By X. Wang, D. M. Zhu, M. A. Hang, Dingjun Luo

ISBN-10: 9810220944

ISBN-13: 9789810220945

Dynamical bifurcation conception is worried with the alterations that ensue within the worldwide constitution of dynamical platforms as parameters are diversified. this article makes contemporary learn in bifurcation idea of dynamical platforms obtainable to researchers drawn to this topic. particularly, the proper effects got by way of chinese language mathematicians are brought in addition to a number of the works of the authors that can now not be well known. the point of interest is at the analytic method of the speculation and strategies of bifurcations. The ebook prepares graduate scholars for additional research during this region, and it serves as a prepared reference for researchers in nonlinear sciences and utilized arithmetic.

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**Additional info for Bifurcation Theory and Methods of Dynamical Systems (Advanced Series in Dynamical Systems 15) **

**Example text**

8) Consider the Poincare map P(r, A) of the limit cycle La : x 2 + y2 along the ray 0 = from the origin, ° = 1 = {(r, 0) : r > 0,0 = O}. ~ We have the successive function d(r, A) = P(r, A) - r. The bifurcation diagram is given by the graph of the relation d(r, A) = 0, that is, (r2 - 1)2 - A = in the (A, r)-plane. The points rt = J1 ± v0.. are fixed points of the Poincare map. sin t, with parameter A. For A = 0, there is a multiple two limit cycle La represented by (xo(t), Yo( t)) = (cos t, sin t).

9. We will meet specific systems which display these behavior later. 3. Codimension and Unfoldings 27 Fig. 9 play an important role in the study of global behavior of dynamical systems. The bifurcations mentioned above will be discussed in more detail in Chapters 2 and 3 for two dimensional systems. In the higher dimensional cases, there might be degenerate critical point with both zero and pure imaginary eigenvalues and more complicated bifurcations might occur, such as bifurcation to invariant torus, which will be discussed in Chapters 4 and 5.

1. Main theorems on center manifolds As mentioned in Sec. 2) there are invariant su bspaces ES, EU, and EC, corresponding to the eigenspaces spanned by eigenvectors which in turn correspond to eigenvalues with negative, positive, and zero real parts respectively (resp. for maps, with modulus less than, greater than, and equal to one). If we suppose that EU = 0, then we find that any orbit of the system will decay exponentially to E C as t --t +00. Thus for nonlinear systems, if we are interested in long-time behavior, we need only to study systems restricted to E C • That is one of the main reason to introduce the concept of center manifolds.

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