inasynapticallycoupledfhnneuronmodel

inasynapticallycoupledfhnneuronmodel内容摘要：

, then the system (1) undergos a Hopf bifurcation at (0,0,0,0) when 02  0r 0r 00r 0  321* , yyyy 0* y 0)( * yh01 ],0[ 01  0)( *39。 kyh).,2,1,0(,11  jjk and Hopf bifurcation for FHN neuron model with two delay  Now let ,*11   ,02  )0(  wiw be a root of Eq.(2) Then we get .0s i nc o s,0s i nc o s21223222124wEFwEFCwAwwEFwEFDBww(16) Where ],s i n)(c o s)[( *121*12121  wwbbwbbwF ].s i n)(c o s)[( *1212*1212  wbbbbF  Taking square on the both sides of the equations of (14), we get (15) 02)22()2( 222212222342628  FEFEDwCB D wwACDBwBAw (15)  If Eq.(15) has positive root, without loss of generality , we assume Eq.(15) has N positive roots, denoted by。 Notice Eq.（ 12）we get 2,1,0,0s i n),2)a r c c o s ( c o s2(10s i n),2)s( a r c c o s ( c o122222  jwjwwwjwwiiiiiiiiiiji (16)  Define  .Let be the root of Eq.(4)  Satisfying  . By putation, we get }{m in )0(2},...2,1{)0(202 iNii   00 i )()()( 222  ivijij i ww  )(,0)( 221    1202220214002*10021302121502102*10202121402121602)(2102)(2123)(210239。 )2()(s i n]3)(2[)](43[)(c o s)](2[]42)(3[4)))(()2(234))((R e ()(02*102*102*1EwbwbwwwbCbwCbAbbbBwbbAwwbbCbBbwbbBbbAwebbEebbECBAebbE Where  Summarizing the discussions above, we have the following conclusions. 0)(,0])([ 0239。 202221402221602   wbbwbbwE Theorem Suppose that (H), hold and Eq.(14) has positive roots. and have the same meaning as last definition. We get  (1) All root of Eq.(4) have negative real parts for  and the equilibrium of system (2) is asymptotically stable for .  (2) If hold , then system (2) undergos a Hopf bifurcation at the equilibrium E, when . I *11 02)( 0239。 ),0( 022   )0,0,0,0(E0)( 0239。 022   and direction of the Hopf bifurcation  In the previous section, we obtained conditions for Hopf bifurcation to occur when  . In this section we study the direction of the Hopf bifurcation and the stability  of the bifurcation periodic solutions when , using techniques from normal form and center manifold theory. 022  022   We assume 02*1   Letting R  ,022and dropping the bars for simplification ),(dt )(d  tt XFXLtX （ 17）  Where CtXX t  )()(  and 3: RCL 3: RCRF ),()()0( 022*111  tttt XBXBXAXL  (18) Where TTtttt txtxtxtxxxxx ))(),(),(),(())(),(),(),(( 43214321   .0)(3)0(0)(3)0(0)(3)(0)(3)(),(。