Quasi-periodic solution

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

A quasi-periodic solution to a system of ODEs is characterized by at least two frequencies that are incommensurate (their ratio is an irrational number) (Bohr, 1947 Besicovitch, 1954). Several such frequencies may be present on high-order tori, but for the two-dimensional forced systems we examine, we may have no more than two distinct frequencies (a two-torus, T2). A quasi-periodic solution is typically bora when a pair of complex conjugate FMs of a periodic trajectory leave the unit circle at some angle , where /2ir is irrational. Such a solution is also expected when we periodically perturb an autonomously oscillating system with a frequency incommensurate to its natural frequency. 

The elimination of secular terms from the power series expansion of the solution is achieved by the method of Lindstedt. The underlying idea is to pick a fixed frequency p, and to look for a quasi-periodic solution with basic frequencies /i and v. This is actually the same thing as looking for a quasi-periodic orbit on an invariant 2-dimensional torus. The process of solution is the following. Write the Duffing s equation as... 

There are many types that represent a set of measure zero in phase space orbits with an original and/or final parabolic escape, orbits with an asymptotic motion to some periodic or quasi-periodic solution, orbits with an oscillating motion of the first type that we have already met in Section 9, orbits open in the past and bounded in the future (complete capture) or inversely, motions leading to a collision of the two point-masses of the binary, etc. and there are also three main types, three types that represent sets of positive measure in phase space ... 

We have determined steady solutions of (2.9). They describe periodic solutions of (2.1). We now wish to describe quasi-periodic solutions of these equations. To do so, we consider equations (2.9) with = pexp(i9), 3] = <5 exp(ij)). Then (2.9) becomes... 

The analysis of the bifurcation equations in this case indicates that in addition to the primary states given by (2.11) and (2.12), there exists a secondary bifurcation to quasi-periodic solutions which satisfies (2.16) with P e a2 P+ bj o). The condition for its existence and its location depends on higher order terms in the bifurcation equations. This analysis is carried out by Erneux and Matkowsky in [6]. Figure 3 exhibits a typical bifurcation diagram of the amplitude as a function of X. 

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