Approximate Periodic Solution

Suppose that the system (1.1) possesses a periodic solution. Then, in order to construct this solution, one should compute the sequence of functions x (f, x given by (2.3) and find the point Xq through which this solution passes at the initial time t = tQ = 0. When finding x (r, xq), one must integrate different pericxlic functions, and this can be (ione by employing the following computation scheme. 

These scheme is quite convenient for calculation, because for all xo, only trigonometric polynomials are to be integrated in order to find x] (t). 

We assume that the function Jc (t) is the mth approximation (calculated according 

The determination of the point co, through which the periodic solution passes at t = o = 0, is a quite complicated problem. However, in some special cases this problem can be solved easily. One of these cases is given by the following theorem. 

Theorem 1.5. (Martinyuk and Samoilenko, 1967a). Suppose that the right-hand side of a system 

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Two comments (1) This exercise shows that the Duffing oscillator has a frequency that depends on amplitude w = 1-i- a -I-O( ), in agreement with (7.6.57). (2) The Poincare-Lindstedt method is good for approximating periodic solutions, but that s all it can do if you want to explore transients or nonperiodic solutions, you can t use this method. Use two-timing or averaging theory instead. 

We want to find an approximate periodic solution of the system (1.1) in the form of a trigonometric polynomial... 

Finally, the approximate periodic solution of the limit cycle type is obtained in the form... 

Fig. 5 illustrates a peculiar kinetic phenomenon which occurs when an initially disordered alloy is first annealed at temperature T corresponding to area b in Fig. 1 and then quenched to the final temperature T into the spinodal instability area d antiphase boundaries "replicate , generating approximately periodic patterns. This phenomenon reflects the presence of critical, fastest growing concentration waves under the spinodal instability (the Calm waves ). Lowering of the temperature to T < T results in lowering of the minority concentration minimum ("c-well ) within APB, while the expelled solute atoms build the c-bank adjacent to the well . Due to the... 

Finally, in the first approximation the amplitude (i.e., the radius vector) p0 of the singular point gives the radius of the periodic solution (which is a circle in the first approximation). 

Hyclic elimination method. We now focus the reader s attention on periodic solutions to difference schemes or systems of difference schemes being used in approximating partial and ordinary differential equations in spherical or cylindrical coordinates. A system of equations such as... 

In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

In order for an undamped periodic solution to exist it is necessary that the constant term in this equation and the coefficient of tf> be equal to zero, so that, to the first approximation,... 

Fig. 20 The forcing amplitude as above which the periodic solution A i is linearly stable. The solid line is the approximate analytical result according to (59) and the solid circles are obtained numerically. The critical amplitude ac above which the evolution of the initially homogeneous system after the quench is locked to the 2%/q-periodicity of the forcing (solid squares, obtained numerically). The results are given for e = 1...

It must be stressed that the hypotheses of Corollary 5.2 give sufficient, but not necessary, conditions for the existence of a positive periodic solution possessing strong stability properties. Furthermore, since the singlepopulation periodic solutions Eft) and 2(0 are not explicitly computable, as the corresponding rest points were in Chapter 1, it does not seem possible to obtain explicit formulas for A,2and A21. However, these crucial Floquet exponents can be easily approximated numerically. One must... 

It s time to change gears. So far in this chapter, we have focused on a qualitative question Given a particular two-dimensional system, does it have any periodic solutions Now we ask a quantitative question Given that a closed orbit exists, what can we say about its shape and period In general, such problems can t be solved exactly, but we can still obtain useful approximations if some parameter is large or small. 

We conclude with a few comments about the validity of the two-timing method. The rule of thumb is that the one-term approximation Xq will be within 0(e) of the true solution x for all times up to and including t O( /e), assuming that both x and Xq start from the same initial condition. If x is a periodic solution, the situation is even better Xq remains within 0(e) of x for all t. 

Find the approximate relation between amplitude and frequency for the periodic solutions of x - exx + x = 0. ... 

Hands on the functional equation) The functional equation g(x) = ag xja) arose in our renormalization analysis of period-doubling. Let s approximate its solution by brute force, assuming that g(x) is even and has a quadratic maximum at x = 0. 

To begin the study of motions in the vicinity of a periodic solution, we use the linear equations of variation, which are the linear approximation to the equations of motion for the differences between the nearby motion and the periodic solution. Suppose that we are investigating motions in the vicinity of the periodic solution... 

The general procedure to be followed is to put each of these equal to zero, and so solve for an approximation for the fii, to first order in e, and then proceed by successive approximation, using the complete expressions for the fii, to obtain expressions for the fii in powers of e, to correspond to a periodic solution of the full perturbed motion. This could be done, taking r = 0, and so obtaining a solution of period T, provided that the Jacobian determinant,... 

To summarize, there exists for = 0 a set of amplitude-synchronous periodic solutions for <5 > 2r]. We believe that approximate chaotic synchronization of intensities occurs as a result of further bifurcations of these solutions when cp is close to 0. As we will see in Sec. 6.4.7 there is a complicated dynamical mechanisms leading to the destabilization of selfpulsations. An example of such chaotic motion is shown in Fig. 6.4.6. 

In the supercell approach, the defect is instead enclosed in a sufficiently large unit cell and periodically repeated throughout space. A common problem with both approaches is the availability of high-level quantum-mechanical periodic solutions, because, as already mentioned, it is difficult to go beyond the one-electron Hamiltonian approximations (HF and DFT), at present. 

Proof. Assume that Xqs D-M (0/2. Then the first approximation to the periodic solution of (4.12) is given by the formula... 

As / -> oo and m o, the set tends to the set of the initial values of periodic solutions. Therefore, each point Xi e can be chosen as the mth approximation to the initial value Xq of the periodic solution, and the accuracy of this approximation is given by the inequality... 

Thus, the problem of existence of periodic solutions to the system of difference equations (11.1) is reduced to the problem of existence of zeros of the function S(xo). In the general case, it is impossible to rind the limiting function x ( q) of the sequence (11.7), and consequently, it is impossible to determine S(Xq). Therefore, by using the approximate solutions x xq), we calculate the function... 

Bilateral Approximations to Periodic Solutions of Systems with... 

Section 12 Bilateral Approximation to Periodic Solutions of Systems with Lag 65 this solution is unique and, consequendy, that the following equality holds... 

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