By Dongming Wang

This booklet offers the state-of-the-art in tackling differential equations utilizing complex tools and software program instruments of symbolic computation. It specializes in the symbolic-computational facets of 3 types of basic difficulties in differential equations: reworking the equations, fixing the equations, and learning the constitution and houses in their strategies. The 20 chapters are written via top specialists and are dependent into 3 parts.

The ebook is worthy studying for researchers and scholars engaged on this interdisciplinary topic yet can also function a worthwhile reference for everybody drawn to differential equations, symbolic computation, and their interaction.

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**Example text**

5). However, the procedure becomes much slower as the degree of the terms increases. In general, as soon as higher order terms are taken into account, the situation becomes much more complex. However, we shall give one result here where we can say something concrete under some generic assumptions. We apply this result to the quartic system considered in the previous section and show that in fact 17 limit cycles can bifurcate from this center when we consider the quadratic terms. We also show that the strata of symmetric centers C46 can generate 11 limit cycles under cubic perturbations.

Thus we have a weak focus of order k − 1. Furthermore, the rank of the the other L(i) will be equal to k − 1, by hypothesis. Thus we can move away from this curve in a direction which produces k − 1 limit cycles. 2. There is a class of quartic system with 17 limit cycles bifurcating from a critical point. Proof. We calculate the linear and quadratic terms of the ﬁrst 18 Liapunov quantities with respect to a general perturbation of a quartic system which has no 32 Christopher quadratic or cubic terms in the perturbation of y, ˙ so that we have 18 parameters (including λ).

2. 1. For a real planar diﬀerential system, inﬁnity Γ∞ is said to be an isochronous center if the trajectories of the system in the neighborhoods of inﬁnity (the equator) are all closed and their periods are constant. Conditions of Inﬁnity to be an Isochronous Center 39 The following theorem is obvious. 2. For two diﬀerential systems (A) and (B), if there exists a homeomorphism taking inﬁnity of system (A) to the origin of system (B), then the inﬁnity of system (A) is a center (an isochronous center) if and only if the origin of system (B) is a center (an isochronous center).