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The Dynamics of Coupled

Oscillators

Nigel Lawrence Holland

A thesis

submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of

Master of Science

in Mathematics.

Victoria University of Wellington

2008

Abstract

The subject is introduced by considering the treatment of oscillators in

Mathematics from the simple Poincar

´e oscillator, a single variable dynam-

ical process defined on a circle, to the oscillatory dynamics of systems of differential equations. Some models of real oscillator systems are consid- ered. Noise processes are included in the dynamics of the system. Cou- pling between oscillators is investigated both in terms of analytical sys- tems and as coupled oscillator models. It is seen that driven oscillators can be used as a model of 2 coupled oscillators in 2 and 3 dimensions due to the dependence of the dynamics onthephasedifferenceoftheoscillators. Thismeansthatthedynamicsare easily able to be modelled by a 1D or 2D map. The analysis of N coupled oscillator systems is also described. The human cardiovascular system is studied as an example of a cou- pled oscillator system. The heart oscillator system is described by a sys- tem of delay differential equations and the dynamics characterised. The mechanics of the coupling with the respiration is described. In particular the model of the heart oscillator includes the barorecep- tor reflex with time delay whereby the aortic fluid pressure influences the heart rate and the peripheral resistance. Respiration is modelled as forcing the heart oscillator system. Locking zones caused by respiratory sinus arrhythmia (RSA), the syn- chronisation of the heart with respiration, are found by plotting the rota- tion number against respiration frequency. These are seen to be relatively narrow for typical physiological parameters and only occur for low ratios of heart rate to respiration frequency. Plots of the diastolic pressure and heart interval in terms of respiration phase parameterised by respiration frequency illustrate the dynamics of synchronisation in the human cardio- vascular system.

Acknowledgments

I acknowledge my parents, the patience of my supervisor Mark McGuin- ness, and the Victoria University library. i

1 Introduction 1

2 Oscillators in Mathematics 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Relaxation Oscillators . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Poincar

´e Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4 Classical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Oscillating Systems . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Measurement of Oscillator Phase . . . . . . . . . . . . . . . . 12

2.7 Noise in Oscillators . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Coupled Oscillators 23

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Nonisochronous driven oscillator . . . . . . . . . . . . . . . . 24

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Weak Coupling . . . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Quasiperiodic behaviour . . . . . . . . . . . . . . . . 27

3.2.4 Structure of Arnolds tongues . . . . . . . . . . . . . . 28

3.2.5 Strong coupling . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Dynamics of phase locking . . . . . . . . . . . . . . . . . . . . 34

3.3.1 Deterministic isochronous oscillators . . . . . . . . . 34

3.3.2 Example of forced oscillator . . . . . . . . . . . . . . . 37

3.3.3 Chaotic oscillators . . . . . . . . . . . . . . . . . . . . 41

CONTENTSiii

3.4 Perturbed Nonisochronous Oscillator . . . . . . . . . . . . . 43

3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.2 Planar dynamics . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . 44

3.4.4 Global bifurcations . . . . . . . . . . . . . . . . . . . . 45

3.4.5 Dynamics near a Saddle point . . . . . . . . . . . . . 46

3.4.6 Bifurcations near a Saddle point . . . . . . . . . . . . 49

3.5 Method of Averaging . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.2 Co-moving coordinates . . . . . . . . . . . . . . . . . 53

3.5.3 Averaging Theorem . . . . . . . . . . . . . . . . . . . 53

3.5.4 Validity of Averaging method . . . . . . . . . . . . . . 54

3.6 Action/Angle coordinates . . . . . . . . . . . . . . . . . . . . 56

3.6.1 Hamiltonian oscillators . . . . . . . . . . . . . . . . . 56

3.6.2 System action variables . . . . . . . . . . . . . . . . . 57

3.6.3 Generating function . . . . . . . . . . . . . . . . . . . 57

3.6.4 Perturbed integrable systems . . . . . . . . . . . . . . 58

3.6.5 Dynamics near resonant trajectories . . . . . . . . . . 59

3.7 Melnikov"s method . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7.2 Geometry of a homoclinic loop . . . . . . . . . . . . . 60

