Phase-plane analysis

Phase-plaiie Analysis.—For compr ensive accounts of phase-plane analysis the reader is referred elsesdiere. Here we wish to hi dili t the features particularly relevant in a treatment of chemical systems. The case of two variables (one of which may be temperature) is considered in detail for it is firom this syston that the greatest information may be extracted. In the transition to even three variables much qualitative knowledge is lost. Consider two coufded differential equations of the form 

This equation is often termed the phase-plane (or more correctly, state plane) equation of (17). A direct integration of (18) is not usually possible nevertheless there is a great amount of information to be gained without the necessity of an analytical solution. The gradient, dy/dx, will be defined at every point except when 

The system of equations (20) is linear and its solutions are linear combinations of the terms exp kit and exp kzt, where Ai and Az are solutions of the diaracteristic equation 

The solutions x t) and yit) will approach (x.,. ) monotonically if (22) has negative real roots but will diverge from it if either one of the roots is positive. When both roots are complex the solutions are oscillatory. 

Equation (24) represents the condition for a singularity to be a focus. In inactice 

Example 2.2.8 is reviewed here using the phase plane analysis. For this purpose the independent variable is eliminated and temperature T is solved as a function of C, the concentration. 

Values for parameters are entered here Eq =subs(pars,eq)  

Both the jacket temperature and the feed stream temperature are taken to be 298 K. 

The differential equation is solved below and the numerical simulation is stopped when the denominator becomes zero. The simulation is performed for different initial conditions  

Warning, cannot evaluate the solution further right of 8.7992960, stop condition 1 violated 

A general systematic technique applicable to second-order differential equations, of which (11.31) is a particular example, is that of phase plane analysis. We have seen this approach before (chapter 3) in the context of systems with two first-order equations. These two cases are, however, equivalent. We can replace eqn (11.31) by two first-order equations by introducing a new variable g, which is simply the derivative of the concentration with respect to z. Thus 

Dividing one equation by the other gives the phase plane equation 

This determines how the gradient varies with the concentration, but the integral has no closed form analytical solution. 

The singular points of this system, where d/J/dz = dg/dz = 0, are (P,g) = ( 1,0) and (0,0). In fact these are the points corresponding to the boundary conditions (11.32) and (11.33). Thus the reaction wave can be represented by a trajectory passing through the fi-g phase plane, originating from one singular point (1,0) and entering the other (0,0), as shown in Fig. 11.5. 

The local stability and character of the singular points can be determined by the usual analysis of the eigenvalues of the Jacobian matrix 

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From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

It is the objective of this chapter to study this effect by a model-based analysis. The spatially distributed model used here will be derived in the next section. In a first step, the steady-state behavior of the model will be investigated by a phase plane analysis for the case of potentiostatic operation of the cell. In the second step, numerical bifurcation analysis of the model will be used to study the technically more interesting case of galvanostatic operation. 

One purpose of the investigation of the potentiostatic operation is to motivate the numerical results obtained for the galvanostafic operation mode and presented in the next section. Furthermore, the described phase plane analysis does not provide any information on the stability of the found solutions. The stability analysis will be included in the numerical studies of Section 3.4. 

A modern course on bifurcations, chaos, fractals, and their applications, for students who have already been exposed to phase plane analysis. Topics would be selected mainly from Chapters 3, 4, and 8-12. 

Time scale for the rapid transient) While considering the bead on the rotating hoop, we used phase plane analysis to show that the equation... 

The solutions of x = Ax can be visualized as trajectories moving on the (x,y) plane, in this context called thep/iaseplane. Our first example presents the phase plane analysis of a familiar system. 

In the next few sections we ll consider some simple examples of phase plane analysis. We begin with the classic Lotka-Volterra model of competition between two species, here imagined to be rabbits and sheep. Suppose that both species are competing for the same food supply (grass) and the amount available is limited. Furthermore, ignore all other complications, like predators, seasonal effects, and other sources of food. Then there are two main effects we should consider ... 

Poincare-Lindstedt method) This exercise guides you through an improved version of perturbation theory known as the Poincare-Lindstedt method. Consider the Duffing equation x + x + e.v = 0, where 0true solution x(r, e) is periodic our goal is to find an approximate formula for x(z, ) that is valid for all t. The key idea is to regard the frequency co as unknown in advance, and to solve for it by demanding that x(z,e) contains no secular terms. 

The article by Odell (1980) is worth looking up. It is an outstanding pedagogical introduction to the Hopf bifurcation and phase plane analysis in general. 

