Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators

1992 ◽  
Vol 340 (1659) ◽  
pp. 473-501 ◽  
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Phase Locking ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Wide Range

Two timescale harmonic balance is a semi-analytical/numerical method for deriving periodic solutions and their stability to a class of nonlinear autonomous and forced oscillator equations of the form ẍ + x = f(x,ẋ,λ) and ẍ + x = f(x,ẋ,λ,t) , where λ is a control parameter. The method incorporates salient features from both the method of harmonic balance and multiple scales, and yet does not require an explicit small parameter. Essentially periodic solutions are formally derived on the basis of a single assumption: ‘that an N harmonic, truncated, Fourier series and its first two derivatives can represent x(t) , ẋ(t) and ẍ(t) respectively’. By seeking x(t) as a series of superharmonics, subharmonics, and ultrasubharmonics it is found that the method works over a wide range of parameter space provided the above assumption holds which, in practice, imposes some ‘problem dependent’ restriction on the magnitude of the nonlinearities. Two timescales, associated with the amplitude and phase variations respectively, are introduced by means of an implicit parameter Є . These timescales permit the construction of a set of amplitude evolution equations together with a corresponding stability criterion. In Part I the method is formulated and applied to three autonomous equations, the van der Pol equation, the modified van der Pol equation, and the van der Pol equation with escape. In this case an expansion in superharmonics is sufficient to reveal Hopf, saddle node and homoclinic bifurcations which are compared with results obtained by numerical integration of the equations. In Part II the method is applied to forced nonlinear oscillators in which the solution for x(t) includes superharmonics, subharmonics, and ultrasubharmonics. The features of period doubling, symmetry breaking, phase locking and the Feigenbaum transition to chaos are examined.

A theory for phase locking of respiration in cats to a mechanical ventilator

1984 ◽  
Vol 246 (3) ◽  
pp. R311-R320 ◽  
Author(s):  
G. A. Petrillo ◽  
L. Glass
Keyword(s):  
Nonlinear Oscillator ◽  
Phase Locking ◽  
Experimental Studies ◽  
Mechanical Ventilator ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Mechanically Ventilated ◽  
Wide Range ◽  
Forced Nonlinear Oscillator ◽  
Good Agreement

A mathematical model describing the Hering-Breuer reflexes in mechanically ventilated cats is developed. There is good agreement between the properties of the model and experimental studies performed over a wide range of frequencies and volumes of the mechanical ventilator. There is a correspondence between the model and a periodically forced nonlinear oscillator, similar to the van der Pol equation. Brain stem mechanisms underlying the entrainment are discussed.

Quasiperiodicity and Chaos Through Hopf–Hopf Bifurcation in Minimal Ring Neural Oscillators Due to a Single Shortcut

2019 ◽  
Vol 29 (05) ◽  
pp. 1950065
Author(s):  
Yo Horikawa ◽  
Hiroyuki Kitajima ◽  
Haruna Matsushita
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Neural Oscillator ◽  
Quasiperiodic Solution ◽  
Chaotic Oscillations ◽  
Van Der Pol ◽  
Neural Oscillators ◽  
Hopf Bifurcation Point ◽  
Codimension Two

Quasiperiodicity and chaos in a ring of unidirectionally coupled sigmoidal neurons (a ring neural oscillator) caused by a single shortcut is examined. A codimension-two Hopf–Hopf bifurcation for two periodic solutions exists in a ring of six neurons without self-couplings and in a ring of four neurons with self-couplings in the presence of a shortcut at specific locations. The locus of the Neimark–Sacker bifurcation of the periodic solution emanates from the Hopf–Hopf bifurcation point and a stable quasiperiodic solution is generated. Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation, and multiple chaotic oscillations are generated through period-doubling cascades of periodic solutions in the Arnold’s tongues. Further, such chaotic irregular oscillations due to a single shortcut are also observed in propagating oscillations in a ring of Bonhoeffer–van der Pol (BVP) neurons coupled unidirectionally by slow synapses.

scholarly journals Quasi-Periodic Bifurcations of Higher-Dimensional Tori

2016 ◽  
Vol 26 (07) ◽  
pp. 1630016 ◽  
Author(s):  
Motomasa Komuro ◽  
Kyohei Kamiyama ◽  
Tetsuro Endo ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Solutions ◽  
Phase Locking ◽  
Period Doubling ◽  
Double Covering ◽  
Heteroclinic Cycle ◽  
One Dimensional ◽  
Local Bifurcations ◽  
Saddle Node ◽  
Higher Dimensional ◽  
The Relationship

We classify the local bifurcations of quasi-periodic [Formula: see text]-dimensional tori in maps (abbr. MT[Formula: see text]) and in flows (abbr. FT[Formula: see text]) for [Formula: see text]. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT[Formula: see text] can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT[Formula: see text] can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT[Formula: see text] and FT[Formula: see text]: namely, saddle-node cycle and heteroclinic cycle bifurcations of the [Formula: see text]-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST[Formula: see text]) and zero-dimensional tori in sections (abbr. ST[Formula: see text]). The bifurcations of ST[Formula: see text] can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST[Formula: see text] can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST[Formula: see text]/ST[Formula: see text] and the bifurcations of MT[Formula: see text]/FT[Formula: see text]. We present examples of all of these bifurcations.

