Fixed Points and Linearization

In this section we extend the linearization technique developed earlier for onedimensional systems (Section 2.4). The hope is that we can approximate the phase portrait near a fixed point by that of a corresponding linear system. 

To simplify the notation, we have written /dx and df /dy, but please remember that these partial derivatives are to be evaluated at the fixed point (x, y ) thus they are numbers, not functions. Also, the shorthand notation (9(m , v ,mv) denotes quadratic term in u and v. Since and v are small, these quadratic terms are extremely small. 

Now since the quadratic terms in (1) are tiny, it s tempting to neglect them altogether. If we do that, we obtain the linearized system 

Is it really safe to neglect the quadratic terms in (1) In other words, does the linearized system give a qualitatively correct picture of the phase portrait near (x, y ) The answer is yes, as long as the fixed point for the linearized system is not one of the borderline cases discussed in Section 5.2. In other words, if the linearized system predicts a saddle, node, or a spiral, then the fixed point realty is a saddle, node, or spiral for the original nonlinear system. See Andronov et al. (1973) for a proof of this result, and Example 6.3.1 for a concrete illustration. 

The borderline cases (centers, degenerate nodes, stars, or non-isolated fixed points) are much more delicate. They can be altered by small nonlinear terms, as weTl see in Example 6.3.2 and in Exercise 6.3.11. 

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The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

Conveniently, an investigation of the dynamic behavior of a set of differential equations starts out with the determination of the fixed points and their stability. The latter is studied hy linearizing the system s equation about the steady state and then evaluating the temporal evolution of small perturbations. Denoting the perturbations by 8( )ql and 8c, in our case the equations read ... 

Figure 4.6 a) The function F(C) in (4.22). The three zeros are at Ci = 0, Cu = 0.4, and Co I. The arrows indicate the direction of motion of the dynamical system C = F(C) in each C-interval. Cu is a unstable fixed point and C and C2 are linearly stable ones, b) The potential U(g) = — f dgF(g). Front solutions of (4.20) joining the two stable states correspond to trajectories of a particle which starts at g = (J 2 with infinitesimal velocity, falls down under friction v and stops precisely at C. Such solutions exist just for a particular value of v. 

A, B same as for the DS fixed point. The linearization of recursion relations about this fixed point gives two eigenvalues Ai = 3.1319 and A2 = 2.5858 greater than one. The line 0Jc u, t) is therefore a tricritical line. The crossover exponent (j> = 0.8321. 

Figure 3 illustrates the phase-plane dynamics for Equations (3). Shown for reference are the nullclines defined by the curves /(n, u) = 0 and g u, v) = 0. On these curves u = 0 and v = 0, respectively. The u-nullcline is a straight line and the u-nullcline has a backward N shape. The middle branch u = Uth has slope a and intercept —b. The nullclines intersect at the origin and the system has a fixed point there since there both u = 0 and u = 0. For a > 0 and b > 0 this fixed point is linearly stable, and hence all initial conditions sufficiently close to the fixed point decay directly to it. 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

The KTTS depends upon an absolute 2ero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equiUbrium, together with specification of an interpolation instmment and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

A parabola is the set of points that are equidistant from a given fixed point (the focus) and from a given fixed line (the directrix) in the plane. The key-feature of a parabola is that it is quadrilateral in one of its coordinates and linear in the other. 

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Weak acid with a strong base. In the titration of a weak acid with a strong base, the shape of the curve will depend upon the concentration and the dissociation constant Ka of the acid. Thus in the neutralisation of acetic acid (Ka— 1.8 x 10-5) with sodium hydroxide solution, the salt (sodium acetate) which is formed during the first part of the titration tends to repress the ionisation of the acetic acid still present so that its conductance decreases. The rising salt concentration will, however, tend to produce an increase in conductance. In consequence of these opposing influences the titration curves may have minima, the position of which will depend upon the concentration and upon the strength of the weak acid. As the titration proceeds, a somewhat indefinite break will occur at the end point, and the graph will become linear after all the acid has been neutralised. Some curves for acetic acid-sodium hydroxide titrations are shown in Fig. 13.2(h) clearly it is not possible to fix an accurate end point. 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

Modern temperature scale proposed by G. Fahrenheit, defined by a thermometer, a law and three fixed points. Fahrenheit s thermometer was a mercury-in-glass one. Thermal expansion versus temperature was assumed linear. Three fixed points were defined 0°F temperature of a mixture of water, ice and ammonium chloride 32°F temperature of melting ice 96°F temperature of human body... 

Ellingsrud and Strpmme have constructed the cell decomposition of P using the following results of [Bialynicki-Birula (1),(2)]. Let X be a smooth projective variety over k with an action of the multiplicative group Gm. We will denote this action by Let x X be a fixed point of this action. Let T x C Tx,x be the linear subspace on which all the weights of the induced action of Gm are positive. 

The problem can be solved using the successive linearization technique until convergence is achieved. The fixed point in the iteration is denoted by z. It is the solution of (9.24) and satisfies... 

We define the linear growth rate Vg as the linear velocity of displacement of a crystal face relative to some fixed point in the crystal. vg may be known as a function of c and c , derived from the theory of transport control, and as a function of c and cs as well, derived from the theory of surface control. Then c may be eliminated by equating the two mathematical expressions... 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

Let us assume that the auxiliary dynamical system is acyclic and has only one attractor, a fixed point. This means that stoichiometric vectors form a basis in a subspace of concentration space with — 0. For every reaction A,- A the following linear operators Qu can be defined ... 

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