Vector field on the line

We think of t as time, x as the position of an imaginary particle moving along the real line, and x as the velocity of that particle. Then the differential equation x = sin x represents a vector field on the line it dictates the velocity vector X at each x. To sketch the vector field, it is convenient to plot x versus x, and then draw arrows o n the x-axis to indicate the corresponding velocity vector at each X. The arrows point to the right when x > 0 and to the left when x < 0. 

As we ve seen in Chapter 2, the dynamics of vector fields on the line is very limited all solutions either settle down to equilibrium or head out to . Given the triviality of the dynamics, what s interesting about one-dimensional systems Answer Dependence on parameters. The qualitative structure of the flow can change as parameters are varied. In particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in the dynamics are called bijurca-tions, and the parameter values at which they occur are called bifurcation points. 

So far we ve concentrated on the equation x = f(x), which we visualized as a vector field on the line. Now it s time to consider a new kind of differential equation and its corresponding phase space. This equation,... 

Actually, we ve seen this example before— it s given in Section 2.1. There we regarded X = sin X as a vector field on the line. Compare Figure 2.1.1 with Figure 4.1.1 and notice how... 

Of course, there s no problem regarding 0 = 0 as a vector field on the line, because then 0 = 0 and 0 = 2zr are different points, and so there s no conflict about how to define the velocity at each of them. 

In practice, such vector fields arise when we have a first-order system 0 = /(0), where /(0) is a real-valued, 27t-periodic function. That is, f d + 2it) = /(0) for all real 0. Moreover, we assume (as usual) that f(Q) is smooth enough to guarantee existence and uniqueness of solutions. Although this system could be regarded as a special case of a vector field on the line, it is usually clearer to think of it as a vector field on the circle (as in Example 4,1,1). This means that we don t distin-... 

In Exercises 2.6.2 and 2.7.7, you were asked to give two analytical proofs that periodic solutions are impossible for vector fields on the line. Review these arguments and explain why they don t carry over to vector fields on the circle. Specifically which parts of the argument fail ... 

We ve already seen a simple instance of hyperbolicity in the context of vector fields on the line. In Section 2.4 we saw that the stability of a fixed point was accurately predicted by the linearization, as long as f x ) 0. This condition is the... 

The trouble with Theorem 7.2.1 is that most two-dimensional systems are not gradient systems. (Although, curiously, all vector fields on the line are gradient systems this gives another explanation for the absence of oscillations noted in Sections 2.6 and 2.7.)... 

Show that all vector fields on the line are gradient systems. Is the same true of vector fields on the circle ... 

The ideas developed in the last section can be extended to any one-dimensional system x = f(x). We just need to draw the graph of /(x) and then use it to sketch the vector field on the real line (the x-axis in Figure 2.2,1),... 

Fixed points dominate the dynamics of first-order systems. In all our examples so far, all trajectories either approached a fixed point, or diverged to °o. In fact, those are the only things that can happen for a vector field on the real line. The reason is that trajectories are forced to increase or decrease monotonically, or remain constant (Figure 2.6.1). To put it more geometrically, the phase point never reverses direction. 

These general results are fundamentally topological in origin. They reflect the fact that X = f(x corresponds to flow on a line. If you flow monotonically on a line, you ll never come back to your starting place—that s why periodic solutions are impossible. (Of course, if we were dealing with a circle rather than a line, we could eventually return to our starting place. Thus vector fields on the circle can exhibit periodic solutions, as we discuss in Chapter 4.)... 

Analyze the following equations graphically. In each case, sketch the vector field on the real line, find all the fixed points, classify their stability, and sketch the graph of x(z) for different initial conditions. Then try for a few minutes to obtain the analytical solution for x(z) if you get stuck, don t try for too long since in several cases it s impossible to solve the equation in closed form ... 

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

Solution This is one of the few time-dependent systems we ve discussed in this book. Such systems can always be made time-independent by adding a new variable. Here we introduce d = cot and regard the system as a vector field on a cylinder d = co, x + x = Asind. Any vertical line on the cylinder is an appropriate section 5 we choose S = (0, x) 0 = 0 mod 2jc. Consider an initial condition on 5 given by 0(0) = 0, x(0) =. Then the time of flight between successive inter-... 

Consider first the case when the external electric field Eo is parallel to the direction of the gravity force, that is, to the vector g. Then the motion of drop S2 relative to drop Si can be considered as planar motion in a meridian plane spherical system of coordinates (r, 6, angle between vector Eo and the line of centers of drops coincides with the angle 0 of spherical system of coordinates. The expressions for electric forces acting on two conducting charged drops have been obtained received in Section 12.3. The components of these forces along axes r and 9 for each drop are... 

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