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Page 1: School of Mathematics and Physics - University of QueenslandKA@7I7q@Ft2 685L68341/KA?#=`XOht6PB815á=z1g/6¦=IY 1/=GBW@ 5&7U=415 =\=I@sDo6¦âW683I@%17q?_=qB8@ ;LKAN Do6Z1DL7I@ 3UDo6WNã?#5

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Page 2: School of Mathematics and Physics - University of QueenslandKA@7I7q@Ft2 685L68341/KA?#=`XOht6PB815á=z1g/6¦=IY 1/=GBW@ 5&7U=415 =\=I@sDo6¦âW683I@%17q?_=qB8@ ;LKAN Do6Z1DL7I@ 3UDo6WNã?#5

Æ �<������Ç�­���È�­����*�8ÄÅ�­>�]�L�������É 5&6�HZ?A2�Y =*5�1�?_^�6WK#XG=4Y&?#5&gV=4Y&1/=�=IY&683U6d1�3U6tB81�7I6W7ThiY&683U6r@�5&6tB�1�5L5&@�=�N&683U?#^�6t=4Y&6�uR1�HZ?AK_=4@�5&?A1�5F@�3UHÊF3U@�HË=4Y&6G0�1�2�3I1�5&2�?A1�5�@�5&6�alu9@�ht6W^�683d=4YL6VB8@�5 ^�683I7U?A@�5sF3U@�HÌ@�5&6RF@�3UHÊ=I@�=4Y&6q@/=4Y&6W3r?A7�1�56Q´L1�H p K#6q@�F°1¦Æ �<�����*Ç*­���È�­����*�8ÄÅ�­>�]�L�)���� 1�5LNOB81�5%1�K#h�1�Xw7iD�6 p 6W3UF@�3UHP6WN�aÍ @�5&7U?AN&6W3C1ªF;&5&BQ=4?#@�5OÎ �Ï�Ð ¨ Ï ¹ � 1�5&NON&6WS�5L6Ñ Ð � � Î� Ï�Ð ¨ Ñ ¹ � � Î� Ï ¹¶`7i?#= p @�7I7U?AD&K#6V=4@ªhi3I?_=46V=IY&6G?A5 ^�6W3I7I6GF@�3UHJ;&KM1/7r?A5s=4YL6GF@�3IHÏ�Ð ��� ¼� Ñ Ð ¨ Ï ¹ ��� ¼� Ñ ¹F@�3d7U@�HP6VF;L5&BW=I?A@�5%¼ � Ñ Ð ¨ Ñ ¹ �UÒ}�@�3i1ª7I?AH p K#6VB�1�7U6�f�741�XOÎ � · Ï ¹ Ð ºÔÓ Ï�ÐyÏ ¹ º Ï ¹¹À Ñ Ð ��Õ Ï�Ð ºÔÓ Ï ¹ ¨ Ñ ¹ ��Ó Ï�Ð º · Ï ¹Â @�K_^�?A5&2JF@�3 Ï�Ð 1�5&N Ï ¹ 2�?#^�6W7Ï�Ð � � · Ñ Ð ºÔÓ/Ñ ¹ ¨ Ï ¹ �ÁÓ�Ñ Ð � Õ�Ñ ¹Â @ª=4Y&1/=C¼ � � Ñ ¹Ð ºÔÓ/Ñ Ð Ñ ¹ � · Ñ ¹¹ f&ht@�3Ig<78Öt×�;L=ihi?#KAKØ?_=91�K#h�1�X�7 Ò ¶`5sF�1�BW=9?_=rhi?AK#KÙaÍ @�5&7U?AN&6W3i=IY&6G6Q´ p 3U687I7U?A@�5Ú � Î �Ï�Ð ¨ Ï ¹ � º ¼ � Ñ Ð ¨ Ñ ¹ � � ��Ï�Ð Ñ Ð º Ï ¹ Ñ ¹ � ¨¶ÅFTht6q=4Y&?#5&g�@�F Ñ ÐW��Ï�Ð ¨ Ï ¹ � 1�5&N Ñ ¹ ��Ï�Ð ¨ Ï ¹ � =4YL685cÚ ��Ï�Ð ¨ Ï ¹ � a  @Z=IY�1/=

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Page 3: School of Mathematics and Physics - University of QueenslandKA@7I7q@Ft2 685L68341/KA?#=`XOht6PB815á=z1g/6¦=IY 1/=GBW@ 5&7U=415 =\=I@sDo6¦âW683I@%17q?_=qB8@ ;LKAN Do6Z1DL7I@ 3UDo6WNã?#5

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R0

Poincare Recurrences

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D finite

Eventually

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0

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00.5

z

Σ x0

x1

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xy

z

Σ x0

Periodic orbit

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ω0 =1, ω =1.9 and ε =0.025

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-3.142

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ω = 1.175, ε = 0.3

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