NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-2 HOMOCLINIC BIFURCATIONS

被引:126
作者
CHAMPNEYS, AR
KUZNETSOV, YA
机构
[1] RUSSIAN ACAD SCI,INST MATH PROBLEMS BIOL,PUSHCHINO 142292,RUSSIA
[2] CTR WISKUNDE & INFORMAT,DYNAM SYST LAB,1090 GB AMSTERDAM,NETHERLANDS
来源
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS | 1994年 / 4卷 / 04期
关键词
D O I
10.1142/S0218127494000587
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
A numerical procedure is presented for the automatic accurate location of certain codimension-two homoclinic singularities along curves of codimension-one homoclinic bifurcations to hyperbolic equilibria in autonomous systems of ordinary differential equations. The procedure also allows for the continuation of multiple-codimension homoclinic orbits in the relevant number of free parameters. A systematic treatment is given of codimension-two bifurcations that involve a unique homoclinic orbit. In each case the known theoretical results are reviewed and a regular test-function is derived for a truncated problem. In particular, the test-functions for global degeneracies involving the orientation of a homoclinic loop are presented. It is shown how such a procedure can be incorporated into an existing boundary-value method for homoclinic continuation and implemented using the continuation code AUTO. Several examples are studied, including Chua's electronic circuit and the FitzHugh-Nagumo equations. In each case, the method is shown to reproduce codimension-two bifurcation points that have previously been found using ad hoc methods, and in some cases, to obtain new results.
引用
收藏
页码:785 / 822
页数:38
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