−1.5 −1 −2/3 0 2/3 1 1 - math.ubc.caward/teaching/m605/every_6.pdf/nc BHs 7:9,o\1?5`

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� �� "!$#&%')(*�"+,#-�.0/21436587:9,;*<>=?9A@?;>/2B?CD9,E FHGJIKF L�MN F:OQPSR T U M2VDMXWY =?583Z=\[23Z9X]_^?5`CD5DB?;a5D^?Eb3dcd=?9�1 ReIfLdF�P MN F:O g U M2Vih2Wcd=?58jk=e[l;>9m3a=?/ncd1&5D1&o\pq^?;>9 M2VDM2V Y =?9r;a9X[23a/21bcd=Hstc09r/2B:<k[l5D1&=HsH36<a9�;a9X36583u583Q<a=\[l<Z[q3 R5D1\jv;>9�[q369X3$wx;>/2Ey3a/2Et9mC8[l;>p29r1?9,p_[n<a5Dz29mz{[2C`^\92|:5`<Z36<>[{s:30/21b<>=?9�1?9,p_[n<a5Dz29r3}<k[lB?CD9rB\;>[21\jk=^?1_<a5DC R~I�R4��I h_� N |Jcd=\9,;>9�[23�[q3 R 7:9�j,;a9X[23a9�3mwx;a/qE�36/qEt9-C8[l;>p29 @4/_365`<a5Dz29tzn[lCD^?9q|�5�<36<>[{s:3*/21�<a=?9A@J/q3a5`<a5Dz29�B?;k[l1\jk=�^?1_<a5DC R�IfLmR4� V$� w"c09ACD9v<R�I�RJ�$� M T h � 365D1 U��\T M��6W g U M2V N W<a=\9,1�<>=?9�3a/2CD^:<a5D/21�cd5DCDC�j,/21_<a5D1H^?/2^\3aCDsSpq/�<a=?;>/2^?pq=K<a=?583�=_s:36<a9�;a9X36583&CD/H/2@ V Y =\5D3e5D33a=?/ncd1�5D1�o\p2^?;>9 M2Vih:V�� m�v��+,��+v�2')!$(d��$+v#-�� /2<a9m<a=\[l<05D1&o\pq^?;a9 M2Vih |q<a=?9�;a9r5D3*[�3aEt[2C`C�7:9�CD[{s B49,<}c09�9,1e<a=?9�7:5D3>[l@\@49X[l;k[l1\j,9�/lw <>=?92|wx/2;Z9,¡:[2Et@?C`9q|:@4/_365`<a5Dz29�36<>[lB\C`9A36<>[n<>5`/q1\[l;>sb36/qC`^?<a5D/21)[l1\7&<a=?9�<>5Èt9�cd=?9,1�<a=\9�36/qC`^:<>5`/q1Et/nz29X3�<>/e<>=?9t1?9,p_[n<a5Dz29 3}<k[lB?CD9t3}<k[n<a5D/214[l;>s�3a/2CD^:<a5D/21 V Y =?9 /2B:¢}9Xj�<�/lwQ<>=?583�3a9�jv<a5D/21�5D3<a/ o\147e<>=?9�[l@\@?;a/{¡:5DEb[n<a9�CD9,1?p2<a=)/lw£<a=\[l<m7:9,C8[{s V¤ 9,< R¥I¦R4�$P¨§ ��V�© ¡:@\[l147:5`1\p&9,zq9,;>sq<>=?5D1?pe1?9X[l;>BHs2| FªI«Fq�$P¨§6¬nF_ [21\7 � I�§�®k¯ V° 3d[t;>9�3a^?C`<�|F:G4I�± U F g R W U M2V ²_WI�±$³ ´�µ>¶ ·{µ P¦± ´ ³ ´ µ ¶ · µ U F�L¸F � W P�± ´v´ ³ ´ µ ¶ · µ U F�L¸F � W}¹h P�± · ³ ´�µ>¶ ·{µ U R&L�R � W T�U M2Viº2WM

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StableUnstable» 5`pq^?;>9 M2VDM2¼ Y =?9)3}<k[lB?CD9�[21\7S^?1\36<>[lB\C`9�B?;k[l1\jk=?9X3�/2w�<a=\9�B\5�wx^?;kj,[l<a5D/21¨7:5D[2p2;k[lE�wx/2;9�]_^\[l<a5D/21 M2VDM2V

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steady statessolution» 5`pq^?;>9 M2Vih:¼ Y =?9*36/qC`^?<a5D/21A<>/d<a=?9*9�]_^\[l<a5D/21 MqV`M cd=\9,;>9 R z{[2;a5D9�3½5`1�<a5DE 9*[23£5`1�9�]_^\[n<>5`/q1M2V N V Y =?9�5D1?5`<a58[lC½j,/21\7:5`<a5D/21�<>=\[n<�cZ[23*^\3a9�7�583 F U¾� W IfL �?T M2Vh

