nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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70 A Brief Introduction to Bifurcation Analysis and Numerical Methods of the question, and so is very large positive values. To determine these intervals we must naturally look at the magnitude of the parameters. We have already discussed the values and uncertainties of f1 and f2; cf. section 5.8. So just as a reminder the resting and activated macrophage phagocytosis rates were found to be the same for NOD-mice, f1 = f2 = 1×10 −5 . 5 While for Balb/c-mice the resting phagocytosis rate was estimated to be f1 = 2 × 10 −5 , whereas the phagocytosis rate of the activated macrophages was f2 = 5 × 10 −5 . These estimated values are associated with uncertainties in the area of 10-20%. The scale (10 −5 ) defines the magnitude of our interval, and the uncertainties tell us what parameter range to investigate as a minimum. This minimum range is very small, and since we are interested in the bigger picture we have decided to let f1 take values in interval from 0 to 4 × 10 −5 while f2 is in the interval from 0 to 10 × 10 −5 . Naturally we need fixed points to conduct a bifurcation analysis. How we came about these is the subject of the next section. 7.5 Locating the Fixed Points The application of the Newton-Raphson method has proven fruitful for finding fixed points. The Newton-Raphson method is a numerical method that, through iterations, computes the zeros of a vector-valued function x, based on the following equation x k+1 = x k − (J(f(x))) −1 f(x k ) (7.6) where x k and x k+1 are the approximations of the zero, x ∗ , after k and k + 1 iterations respectively, (J(f(x))) −1 is the inverted Jacobian 6 , and f(x) = ˙x. The iteration continues until ||x k+1 − x k || = ||∆x|| < ε where ε is a user defined error-tolerance on the estimated value of x ∗ compared to the analytical value. The partial derivatives of the Jacobian matrix J(f(x)) are approximated as the finite difference. 7 The procedure of Newton-Raphson requires one to make a guess on the initial conditions from where to start the iteration. This determines what fixed point the function converges to – if it converges at all! However, to begin with, it is not in anyway evident what initial conditions to use when searching for nontrivial fixed points, so this tends to be a process of trial and error based on an “educated guess”, that is based on simulations of the DuCa model. Despite this guideline, there is no guarantee that we see the “full picture” when less than five fixed points are found – there may be non-relevant fixed points located outside 1. quadrant, but undetected fixed points in the 1. quadrant may also exist. The procedure is easiest in the case where the first fixed point for a given value of the bifurcation parameter is found, and the position of the fixed point changes very little for small changes of the bifurcation parameter. Then it is possible to use the coordinates of the located fixed point as an initial guess, when searching for fixed points at values of the bifurcation parameter close to the initial value. The stability of the fixed points has been determined by computing the eigenvalues, of the Jacobian, in each point. Plots of the eigenvalues with respect to f1 and f2 can be seen in appendices B.1 to B.5. 5 From here on we will drop the dimensions for the sake of convenience. 6 We have tacitly assumed that the Jacobian has a non-zero determinant. 7 For a derivation of the Newton-Raphson method and a more detailed description of the procedure applied see appendix A.5.
7.5 Locating the Fixed Points 71 Evaluation of stability of the fixed points and detection of bifurcations The method we have used for detection of bifurcations is to look for fixed points and evaluating their stability for different values of the chosen bifurcation parameter. 8 E.g. when we have looked for Hopf bifurcations we have made use of their defining characteristic, i.e. two complex eigenvalues that cross the imaginary axis. Using these traits of the Hopf bifurcation we use the numerical estimates of the fixed points found by the Newton-Raphson method. This eases the job of evaluating the Jacobian in each fixed point since this is already done, albeit as an approximation, by the Newton-Raphson method. The eigenvalues of the Jacobian were computed in order to plot the real and imaginary part of the eigenvalues as a function of the bifurcation parameter. Furthermore a plot of the imaginary part as a function of the real part will be shown. This is the “traditional” way of presenting the behavior of eigenvalues during a Hopf bifurcation. 8 It should be mentioned that this approach limits us to the detection of only local bifurcations since global bifurcations involve large regions of the phase space, and therefore cannot be detected solely by looking at the stability of local fixed points (Lynch, 2004, p.330-333).
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- I, OM OG MED MATEMATIK OG FYSIK E
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Exploring Treatment Strategies for
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Preface iii questions that were out
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vi Contents Significance of the Apo
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List of Tables 5.1 summarizes the m
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x List of Figures 8.3 Phase space p
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xii
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2 Introduction In the following dec
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4 Introduction There are several sc
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6 Introduction because I had to mak
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8 Type 1 Diabetes and Its Etiology
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10 Type 1 Diabetes and Its Etiology
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3 Prospects for Therapy - A Mini Re
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3.3 Gastrin 15 eration in situ Urus
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4 Mathematical Modelling The langua
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120 Discussion of the Expanded Mode
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126 Bibliography Christen, U. and M
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128 Bibliography Notkins, A. L. and
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130 Bibliography Yu, J. and G. S. E
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132 Mathematical Appendices the com
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134 Mathematical Appendices in that
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136 Mathematical Appendices A.6 Dim
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138
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140 Additional Figures Figure B.2 P
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142 Additional Figures B.2 Eigenval
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144 Additional Figures Figure B.8 P
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148 Additional Figures Figure B.14
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158 matlab Code w(r+1)=w(r)+dw; g=w
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160 matlab Code end V_11(r)=E(1); V
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162 matlab Code n=675; w_5=0; dw_5=
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164 matlab Code M=M’; % end s_7(r
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166 matlab Code w_7,V_34,’.b’,w
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168 matlab Code M=M’; % M must be
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170 matlab Code %options = odeset(
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172
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174 Index value, 73 Bifurcation Ana
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176 Index of diabetes, 2 Michaelis
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nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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