3.7.3 Bifurcations of a homoclinic loop . . . . . . . . . . . . 62

4 Mathematical Modelling of the Cardiovascular System 64

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Studying Synchronisation in the Cardio-

vascular System . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.1 Numerical solution of system . . . . . . . . . . . . . . 73

4.3.2 Behaviour of heart system model . . . . . . . . . . . . 76

4.3.3 Frequency response of heart system . . . . . . . . . . 83

CONTENTSiv

5 Coupling in the Cardiovascular System 90

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 A Phase locked loop model of

Cardiovascular system synchronisation . . . . . . . . . . . . 92

5.3 The Synchronization mechanism in

the Cardiovascular system . . . . . . . . . . . . . . . . . . . . 100

5.4 RSA Synchronization detection . . . . . . . . . . . . . . . . . 106

5.5 RSA locking regions of the Cardiovascular system . . . . . . 116

5.6 Dynamics of RSA phase locking . . . . . . . . . . . . . . . . . 123

5.6.1 Step response of heart system . . . . . . . . . . . . . . 123

5.6.2 Phase relationship across locking region . . . . . . . . 124

5.6.3 Maps of the cardiovascular system . . . . . . . . . . . 124

6 Conclusions 135

A Physiology of the Cardiovascular System 146

Chapter 1 Introduction

Coupling between oscillators was first studied by Huygens in 1673 who noticed that pendulum clocks in the same room became synchronised. In

1889 Henri Poincar

´e laid the foundations of dynamical systems theory in his study of the stability of the solar system. His work was an entry in a contest to commemorate the 60th birthday of King Oscar II of Sweden.

Poincar

´e uncovered the significance of the homoclinic points at the cross- ing of the unstable and stable manifolds and also introduced the Poincar ´e map. Coupled and synchronised oscillators are significant in electronics also. The ubiquitous phase-locked loop is an example. The dynamics of a phase-locked loop can be highly nonlinear and are not fully understood. For example chaotic dynamics and locking region hysteresis are possible. Commonly only a simplified linear model of the dynamics is used. Coupled oscillators also occur in biological systems. One early use of mathematics to model biological systems was Van der Pol"s use in 1928 of a driven Van der Pol oscillator to explain some normal, and patholog- ical, rhythms of the heart. However Van der Pol devised his differential equation to model an electronic oscillator based on the triode valve. In the human cardiovascular system coupling between respiration and the heart results in synchronisation at a fixed mn ratio for some respiration frequencies. This results in the heart rate being entrained by the respi- 1

CHAPTER 1. INTRODUCTION2

ration so over a zone of respiration frequencies the heart rate maintains the fixed ratio. This process of entrainment is common to all systems of coupled oscillators. The zones, which correspond to the rational numbers, are known as Arnold Tongues after V.I. Arnold who discovered them in

1963. While the dynamics are quite simple at low coupling, consisting

of zones of synchronisation the boundaries of which are a tangent bifur- cation, interspersed with quasiperiodic regions where the frequencies are incommensurate, at higher coupling chaotic behaviour is possible with a more elaborate bifurcation structure. Arnold Tongues are generic to coupled oscillators and there is a range of literature on their theory and the results of mathematical modelling. follows explains the results of modelling the human cardiovascular sys- tem using Matlab and the DDE23 differential delay equation solver. The locking zones found in the model should be comparable to physiological data. The dynamics of cardiovascular synchronisation and the mechanics of synchronisation in the human cardiovascular system are explored.

Chapter 2 Oscillators in Mathematics

2.1 Introduction

The progression of states is measured by the phase of the oscillator. For a mapxi+1=f(xi)an ordernfixed point exists whenxi+n=xiwherefis a C

1homeomorphism. The iterationiindicates the progression of the phase

wrt the initial point. As a dynamical system defined by a vector field on a differential manifold, an oscillator is a piecewise continuous diffeomor- phismf(θ)onT1orS1which has the range[0...1)or[0...2π). What are the possible dynamics of a single dimensional dynamical system? The dynamics could tend to an equilibrium point or they must tend in one direction for all time. So for an oscillator the vector field onR1must be piecewise continuous, periodic inθ, and there must be a constant term so that an equilibrium point is not possible.