� /nc�BHs�7:9,o\1?5`<a5D/21�|4cd=?9,1�<a=\9�3}<k[lB?CD9A3}<k[n<>5`/q1\[l;>se3a/2CD^:<a5D/21�o\;k3}<�¿i7:583a[2@?@49X[l;k3,ÀD| ±)I �3a5`1\j,9dc09qÀ ;>9�36<a5DC`CJ[n<$<a=\9m3}<k[n<>5`/q1\[l;>s�3a/2CD^:<a5D/21 V ¤ /:j�[lCDC`s�[l<u<a=\9�@4/q5`1_<ucd=?9�;a9Z<>=?9m36<>[2B?C`936<>[n<>5`/q1\[l;>s~36/qC`^?<a5D/21Áo4;>36<�¿i7:583a[2@?@J9�[l;k3�À`|$<>=?9ªjv^?;>z29�7:9v<>9,;>Et5`1?9X7ÂBHs ± U F g R W I � 5D3z29�;6<>5Dj�[lC�5D1�<>=?9 R?ÃÄF @\CD[21?92|\[21\7�36/ ± ´ I � V ° 3�[ ;>9�3a^?C�<dc*9�/2B:<k[l5D1F G IÅ± ´v´ ³ ´�µ>¶ ·{µ U F-L�F � W ¹h P¦± · ³ ´ µ ¶ · µ U RbL~RJ� W U M2V ÆqW§ ¬nÇ�® F È IÉ§ ¹ ¬ ± ´v´h F ¹ PÁ§ 6Ê ® ± · ¯ T U M2VÌËlW� 1)/2;k7:9,;*wx/q;Z<a=?9X369�<a/&[2C`C BJ9A/lwÍ<a=\9�3a[2E 9�/q;>7:9�;�|Hc*9�E�^\36<�=\[{z29Î L�ÏAI h Î I M PÁÏ U M2V ÐqW[l147�[q3d[-;>9�3a^?C`<�|?c*9�E�^\36<�=\[{zq9 Î I M{� N | Ï�I�L M{� N V"Ñ 9�<>=?9,1)/qB:<>[25`1�<a=\[l<F È I ± ´,´h F ¹ PS± · ¯ g U M2V ÒqWcd=?58jk=~5D3�[l1~9�]_^\[l<a5D/21�/lwd/q;>7:9�; M2V�Ñ 9&E�^\36<�1?/ncÓ3a/2CDz29 <a=\5D3-9�]_^\[l<a5D/21 V�� wdc09&CD9v<F I�Ô�Õ [21\7 ¯eIÉÖl× |?<a=?9�1�c*9�/2B:<k[l5D1Ô Ö Õ G I ± ´v´ Ô ¹h Õ ¹ PÂ± · Öl× U M2VDM � WÕHGJI ± ´v´ Ô½Öh Õ ¹ P ± · Ö ¹Ô × T U M2VDM2M{W� /nc�c09Ajk=?/H/_369r<a/&3a@49Xjv5`wxs&<>=\[n<± ´v´ Ô½Öh IfL M g ± · Ö ¹Ô I M g U M2VDMXhqWcd=?58jk=�Et9X[l1\3*<>=\[n<Ö�I«L�Ø h± ´v´ ± ·�Ù6Ú O g Ô�IÛØ ² ± ·± ¹´v´ Ù

aÚ O g U M2VDM N W[l147�3a/ Õ_G�I«LdÕ ¹ PS× T U M2VDM,²HW� w"c09�<a=\9,1)C`9,< Õ U × W I�Ü G U × Wa� Ü U × W |4c09�/2B?<>[l5D1�<a=\[l<Ü"G G U × W IÉ×XÜ U × W g U M2VDMXºqW[l147�3a/ Ü U × W I�Ý_Þ ° 5 U × W PÂÝ?�ß 5 U × W g U M2VDM�Æ_WÕ U × W I Ý_Þ ° 5 G U × W PÁÝ?�ß 5 G U × WÝ Þ ° 5 U × W PÁÝ ß 5 U × W T U M2VDM{Ë2WN