Experimental and theoretical investigation of an equation of dendritic crystal growth

1991 ◽  
Vol 2 (3) ◽  
pp. 199-222
Author(s):  
J. N. Dewynne ◽  
F. N. H. Robinson
Keyword(s):  
Harmonic Balance ◽  
Large Time ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Dendritic Crystal ◽  
Method Of Harmonic Balance ◽  
Electronic Model ◽  
Wide Range ◽  
Periodic Form ◽  
Dissipative Terms

An experimental study using an analogue electronic model of the equation x‴ + x′ = є sin x, modified by the addition of small terms ax″ and βx with 0 < β < α ≫ є shows that these dissipative terms have a profound effect on the solutions for large time. If ∈ is not too large, experimental solutions tend to a simple periodic form, unlike the case α=β = 0. The existence of this limiting periodic form suggests the possibility of a simple analytic treatment using the method of harmonic balance, and this treatment leads to excellent agreement with the experimental results for a wide range of initial conditions and values of the parameters. The approach towards attracting limiting periodic solutions is analysed by using the method of multiple scales.

Bifurcation Analysis of a Power System Model with Three Machines and Four Buses

2016 ◽  
Vol 26 (05) ◽  
pp. 1650082 ◽  
Author(s):  
Yu Chang ◽  
Xiaoli Wang ◽  
Dashun Xu
Keyword(s):  
Power Systems ◽  
Power System ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Chaotic Behavior ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Bifurcation Phenomena ◽  
Fold Bifurcation ◽  
Approximate Expressions

The bifurcation phenomena in a power system with three machines and four buses are investigated by applying bifurcation theory and harmonic balance method. The existence of saddle-node bifurcation and Hopf bifurcation is analyzed in time domain and in frequency domain, respectively. The approach of the fourth-order harmonic balance is then applied to derive the approximate expressions of periodic solutions bifurcated from Hopf bifurcations and predict their frequencies and amplitudes. Since the approach is valid only in some neighborhood of a bifurcation point, numerical simulations and the software Auto2007 are utilized to verify the predictions and further study bifurcations of these periodic solutions. It is shown that the power system may have various types of bifurcations, including period-doubling bifurcation, torus bifurcation, cyclic fold bifurcation, and complex dynamical behaviors, including quasi-periodic oscillations and chaotic behavior. These findings help to better understand the dynamics of the power system and may provide insight into the instability of power systems.

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

2017 ◽  
Vol 78 (4) ◽  
pp. 1217-1231 ◽  
Author(s):  
Jifeng Cui ◽  
Wenyu Zhang ◽  
Zeng Liu ◽  
Jianglong Sun
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Duffing Oscillator ◽  
Period Doubling ◽  
Van Der Pol

Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms

2014 ◽  
Vol 24 (03) ◽  
pp. 1430011 ◽  
Author(s):  
Zhiyan Yang ◽  
Tao Jiang ◽  
Zhujun Jing
Keyword(s):  
Numerical Simulation ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Original System ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
External Forcing ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Period Doubling Bifurcations

Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.

On the modeling of some triodes-based nonlinear oscillators with complex dynamics: case of the Van der Pol oscillator

2021 ◽  
Author(s):  
Joakim Vianney Ngamsa Tegnitsap ◽  
Merlin Brice Saatsa Tsefack ◽  
Elie Bertrand Megam Ngouonkadi ◽  
Hilaire Bertrand Fotsin
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Oscillator ◽  
Phase Portraits ◽  
Van Der Pol ◽  
Physical Consideration

Abstract In this work, the dynamic of the triode-based Van der Pol oscillator coupled to a linear circuit is investigated (Triode-based VDPCL oscillator). Towards this end, we present a mathematical model of the triode, chosen from among the many different ones present in literature. The dynamical behavior of the system is investigated using classical tools such as two-parameter Lyapunov exponent, one-parameter bifurcation diagram associated with the graph of largest Lyapunov exponent, phase portraits, and time series. Numerical simulations reveal rather rich and complex phenomena including chaos, transient chaos, the coexistence of solutions, crisis, period-doubling followed by reverse period-doubling sequences (bubbles), and bursting oscillation. The coexistence of attractors is illustrated by the phase portraits and the cross-section of the basin of attraction. Such triode-based nonlinear oscillators can find their applications in many areas where ultra-high frequencies and high powers are demanded (radio, radar detection, satellites communication, etc.) since triode can work with these performances and are often used in the aforementioned areas. In contrast to some recent work on triode-based oscillators, LTSPICE simulations, based on real physical consideration of the triode, are carried out in order to validate the theoretical results obtained in this paper as well as the mathematical model adopted for the triode.

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