2.2 Relaxation Oscillators

The integrate-and-fire relaxation oscillator explicitly models episodic be- 3

CHAPTER 2. OSCILLATORS IN MATHEMATICS4

[1]). Mostgenerallytheintegrate-and-fireoscillatorisdescribedbyaquan- tityx(t)which increases dynamically according to a charging process. At a fixed upper threshold the process changes to a discharge to a lower thresh- old level where the cycle repeats itself. It is a useful simplification to as- sume that the discharge process occurs instantaneously. C.S. Peskin [2] described the dynamics of the charging process by dxdt=S0-γxwhich can also model the charging of a capacitor through a resistor by a con- stant voltage source

S0γ

. Mirollo and Strogatz [3] formalised the idea of the state variablex(t). The state of the oscillator is completely determined by the single variablexwhich increases monotonically in time, from a lower threshold of zero to an upper threshold of one, at which point the oscil- lator fires andxis instantly reset to zero. The phase of the oscillatorφ, is related to the state variable byx=f(φ). The functionf(φ)is defined such that dφdt= const =ν, the free running frequency of the oscillator in cycles per second. The phase is defined to be zero when the statexis zero. A fur- ther assumption is thatf(φ)is concave downwards as it is for the voltage vs time curve of the example of the resistor and charging capacitor. For the example of the resistor and capacitorx(t) =S0γ (1-e-γt). The function f(φ) =xis found by substitutingφ=νtso thatf(φ) =1-e-φγν

1-e-γν

, where the denominator is found from the upper threshold ofx(t) = 1att=1ν

2.3 Poincar´e Oscillator

Wanzhen, Glass, and Shrier [4] describe a 2D system of differential equa- tions which is called the Poincar

´e oscillator as probably the simplest stable

limit cycle oscillator, which has been considered many times as a model of biological oscillations. Radial coordinates are natural, so dφdt =1T

0,(2.1)

drdt=kr(1-r),(2.2)

CHAPTER 2. OSCILLATORS IN MATHEMATICS5

where the two variablesrandφrepresent the distance from the origin, and the angle or phase, respectively. Any value ofrexcept0evolves to r= 1. Evidentlyr= 0is an unstable equilibrium point.φincreases at a constant rate and is evaluated modulo 1.φis not defined at the origin and is a phase singularity there. What is the effect of an external perturbation on the trajectory?rde- cays back tor= 1, depending onk, andφis displaced to a new phase ?. Wanzhen et al. consider as a perturbation a horizontal translation of magnitudeb. This causes a shift of variables(r0,φ0)to(r?,φ?)where ?=12πcos-1?b+r0cos(2πφ0)r ,(2.3) r ?=?r

20+b2+ 2r0bcos(2πφ0).(2.4)

Two cases can be distingushed depending on the winding number of the resetting. The winding number is the relative change in phase ofφ?while

0changes in phase from 0 to 1. Thus for type 1 or "weak" resettingφ?goes

through the equivalent change of phase asφ0, and for type 0 or "strong" resetting, the change in phase ofφ?is zero. The type of the resetting is syn- onomous with the winding number. Geometrically the two cases can be distinguished by whether or not the originalr= 1trajectory is displaced past the phase singularity at the origin. Wanzhen et al. further consider the effect of multiple pulse resetting and naturally find a dependance on k,b, andδ,the time between the reset pulses. The model is compared with experimental results of stimulation of chicken heart cell aggregates and good agreement is found but it is noted that this does not mean that the Poincar

´e oscillator represents an accurate

model of the cardiac preparation but that it captures the essential topolog- ical properties of biological oscillators such as the heart cell aggregate. Further understanding of the effect of phase resetting on an oscillator can be found through the notion of isochrons. An isochron is a line or sur- face in the phase space of the system from which the trajectories evolve to equal phase. In the case of the Poincar