� /ncà[23 ��á Lrâ |ZBHsS7:9,o\1?5`<a5D/21�| × á â V � /ncã[q3 × á â | ° 5 U × Wªá � [l147ß 5 U × Wuá â |4[l1\7�36/ Õ U × W�ä}å�æLHL?L áèç ß 5 G U × Wa� ß 5 U × W 5`w Ý -éI �° 5 G U × Wa� ° 5 U × W 5`w Ý I � U M2VDM�Ð_Wä}å�æLHL?L á ç ê × 5`w Ý: éI �L ê × 5`w Ý:dI � U M2VDM�Ò_W� /ncS5`1t/q^?;uj,[q369q|n/2^?;$5D1?5`<a58[lC\j,/21\7:5`<a5D/2143 U <a=4[n<Qc*9d3}<k[l;a<a9�7�wx;>/2E R�I«Lrâ | F IfLrâ W;>9�]_^?5D;a9-<a=\[l<�[23 × á â | Õ U × W E-^\36<�BJ9t1?9,p_[n<>5`zq9 V ° 3A[�;a9X36^\C�<X|�c09bj,/21\j,C`^\7?9-<>=\[n<Ý I � V"Ñ 9�<>=?9,1)/2B?<>[l5D1�<a=\[l<FtIÉFq�ÍPÁ§ ¬ F_ U M2Vih � WIÉF � PÁ§ aÚ O,Ô�Õ U × W U M2Vih:M{WIÉF � PÁ§ aÚ O,Ô�Õ U × W U M2Vih2hqWIÉF � PÁ§ aÚ O ØQ² ± ·± ¹´v´ ÙaÚ O ° 5 G U × W° 5 U × W g U M2Vih N Wcd=?9�;a9 ×�I ¯ Ö U M2Vihn²HWI�Ld¯ªØ ± · ± ´,´h Ù6Ú O U M2Vih2ºqWI�L � § Ç ® Ø ± · ± ´v´h ÙaÚ O U M2VihlÆ_WI�L � § aÚ O Ø ± · ± ´v´h ÙaÚ O T U M2VihqË2W� /ncSc09d9,¡:@49Xj�<$<a=\5D3u<a/ABJ9d5`1Hzn[lCD5D7 cd=?9�1 F B49Xjv/qE 9X3$C8[l;>p29 V ° 3u[�;>9�3a^?C`<�|lc*9d@?;>9�7:58j�<<a=4[n<�<>=?9�36/qC`^?<a5D/21)cd5`CDC£[2@?@?;>/q[qjk=e<>=?9�/2@\@4/_365`<a9�36<>[l<a5D/21\[2;as&B\;>[21\jk=�cd=\9,1 ° 5 U × W I �wx/2;*<>=?9�o\;k3}<Z<a5DEt92|\[l<dcd=?5Djk=�@J/25D1q< F B\C`/nc�3Z^?@ V Y =?5D3Z/:j�jv^?;k3*cd=?9,1 ×rë�L h T N2N Ð?M2MqVìªí�î�ï�ðÂñ�ò�ñ_óQñHô£õÍñHô£öqñ÷H9,< RÏãRJ�0P�§ U cd5�<>=Á1?/ª<>5Èt9�7:9,@J9,1\7?9,1\j,9 W�V Y =?9,1Á9,¡:@\[l1\7 F [l147 � ^4365D1?p FÁIFq�ÍPÁ§ ¬ F_ [l1\7 � IÉ§ ® ¯ V Y =\9,1�|:@J9,;awx/2;>Et5`1?pb[21\[lCDs:36583Z[23ZBJ9vwx/2;>92|± U F g R W IKF L�MN F:OQPÂR U M2VihlÐ_WF G I ± ´v´h U § ¬ FH W ¹ PÂ± · § U M2VihlÒ_W§ ¬nÇ�® F_ È I�§ ¹ ¬ ± ´v´ F ¹h PÂ± · § U M2V N � W²

3a/-<>=\[n< Î L�ÏAI h Î I M Et9�[l143*<a=\[l< Î I M{�2h | Ï�I«L Mn�lh [l1\7F È I ± ´v´h F ¹ PÂ± · T U M2V N M{W÷H/qC`zH5D1?p <a=?583,|\c09�o\1\7�<>=\[n<FtI�F � P�ø h ± ·± ´v´ <>[l1¥ù ¯�ú± ´,´ ± ·h û g U M2V N hqWcd=?58jk=�c*9�9v¡:@J9�j�<d<>/bB49Azn[2C`587�^?1_<a5DC¯eë üý h ± ´,´ ± · U M2V N2N W� ë üý h ± ´,´ ± · �ÂMê § U M2V N ²HW° 3Í[�;>9�3a^?C`<�|{9,zq9,1�cd=\9,1�c*903a@49�9�7A^\@�<a=\5D3£@?;>/Hj,9�3>3�BHs�C`9,<6<>5`1?p R$L�R � 5D1\j,;a9X[23a9uC`5D1?9X[l;>C`scd5`<a=b<>5Èt92|_<>=?5D301\[l<a^?;k[lCJ7:9,C8[{s j�[l^\3a9�3u<a=?9�3a/2CD^:<a5D/21t<>/�<>[2þ29m[�cd=?5`CD9�B49,wx/2;>9�5�<*Et/nz29�3/nz29�;*<a/ <a=?9A/q@?@J/q3a5�<>9�36<>[l<a5D/21\[2;asbB\;>[21\jk= Vÿ��6ÿ �� w"c09Ajv/q1\3a5D7:9�;Z<a=?9A9,¡?[lEt@?CD9rwx;>/2Ey<a=?9A5D1_<a;>/H7?^\j�<>5`/q1 U�� M2VDMXW |:<a=4[n<FHGJIKF L�MN F:OQPSR g U M2V N ºqW<a=\9,1 ± · I M [l1\7 ± ´v´ I«L h F � I h:V » ;>/2Ey9X]_^\[n<>5`/q1\3 M2Vih N [l1\7 MqV h_Ë c*9�=\[{z29�<a=4[n<FtIfL M PÁ§ aÚ O ° 5 G U × W° 5 U × W g U M2V N Æ_W×rIfL � § 6Ú O T U M2V N Ë2WÑ 9�9v¡:@J9�j�<�[t<a;k[l14365`<a5D/21�<a/e/Hj�jv^?;rcd=?9,1 ° 5 U × W I � |4cd=\5Djk=ª583�cd=?9�1 ×�ë�L h T N2N Ð?M2M |/2; � ë�§ Ç aÚ Od� h T NqN Ð?MqM U M2V N Ð_WY =?583-[2@?@?;>/{¡:5Èb[n<>9&36/qC`^?<a5D/21~583-3a=?/ncd1~5D1~o\p2^\;a9 MqV ² wx/2; §tI �\T � M |"5`1Sjv/2Et@\[2;a583a/21cd5`<a=�<a=?9A1H^?Et9�;a58j,[2C�36/qC`^:<>5`/q1�[2CD3a/t36=?/ncd1�5D1�o4p2^?;>9 M2V N Vÿ�� "!# ��$�%�&�%��'(�)vô+*-,{í½õ/.Íö0*Xï�íJô° Et/:7:9,C�wx/q;d<a=?9A3a@?;>^\jv9�B?^\7?c0/q;aEy5D1:wx9X3}<k[n<a5D/21)5831�21 � I 3 2 Ø M L 2 4 Ù L 5 U 2 W U M2V N Ò_Wº

