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+ ry" + y3 generate, create pdf-417 2d barcode none on java projects Visual Studio Development Language = b cos t,. 3. AUTONOMOUS SYSTEMS FIGURE 27. A strange attractor for the Lorenz system. 10 5. FIGURE 28. A graph of x for the strange attractor. where y is the fl ux in the inductor. Of course, this is a second-order differential equation, but it is nonautonomous, so it tranforms into a threedimensional autonomous system. Let x = t, y = y, and z = y":.

bcosx - y3 - rz,. where y is the fl ux in the inductor. For this example, we will be content to look at some numerical solutions of this system in the y, z-plane. First, choose r = .

1, b = 12 and initial conditions x = 0, y = 1.54, z = O..

3.7. THREE-DIMENSIONAL SYSTEMS 15 10. -5 -10. FIGURE 29. Sensitive dependence on initial conditions. FIGURE 30. A limit cycle for the Lorenz system. Figure 33 shows a solution that is periodic in the y, z-plane (but not in three dimensions). A slight change in the initial value of y from 1.54 to 1.

55 sends the solution spinning into what looks like a chaotic orbit (see Figure 34, in which the orbit is plotted for 300 ::; t ::; 400). Figure 35 is a graph of y versus t for this solution on an initial interval. Note that the solution first appears to be settling into a periodic orbit, but around t = 40 the regular pattern is lost and an erratic pattern of oscillation emerges.

Alternatively, we can retain the original initial conditions, and change the value of the parameter b from 12 to 11.9 to obtain an intricate orbit (see Figure 36)..

3. AUTONOMOUS SYSTEMS FIGURE 31. The first doubling of the period. FIGURE 32. The second doubling of the period. This system displ pdf417 2d barcode for Java ays a wide variety of periodic solutions. For example, if we reduce the parameter b to 9, while keeping r = .1 and the original initial conditions, we obtain the elaborate periodic solution in Figure.

3.7. THREE-DIMENSIONAL SYSTEMS FIGURE 33. An orb barcode pdf417 for Java it that is periodic in y and y"..

FIGURE 34. A chaotic ( ) orbit for a nonlinear oscillator. 37. Now increase b to 9.86 to find a doubling of the period (Figure 38).

This appears to be another example of the period doubling route to chaos. Another example of the period doubling phenomenon can be obtained by setting b = 12 and then choosing r = .34 (periodic), r = .

33 (period doubles), and r = .318 (period doubles again). There has been an explosion of interest and research activity in the area of complicated nonlinear systems in the last 25 years, leading to a host of new ideas and techniques for studying these problems.

A thorough discussion of these would fill several large volumes! Some of the important terms are Smale horseshoes (chaos producing mechanisms), discrete dynamics (nonlinear dynamics in discrete time), fractals (complicated sets with nonintegral dimension), Liapunov exponents (a measure of contraction of a set of or bits near a strange attractor), and shadowing (existence of real. 3. AUTONOMOUS SYSTEMS -2 -4. FIGURE 35. A plot of y versus t for a nonlinear oscillator. FIGURE 36. Another chaotic ( ) orbit for a nonlinear oscillator. orbits near compu ted orbits). Here is a partial list of the fundamental references on nonlinear dynamics: Guckenheimer and Holmes [16], Wiggins [51], Devaney [11], and Peitgen, Jurgens, and Saupe [38]..

3.8 Differential Equations and Mathematica In this section, we will indicate briefly how a computer algebra system can be used to find solutions of elementary differential equations and to generate information about solutions of differential equations that cannot be solved. Although we restrict the discussion to Mathematica, version 4.0, other popular computer algebra systems such as Maple and MATLAB have similar capabilities.

. The basic command Java PDF-417 2d barcode for solving differential equations is DSolve. For example, to solve the logistic differential equation (see Example 1.13), we enter.

3.8. DIFFERENTIAL EQUATIONS AND Mathematica FIGURE An elaborate periodic soution of a nonlinear oscillator. FIGURE A doubling of the period. DSolve [x"[t] ==r *x [t] * (1-x[t] /K), x [t], t]. Mathematica responds with Here e[l] stands for an arbitrary constant, and we have the general solution of the logistic equation. If we want the solution that satisfies an initial condition, we simply supply Mathematica with a list of the differential equation and the equation for the initial value:. DSolve[{x"[t]==r* swing PDF-417 2d barcode x[t]*(1-x[t]/K), x[O]==xO}, x[t], t].
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