´e oscillator system it can be seen

CHAPTER 2. OSCILLATORS IN MATHEMATICS6

that depending onkthe trajectories of points in the phase plane lying on an isochron will converge to the same phase on the limit cycler= 1as time progresses. If the change in phase dφdtdoes not depend onrthen the isochron is aligned with the component of the phase line inr. The curva- ture of the isochron reflects the change in magnitude of theθcomponent withr. In the case of the Poincar´e oscillator it can be the seen that the isochrons are radial lines. Clearly the isochrons converge at the phase singularity at the origin where the phase is undefined. Evidently the phase of points near the ori- gin can exhibit large changes in response to relatively small perturbations. A.T. Winfree [5] terms the phaseless manifold of the singularity where the phase is undefined a "black hole". In biological systems this can be a re- gion of phase space rather than a point. Winfree gives an example of the construction of isochrons for limit cycle attractor systems with polar sym- metry, that is dφdtdepending only onr, which generalize the Poincar´e os- cillator of Wanzhen et al. Let dφdt=A(r)with the unit of time chosen so that at the radius of the limit cycle attractorA(r0) = 1, anddrdt=B(r)with B(r0) = 0anddBdr<0atr0. The isochrons must have polar symmetry so ?=g(φ,r) =φ-f(r), that is the difference in phase of the isochron at state(φ,r)fromφdepends only onr. The rate of change of the isochron? at the state(φ,r)must then be the same asφon the limit cycle, that is 1, so d?dt≡1 =dφdt-df(r)drdrdt,(2.5) and dfdr=dφdt-1drdt.(2.6) f(r)is obtained by integrating this equation and thusg(φ,r).

For example, consider the system:

drdt= 5(1-r)(r-12 )r,(2.7) dφdt= 1 +ε(1-r).(2.8)

CHAPTER 2. OSCILLATORS IN MATHEMATICS7

Solution is via partial fractions, the identities, dln(x)dx=1x ,dln(1x )dx=-1x ,|x|>

0, and simplifying radicals, so

r(t) =-2+12 Ce52 t±12 (C2e5t-4Ce52 t)12 Ce52 t-4,(2.9)

φ(t) =?εr(t)dt+t+εt+D,(2.10)

?=φ-2ε5 ln(2-1r ).(2.11)

Note thatr= 1is a stable attractor,r=12

,is an unstable periodic solution, andr= 0, is a fixed point, so the regionr <12 is attracted tor= 0, and the regionr >12 is attracted tor= 1. See fig 2.1. A phase space consisting of a periodic attractor and an attracting fixed point occurs often in biological systems. An external signal shifts the system between the two states.

2.4 Classical Oscillators

One classical oscillator is the damped pendulum:

2θdt2+δdθdt+ sin(θ) = 0.(2.12)

Naturally the solution of (2.12) is periodic inθconsisting of fixed points on theθaxis atθ= 2nπ. The phase lines spiral into a fixed point depending on the value of the dampingδ. Ifδ= 0the phase lines orbit a fixed point or monotonically increase inθ, in a narrow range ofdθdt. In this case the system is conservative and the phase lines are the solution of the equation ?φ= 0, whereφis a scalar function.

The well known double well Duffing equation:

2xdt2+δdxdt+x3-x= 0,(2.13)

is a specialization of a larger class of equations given by (2.14), usually referred as Duffing equations: d

2xdt2+δdxdt+dP(x)dx= 0,(2.14)

CHAPTER 2. OSCILLATORS IN MATHEMATICS8Figure 2.1: nonisochronous oscillator with oscillator annihilation in phase

plane

CHAPTER 2. OSCILLATORS IN MATHEMATICS9

whereP(x)is the equation for a potential energy field. In Physics, work equals force times distance, which explains the use of dP(x)dx. It can be seen in the case of (2.12) that the potential function is-cos(θ), and the minimums of potential energy are at2nπ. In the case of (2.13)P(x) = x 4-12 x2+δ, so the-x2term causes a peak in the middle of thex4term leaving two wells. If the potential function-cos(θ)is approximated by its first 3 terms,-1 +12

θ2-124

θ4then this is the negative of the potential

function of the double well Duffing equation and a single potential well exists on the peak of the-θ4term. B. van der Pol [6] devised the Van der Pol oscillator to model an elec- tronic oscillator, using a parallel inductor and capacitor as the resonator,quotesdbs_dbs20.pdfusesText_26

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