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Bifurcation DiagramSolution» 5`pq^?;>9 M2V N ¼ Y =\901H^?Et9,;>58j,[lC_36/qC`^?<a5D/21A<>/r9X]q^4[n<a5D/21 MqV`M cd5�<>= §ZI �?T � M [l1\7 ReI�RJ�{P�§ ��VY =?9�5D1?5�<>5D[2C½jv/q1\7:5`<a5D/21)583 FtI«L h [l< � I�L MXº � V

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Bifurcation DiagramSolutionApproximation» 5`pq^?;>9 MqV ²4¼ °76 /H/2Et9�7 Ã 5D1�zq9,;k365D/21�/lw½<>=?9A1H^?Et9,;>5Dj�[lC 3a/2CD^:<a5D/21�<a/t9�]_^\[l<a5D/21 M2VDM cd5�<>=§0I �?T � M [21\7 R�I�R � P�§ � |H[2305`1eo\p2^?;>9 MqV N V Y =\5D305D3Z3a=?/ncd1b5D1�jv/qEt@\[l;>5D3a/21bcd5`<a=e<a=?9[l@\@?;a/{¡:5DEb[n<a9�3a/2CD^:<a5D/21�p25Dz29�1�BHs&9X]q^4[n<a5D/21 M2V N Æ\VÆ

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» 5`pq^?;>9 M2Viº:¼ Y =?9)3}<k[lB?CD9�[21\7S^?1\36<>[lB\C`9�B?;k[l1\jk=?9X3�/2w�<a=\9�B\5�wx^?;kj,[l<a5D/21¨7:5D[2p2;k[lE�wx/2;9�]_^\[l<a5D/21 M2V ²_h wx/2;�8 I �\T ºqº:Vcd=?9�;a9 2 5D3Z<>=?9A365 6 9A/2w½<>=?9A@4/q@?^?C8[n<>5`/q1�| 3 583*<a=?9AEb[l¡H5DE�^\EÅ;k[n<a9�/lw"p2;>/ncZ<a=�/lwÍ<a=?9@J/2@?^?C8[n<>5`/q1�| 4 5D30<a=\9�j,[l;>;>s_5D1?p j,[l@4[2jv5`<}st/lw£<a=?9A9�1_zH5D;a/q1?Et9,1_<d[l1\7 5 U 2 W 5830<a=?9A;k[n<>9/lw"7:9�[l<a= U @\;a9X7?[n<>5`/q1�;k[n<>9 W /lwÍ<a=?9A@J/2@\^?CD[l<a5D/21 V Y =?9A@\;a9X7?[n<>5`/q1�;k[n<>9�583Zp25Dz29�1�BHs5 U 2 W I 9 2 ¹: ¹ P 2 ¹ g U M2V ² � Wcd=?58jk=�pq/H9�3*<a=\;a/q^?p2=�<a=?9A/q;a5Dp25D1�[21\7�=\[23d[ Eb[n¡:5DE�^?E�[23 2 á â /2w 9&VQ� wÍc*9�C`9,<; I M: 2 g ¯&I<9: � g 8 I :9 3 g = I M: 4 U M2V ²\M{W<a=\9,1)/2^?;d@\;a/qB?C`9�Ey;>9�7:^\j,9�3*<>/-<>=?9A1?/q1 Ã 7?5Èt9,14365D/21\[2C @?;>/2B?CD9,E1 ;1 ¯ I 8 ;�> M L ; =@? L ; ¹M P ; ¹ T U M2V ²_hqWY =?9�9�]_^?5DC`5DB?;>5`^\Ey@4/q5`1_<k3*wx/2;*<a=?583d[l;>9�36=?/ncd1ewx/2;A8 I �?T º2º 5D1eo\pq^?;>9 MqV º?V » /2;A8�o?¡:9X7c*9�zn[2;as = [qj,j,/2;k7:5`1\p�<>/= I�Ø = $P = ¹h Ù P hCB Ø = ¹

L = h Ù 365D1 U § �6W g U M2V ² N Wcd=?9�;a9 = [21\7 = ¹ [2;a9A<>=?9�CD/nc*9,;r[21\7)^?@?@J9,;�BJ/2^?1\7\3r/q1)<a=?9-;a9�p25D/21ªcd=?9�;a9�<a=?9�;a9-5D3[l1¸^?1\36<>[2B?CD9 9X]_^?5`CD5DB?;a5D^?E @4/q5`1_< VtÑ 9t<a=?9�1�/qB:<>[25`1¸<>=?9b36/qC`^:<>5`/q1\3A3a=?/ncd1¥5D1¸o\p2^?;>9�3M2V Æ [l1\7 M2VÌËHVED B4369�;azq9�<>=\[n<�9�z29,1¸<>=?/2^?pq=�5D1¸o\p2^\;a9 M2VÌË <a=?9�;a9&[2;a9t;>9,pq5`/q1\3�/lw*<a5DE 9Ë

