a chaotic Rössler system subject to a delayed feedback control (DFC). We show that the profile of the phase response curve of a periodic orbit stabilized by the DFC algorithm does not depend on the control matrix. This property is universal and holds for any dynamical system subject to the DFC.

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 6, which is a Liénard system of the form x ′ = y , y ′ = Q 1 ( x ) + ε y Q 2 ( x ) with Q 1 and Q 2 polynomials of degree 5 and 4, respectively. It is shown that this system can undergo degenerate Hopf bifurcation and Poincaré bifurcation, from which at most three limit cycles can emerge in the plane for ε sufficiently small. And the limit cycles can only surround an equilibrium inside, i.e. the system can have the configuration ( 3 , 0 , 0 ) of limit cycles for some values of the parameters, where ( 3 , 0 , 0 ) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding any of the two or three equilibria.