Odell, G. M. (1980) Qualitative theory of systems of ordinary differential equations, including phase plane analysis and the use of the Hopf bifurcation theorem. Appendix A.3. In L. A. Segel, ed.. Mathematical Models in Molecular and Cellular Biology (Cambridge University Press, Cambridge, England). 

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

Phase-plane analysis of nonlinear systems, suggested by MacColl (Ml), is based on the characteristics of the equation... 

A succinct appraisal of phase-plane analysis, its possibilities and limitations, is given by Truxal (T3). 

In the model, the appearance of birhythmidty is closely linked to the existence of multiple oscillatory domains as a function of the substrate injection rate, which is taken as the control parameter. In these conditions, the increase or decrease of this parameter from a value corresponding to a stable steady state gives rise to either one of two stable rhythms which markedly differ in period and amplitude. In a very different context, a similar property characterizes neurons of the thalamus. Thalamic neurons are indeed capable of oscillating with a frequency of 6 Hz or 10 Hz when the membrane, initially in a stable resting state, is slightly depolarized or hyperpolarized (Jahnsen LUnas, 1984a,b Llinas, 1988). The phase plane analysis of the biochemical model provides a clue for this behaviour and for the existence of multiple excitability thresholds, which are also observed in these neurons. 

Phosphofructokinase possesses two substrates, ATP and F6P, which it transforms into ADP and FBP. A complete model for this reaction should therefore take into account the evolution of these four metabolites. However, studies carried out in yeast indicate that the couple ATP-ADP plays a more important role than the couple F6P-FBP in the control of oscillations. Indeed, the addition of ADP ehcits an immediate phase shift of the oscillations (fig. 2.8) while the effect of FBP is much weaker (Hess Boiteux, 1968b Pye, 1969). The predominant regulation is thus exerted by ADP. In order to keep the model as simple as possible and to limit the number of variables to only two, which allows us to resort to the powerful tools of phase plane analysis, the situation in which an allosteric enzyme is activated by its unique reaction product is considered (fig. 2.10). This monosubstrate, product-activated. 

Phase plane analysis explanation of the control of oscillations by the substrate injection rate... 

Fig. 2.20. Phase plane analysis explains the effect of the substrate injection rate on glycolytic oscillations. The substrate nullcline is represented for three values of the normalized injection rate v (indicated on the curves labelled 0.1,0.6,1.8, in s" ). The product nullcline is also represented this curve does not depend on the value of v. When the steady state, located at the intersection of the two null-clines, lies in a region of sufficiently negative slope on the product nullcline, this state is unstable and the system evolves towards a limit cycle. Such a limit cycle is represented by a dashed line for the intermediate value of the substrate input V (Venieratos Goldbeter, 1979).

Phase plane analysis thus readily accounts for the main experimental observation on the control of glycolytic oscillations by the substrate injection rate. Below the lower critical value of the substrate injection rate v, a stable steady state is estabUshed, corresponding to a low level of reaction product and to an enzyme predominantly in the inactive T state. Above the higher critical value of v, the stable steady state is associated with a higher level of product and with an enzyme predominantly in the active state R. Sustained oscillations, in the course of which the enzyme switches back and forth between the R and T states, occur in the range delimited by the two critical values of the substrate input. 

The linear stability analysis of these equations, coupled to phase plane analysis, again leads to the instability condition (2.26) obtained for the model in the absence of product recycling. 

A conjecture for the origin of birhythmicity, based on phase plane analysis... 

Phase plane analysis indicates that the two limit cycles possess different sensitivities toward perturbations. It is much easier to pass from the small limit cycle to the large one than to achieve the reverse transition. This differential sensitivity results from the relative sizes of the attraction basins of the two cycles. Moreover, to pass from the large cycle to the small one, the quantity of substrate must be sufficient to cross the border defined by the unstable trajectory, but not so large so as to avoid bringing the system across the basin of the small cycle, into the other side of the attraction basin of the large cycle in such a case, the perturbation would only cause a phase shift of the large oscillations. In... 

A last type of dynamic phenomenon introduced by the recycling reaction is that of a multiplicity of oscillatory domains as a function of a control parameter. This phenomenon is apparent in the bifurcation diagram of fig. 3.6h. Here again, the interest of the phenomenon stems from its relationship to the behaviour of certain neurons the model provides a straightforward explanation for the neuronal behaviour in terms of phase plane analysis. 

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