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steady statessolution» 5`pq^?;>9 M2V Æ?¼ Y =?9&3a/2CD^:<>5`/q1�<>/9�]_^\[l<a5D/21 M2V ²_h wx/q;F8 I �?T º2º [l147�cd=?9,;>9 = zn[l;>5`9X3A5D1<a5DEt9�[q3*5`1�9�]_^\[n<>5`/q1 M2V ² N V&G 9,;>92| §*I �?T � M [l1\7&<a=?9�5D1?5�<>5D[2C jv/2147:5�<>5`/q1&<>=\[n<Zc*[q3Q^4369X7583 ; U�� W I M � V5D1�cd=?58jk=&<>=?9�;a9�p25D/21&/2w£CD/ncÉ@J/2@?^?C8[n<>5`/q1e583Z1?/ C`/q1?p29�;*36<>[2B?C`9q|?7:^?9r<a/ <a=\9r<>5Èt9A7:9�CD[{s;>9�]_^?5D;a9X7 <a/�;>9�[qjk=t<a=?9r;a9�p25D/21b/2w�=?5`pq=&@J/2@?^?C8[n<>5`/q1�|q<a=?9�36/qC`^?<a5D/21b583uwx/q;>j,9�7t<>/�;>9,Eb[l5D15D1�<>=?9ACD/nc Ã @J/2@?^\CD[l<a5D/21�36<>[l<a9 VìH.ÍðJIQñ�,bí�K�L�í�MN./*�ï¾í4ô&OY =?9�9�]_^?5DCD5`B?;>5D^?Eb3d/lwÍ9�]_^\[n<>5`/q1 MqV ²Hh /Hj�jv^?;dcd=\9,1

� I 8 ;�> M L ; =@? L ; ¹M P ; ¹ U M2V ²2²HW8 > M L ; =P? I ;M P ; ¹ T U M2V ²_ºqWß s&CD9v<6<>5`1\p FtI ;M P ; ¹ I 8 > M L ; =(? g U M2V ²qÆ_Wc*9�j�[l1�@?CD/l<d<a=\9�3a9�[21\7�369�9Acd=?9,;>9�<a=?9�s&5`1_<>9,;k369Xj�< V ÷:9,9�o\p2^\;a9 M2V Ð?V » /2;Z<a=\9Az{[2C`^\9�/2w8&3a=?/ncd1�|4<a=?9�;a9t[l;>9A<}c*/ezn[lCD^?9�3r/lw = |�CD[2B49�C`CD9�7 = [l1\7 = ¹ 5`1¥<a=?9�o\p2^\;a9q|�3a^\jk=ª<>=\[n<<a=\9,;>9 [2;a9-<}c0/�9�]_^?5DCD5`B?;>5D^?Eb3 V ° 3A[�;a9X36^\C�<X|½[eB\5�wx^?;kj,[l<a5D/21�/Hj�jv^?;k3�[n< = [21\7 = ¹ wx/2;<a=\5D3�zn[lCD^?9t/lwA8 V » /2; = RQ = Q = ¹ |½<a=?9�;a9b[2;a9-<a=?;>9,9t9X]q^\5`CD5`B\;a5D^?Eb3,|£[l1\7¥wx/2;A9�5�<>=?9,;Ð

0 100 200 300 400 5000

5

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steady statessolution» 5`pq^?;>9 M2VÌËH¼ Y =?9&3a/2CD^:<>5`/q1�<>/9�]_^\[l<a5D/21 M2V ²_h wx/q;F8 I �?T º2º [l147�cd=?9,;>9 = zn[l;>5`9X3A5D1<a5DEt9�[q3*5`1�9�]_^\[n<>5`/q1 M2V ² N V&G 9,;>92| §*I �?T � º [l1\7&<a=?9�5D1?5�<>5D[2C jv/2147:5�<>5`/q1&<>=\[n<Zc*[q3Q^4369X7583 ; U�� W I M � V

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0 3 k_1 10 k_2 200

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One IntersectionTwo IntersectionsThree Intersections

» 5`pq^?;>9 MqV Ð\¼ Y =?9&jv^?;>z29b[21\7¥CD5D1?9twx;a/qE�9X]q^4[n<a5D/21 M2V ²qÆ |�cd5`<a=S8eo\¡H9X7�[21\7�7:5UTJ9�;a9�1q<zn[lCD^?9�3Z/2w = V Y =?583�36=\/nc�3Q<>=?9A@J/q3>365DB?5DC`5`<}s&wx/q; M | h /2; N 9�]_^?5DCD5`B?;>5D^?EÅ@J/25D1_<>3 V= Q = /q; =WVX= ¹ |$<>=?9,;>9e583 /21?CDs�/21?9�9�]_^?5DC`5DB?;>5`^\E V�� 1S/2;k7:9,;�<a/�7:9v<>9,;>E 5D1?9�<a=?9;>9,p25D/21t5D1t<a=?9�8 Ã = @?CD[21?9�5D1&cd=?58jk=t<a=?9�;a9r[l;>9Z<a=?;>9,9�9X]_^?5`CD5DB?;a5D^?Eb3�|qc*9d<a=\9,1e3a9�[2;>jk= wx/2;<a=\9Az{[2C`^\9�3d/2w&8t[l1\7 = 3a^\jk=�<a=4[n<d<a=\9AC`5D1?9�583Z<>[21?p29�1_<*<>/t<a=?9-jv^?;>z29 V Y =?5D3�cd5`CDC wx/2;>E<a=\9AB4/q;>7:9�;Z/lwÍ<a=?9A;>9,pq5`/q1e5D1)]_^?9X3}<>5`/q1 V� w£<>=?9�C`5D1?9A583*<>[21?p29�1q<*<a/ <>=?9�jv^?;>z29�[n<�3a/2Et9�@4/q5`1_< U ; Þ g FlÞ W |H<a=?9�1Y8-[l147 = E-^\3}<3>[n<a5836wxs ; ÞM P ; ¹Þ I 8 > M L ; Þ=H? g U M2V ²HË2WM L ; ¹ÞU M P ; ¹Þ W ¹ I 8= T U M2V ²qÐ_Wß se3a/2CDz_5D1?p wx/2;Z<>=?9�3a92|:c*9A/2B?<>[l5D1�<a=?9�j,^?;azq9r@4[l;k[lEt9v<>;a58j,[2C`CDsb5`1�<>9,;>Et3Z/2w ; Þ ¼8 I h ; OÞU ; ¹ P M{W ¹ g = I h ; O; ¹ L M T U M2V ²qÒ_Wß s&p2;k[l@\=?5`1\p�<>=?5D3�|?c*9�/2B:<k[l5D1�o4p2^?;>9 M2V Ò?VD B\3a9,;>z29d<>=\[n<X|H[q3u5D1bo\pq^?;a9 M2VDM � |l<a=4[n<*<a=?9�36Eb[lCDCD9�36<0zn[lCD^?9m/2w ; Þ wx/2;0cd=?5Djk=�[�CD5`1?9j,[21�BJ9 U [2CÈt/q36< W <k[l1?pq9,1_<Z<a/ <a=\9�jv^?;>z29�5D3 M |:cd=?9,;>9�wx/2;�[ CD5`1?9�<>/tB49�<k[l1?pq9,1_<0<>/t<a=?9jv^\;azq92| 5`<�c*/2^\CD7¥1?9,9X7<>/�=4[{z29C8 I �\T º gZ= IÓâ V ÷H5D1\jv9 <a=\9tj,^?;>z29 =?9,;>9-583Ajv/q1\j,[{zq97:/ncd1�|J[23 ; Þ 5D1\j,;a9X[23a9�3d@\[q3}< M |?<>=?9�zn[2C`^?9-/lw�8tcd5`CDCÍ;a58369A^\1q<>5`C ; Þ ;a9X[2jk=?9X3Z<a=\9�@J/25D1q</lwZ5D1%[\9�jv<a5D/21�| ; � |"[nw <a9�;�cd=?58jk=\8�cd5DC`CQ7?9�3>jv9,147¸[2pq[25`1 V ° 3�[);>9�3a^?C�<X|�<>=?9bEb[n¡:5È-^?Ezn[lCD^?9�/2w]8�/:j�jv^?;k3ucd=?9�1b<a=\9mCD5`1?9m5D3Q<>[21?p29�1q<Q<a/A<>=?9rj,^?;azq9m[n<0<a=?9r@4/q5`1_<Q/2w�5`1^[\9�jv<a5D/21 VM �

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0 0.2 0.4 0.6 r_c 0.80

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» 5`pq^?;>9 M2V Ò?¼ Y =?9�;>9,p25D/21)5D1ª<a=\9F8 Ã = @?C8[l1?9�5D1cd=?58jk=ª9�]_^\[l<a5D/21 MqV ²Hh =\[23Z<>=?;>9,9�9X]q^\5 ÃCD5`B?;>5D^?EÅ@J/25D1_<>3 VÑ 9�j,[l1)j,[2CDj,^?C8[n<a9�<>=\[n<d<>=?5D3�/Hj�jv^?;k3*cd=?9,1; � I ê N g 8 � I N ê NÐ g = � I N ê N T U M2Viº � W

_ óÍó&,{ía`�ï¾ðcb%*Xï�íJôd*Xí�ò�ñaMNb$eG 9�;a9q| ±0f�I 8 ; ¹= ¹ I �?T �2� º:M U M2Viº:M{W±hgigAI L h 8= L h L Æ ; ¹U M P ; ¹ W O I �\T N Ò U M2Viº2hqW� w"c09ACD9v< = I = �£PS§ � wx/q;"8�o\¡H9X7 |:<>=?9�[lBJ/nz29A[l14[lCDsH3a5830=\/2C87?3,|\cd5�<>= F ;a9�@?C8[2jv9X7�cd5�<>=; [l1\7 R ;>9,@\CD[qjv9�7&cd5�<>= = V"Ñ 9r<a=?9�1�9,¡:@49Xj�<d[�<>;>[21\365`<a5D/21&<a/ /Hj�jv^?;*cd=?9�1 ° 5 U × W I � |cd=?58jk=�583dcd=?9�1 ×rë«L h T NqN Ð\M2M |_/2;� ë�§ Ç aÚ O � h N T ²HËnÆqÐ U M2Viº N WY =?9t[l@?@\;a/{¡:5DEt[l<a9�3a/2CD^:<>5`/q1¥/2B?<>[l5D1?9X7)wx;a/qE 9�]_^\[l<a5D/21 MqV h N 583�3a=?/ncd15D1o4p2^?;>9 MqV`M{hwx/2; §&I �?T � M |"5`1¨jv/2Et@\[2;a583a/21�cd5�<>=Á<>=?9�1H^?Et9�;a58j,[2C03a/2CD^:<a5D/21Â[2CD3a/�3a=?/ncd1~5D1Áo\pq^?;>9M2VDM2MqVM2M

0 1x_c k_c 200

y_c

0.5

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x

y

» 5`pq^?;>9 M2VDM � ¼ Y =?9 jv^?;>z29�[21\7ªCD5`1?9�wx;a/qE�9X]q^4[n<a5D/21 MqV ²_Æ |4cd5�<>=ª<a=?9 jv;>5�<>5Dj�[lC£z{[2C`^\9�3�/2w8-[21\7 = p25Dz29�1�5D1)9�]_^\[n<>5`/q1 MqV º � V

−600 −500 −400 −300 −200 −100 0 100 200 3000

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Bifurcation DiagramSolution

» 5`pq^?;>9 MqV`MqM2¼ Y =?9b1H^?Et9,;>5Dj�[lC$36/qC`^?<a5D/21�<a/)9�]_^\[l<a5D/21 M2V ²_h cd5�<>= §�I �\T � M |j8 I �\T ºqº[l147 = I = � PÂ§ ��V Y =?9A5D1?5`<a58[lC½j,/21\7?5�<>5`/q1�583 FtI M [n< � I�L Æ �2� VMXh

−100 −50 0 50 100 1500

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Bifurcation DiagramSolutionApproximation» 5`pq^?;>9 M2VDMXh?¼ °k6 /H/2Et9X7 Ã 5D1Ázq9,;k365D/21Á/2w�<a=\9�1H^?Et9�;a58j,[2C*3a/2CD^:<a5D/21Â<a/¸9�]_^\[l<a5D/21 MqV ²Hhcd5`<a= § I �?T � M |]8 I �\T ºqº [21\7 RSIÅR4�*P�§ � |"[23A5D1�o\p2^\;a9 MqV`MqM2V Y =\5D3A583�3a=?/ncd1�5D1jv/qEt@\[l;>5D3a/21�cd5�<>=�<a=?9�[2@?@?;>/{¡H5DEb[n<>9�3a/2CD^:<a5D/21�p25Dz29�1eBHse9X]_^\[n<>5`/q1 MqV h N V��#l m�+on ��m�v�d� +,�qpe��(d# �)%Er]s�!$%ª��!utwve')��"+,#-�.0/21436587:9,;*<>=?9A9�]_^\[l<a5D/21 1 ¹ ;1 � ¹ P = 1 ;1 � I ;yx�z L h ; ¹|{ T U M2Viºn²HWÑ 9A=\[{z29�9�]_^?5DC`5DB?;>5`^\Et3�cd=?9,1 � I ; x z L h ; ¹i{ U M2Viº2ºqW; I �\g~} ú z h T U M2ViºlÆ_W° 3d[t;>9�3a^?C`<�|:c*9�p29v<m[ B?5`wx^?;kj,[l<a5D/21�cd=?9,1 ; I � | z I � V� /2<a9¸<a=\[l<�9�z29�1É<a=\/2^?pq=É<a=\9�3a9¸3a/2CD^:<a5D/2143�j�[l1�1?/ncÛBJ9�/_3aj,5`CDCD[l<a/q;asq|d<a=?9�s�36<a5DC`C9v¡:=?5DB?5`<0<>=?9A7:9,C8[{stB49,<}c09�9,1e<a=?9�5`;Z/2;>5Dp25D1\[lC�[l1\7&o\1\[2C�36<>[l<a9X3 V ÷H9�9mo\pq^?;>9�3 M2VDM N|� M2VDMXº?V¤ 9,<�^\3�[n<a<a9�E @?<�<>/¨7:/¨[Â3a5Èt5DCD[2;�[21\[lCDs:36583e[q3e5D1É<>=?9�@\;a9�z_5D/2^43e3a9�j�<>5`/q1 V ¤ 9v<z I � PÂ§ � [l147eCD9v<�^\3�9v¡:@\[l147; I � PÂ§ ¬ ; g � IÉ§ ® ¯ T U M2ViºqË2WY =?9�1�c*9�/2B:<k[l5D1 1 ¹ ; 1 ¯ ¹ § ¬nÇ ¹ ® P = 1 ; 1 ¯ § ¬nÇ�® I�¯ ; § }Ê ¬ Ê ® L h ; O §kO ¬ T U M2ViºlÐ_WM N

−600 −400 −200 0 200 400 600−1

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» 5`pq^?;>9 M2VDM N ¼ Y =?9¸36/qC`^:<>5`/q1K<a/S9�]_^\[n<>5`/q1 M2Viºn² |*^4365D1?p~<>=?9wx/qC`CD/ncd5`1\pÁ@\[l;k[lEt9,<a9,;k3 ¼; U L Æ �q� W I �\T � ² | ; G U L Æ �2� W I � | §ZI �\T �q� h | = I �?T �2� º:V

−400 −300 −200 −100 0 100 200 300 400−1

−0.5

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» 5`pq^?;>9 M2VDM,²4¼ Y =?9¸36/qC`^:<>5`/q1K<a/S9�]_^\[n<>5`/q1 M2Viºn² |*^4365D1?p~<>=?9wx/qC`CD/ncd5`1\pÁ@\[l;k[lEt9,<a9,;k3 ¼; U L ² �q� W I �\T M | ; G U L ² �2� W I � | §ZI �?T �2� h | = I � VM,²

−400 −300 −200 −100 0 100 200 300 400−1

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x

» 5`pq^?;>9 M2VDMXº?¼ Y =?9¸36/qC`^:<>5`/q1K<a/S9�]_^\[n<>5`/q1 M2Viºn² |*^4365D1?p~<>=?9wx/qC`CD/ncd5`1\pÁ@\[l;k[lEt9,<a9,;k3 ¼; U L ² �q� W I �\T h | ; G U L ² �2� W I � | §ZI �?T �2� h | = I � VY /�3a5Èt@?CD5`wxs<a=\9b@?;a/qB?CD9,E)| c09tcd5DC`C$Eb[2þ29b36/qEt9 [q3a3a^?Et@:<a5D/2143A[lBJ/2^:<�<a=\9t3a5 6 9t/lw = V� 1b<>=?9�o\;k36<*j,[q369q|qc*9mcd5DCDCJ[q3a3a^?Et9 = 5D30/lw�/q;>7?9,; MqVÍ� 1&<a=?9�369Xjv/q1\7 |_c09�cd5DCDC�[q3a3a^?Et9 =5830[2@?@?;>/{¡H5DEb[n<>9,CDs �eU 5 V 9 V |:36Eb[2C`CJ5`1�jv/qE @4[l;>5D3a/21tcd5`<a= § W [l1\7 ; 583*36Eb[lCDC VÍ� 1eB4/2<a=e/2w<a=\9�3a9-j,[q369X3,|J<>=?9�@?;>/2B\C`9�Eãcd5DCDCÍ1\/l<�wx9�[n<>^?;>9�C8[l;>p29�/_3aj,5`CDCD[l<a5D/21\3 V » 5D1\[lCDCDs2| c*9�/2^?<aCD5`1?9cd=\[l<"/q1?9Zj,[l1t7:/�wx/2; = /2wJ/q;>7:9�; § cd5�<>= 1\/�[q3a3a^?Et@:<>5`/q1\3"[lBJ/2^:<"<a=\9Z3a5 6 9Z/lw ; |lcd=?58jk=5D1_zq/2CDz29X3"9v¡H<a9�1\7:5D1?p�/2^?<u<a=?9r;>[21?p29d/2w z{[2C`587:5`<}s-/2w�<a=?9mj�[23a9�cd=?9�1 = 583Q[l@\@?;a/{¡:5DEb[n<a9�C`s� [l1\7 ; 5D3d3aEb[lCDC Vÿ��6ÿ ��^'/�"��&�^��'(�X!#'@��'@!E'@�� ]� ÿ» /q;�<a=?583�[l@?@?;>/{¡:5Èb[l<a5D/21�|�c*9 [q3a3a^?Et9�<>=\[n< = 583�/lwQ/q;>7:9�; MqV ° 3�[�;>9�3a^?C`<�| c*9-E-^\3}<=\[{zq9r9�5�<>=?9,; Î L h ÏAI N Î g Î L M I M P Î PÂÏ U M2ViºlÒ_W/2; Î L h Ï�I M P Î PÂÏ g Î L�Ï�I N Î T U M2V Æ � W� bÔXñH)� 1�<a=?9Aj�[23a9 Î L h Ï�I N Î g Î L M I M P Î PÂÏ g U M2V Æ?M{WMXº

c*9�/qB:<>[25`1 ÏAI«L Mh g Î I Mh T U M2V ÆqhqW° 3d[t;>9�3a^?C`<�|:/q^?;�7:5UTJ9�;a9�1q<>5D[2C 9�]_^\[l<a5D/21�B49Xjv/2Et9X31 ¹ ; 1 ¯ ¹ §>O Ú ¹ P = 1 ; 1 ¯ §*IÉ¯ ; ,§uL h ; O §kO Ú ¹ T U M2V Æ N W� /nc�^?1\7:9�;�<a=?9&[q3a3a^?Et@:<>5`/q1¥<>=\[n< = 583�/lwZ/2;k7:9,; M |½[l1\7¸/q1?C`s¥þ29,9�@?5D1?pe<>=?9b=?5Dp2=?9X3}<<a9�;aEb3�|:c09�/qB:<>[25`1= 1 ;

1 ¯ IÉ¯ ; U M2V Æl²HW; IW�� È-� Ú ¹ f U M2V ÆqºqW; IW� ê §#�-� �Z� Ú ¹ f U M2V Æ2Æ_W; I ; U¾� W � � �Z� Ú ¹ f T U M2V Æ_Ë2W� bÔXñH)i)� 1�<a=?9Aj�[23a9 Î L h Ï�I M P Î PÂÏ g Î L�Ï�I N Î g U M2V Æ2Ð_Wc*9�/qB:<>[25`1 ÏAI«LªMN g Î IÛMÆ T U M2V Æ2Ò_W° 3d[t;>9�3a^?C`<�|:/q^?;�7:5UTJ9�;a9�1q<>5D[2C 9�]_^\[l<a5D/21�B49Xjv/2Et9X31 ¹ ; 1 ¯ ¹ §kO Ú�� P = 1 ; 1 ¯ § 6Ú ¹ IÉ¯ ; §>O Ú�� L h ; O § 6Ú ¹ T U M2VÌË � W° p_[l5D1)^?1\7:9�;�<a=\9�[23>36^\E @?<a5D/21�<a=\[l< = 583d/2w"/q;>7?9,; M |\[l1\7)/q1?CDseþ29�9,@?5D1?pt<>=?9�=?5Dp2=?9X3}<<a9�;aEb3�|:c09�/qB:<>[25`1

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SolutionApproximation

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U j W�� IfL N �q�

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