y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V...

19

Transcript of y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V...

Page 1: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

� ��������� ����������������������� ������ �"!#�$&%���(') ���*,+-��� .0/1+����*$2� .3�*4�-+5�6���7$8�

93:<;>=�?A@�B�CEDF=�GIHKJE=�L5M

NPORQTSUORVXWYV[Z]\_^a`UQbZcSedIf<N�ORQTSUOgVhfjilknm_oRoIp qIfrW3s<k

t*uwvyxyz {[|RxR}�~�� `��ROR�g��Z]V�SU��Z�Q������r^a`�Z]Q�SUO3�<^yQb�a`UZ]�r^�SU�[^��Kd�V[�g��Z]�,�����r^y�[�,\<Z]O�`�O��jQbZ�V�� � ��� `�r^A`bS � `U��� SUZ]ORV[Q�O����yOg���[�]^a���Kd�V[�g��Z]�,����QTd<QTSU^y��Qy�[s<Z]V[� � ����`��r^a`bS � `U��� SUZ�OgV[Q�OI�a� � `�����^yV��rOg�]^�Z]Q�Z]VKSb`UOK� � �y^y��Z]VKSUOYSU�[^��<d<V[�g��Z]�yQwOg�E�(�rOg�cd�V�OR��Z��g���R��VFSU��Z]QP�[�g�r^a`,f���^��yOgV[QbZ���^a`�SU�[^QbZ]������^aQTS��rOgQbQbZ����]^��,��Qb^R�IQTS���`bSP��ZcSU�*SU��^��*�g�*������_ ����[^a`U^�¡-¢£pY��V���SU�[^aV*�R�[�����rOg�]^� S�SU��^FO�`UZ��gZ]Vh�h¤[Og`�QbZ]���[�]Z]�yZcS�dIf[��^��yOgV[QbZ���^A`(SU�[^��,��Qb^�Og�P�*�g��Q�O��PSU��^��¥Og`U�#�_ 7¦¨§r©g�_ r�ª«^�SU�[^aV«�<^yQb�a`UZ]�r^�QbOR��^�O���SU��^��*�gVIdX���yd<Q7s<Z]^a`U��Z]V[Qb¬KZ6� � `U\_^�­ � �]Z��Qb^aSUQ�� `UZ�Qb^*Z]V�SU��Z]Q�®�g��Z]�cdI�6ª�^X����QbO«�RZ]\_^-���a����QbQbZc¯��,� SUZ�OgV°Og��SU��^X�<d<V[�g��Z]�yQ�OgV±SU�[^yQb^Qb^aSUQ*�gV[�±��^aQb�a`UZ]�r^SU��^(Z]VISb`UZ��y��SU^�QTSb` � �aS � `U^�SU��� S�OI�y� � `UQ�� `UO � V[� SU�[^3i� i � �]�]^yV-��Og�*�gZ]VZ]VSU��^(��� `��g��^aSU^A`�����gV[^��®O�`�SU�[^aQb^��*���[Qy�K¤�Z]V[�g�]�cdIf_��^Y��Z]Qb� � QbQ�QbOR��^�Og�ESU�[^��*�,²TO�`���Zc³j^a`U^aV[�y^yQP�r^aS���^y^yV�SU�[^�y�gQb^yQ�¡X´°o���V�� ¡5¢±oK�

µ7¶ ��^y�gQb^7�R���<`U^yQbQY�g�]�>�aOg`b`U^yQb�rOgV���^aV[�y^3SUO�·�OR�r^a`bSY¸6�[¹(^a\R��V[^adIf<¹(^y�[��`bSU��^yVIS�O��Pil��SU��^aº�*��SUZ]�yQyfIN�ORQTSUOgV WYV[Z]\_^a`UQbZcS�dIf�pRpgp3» � ����Z�V��gSUOgV�sKSb`U^y^aS,fIN�ORQTSUOgVilk°mRoRoKp,qKf�Og`�^aº¼�*�gZ]��rOR��½�� � � ^y� � �

Page 2: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

o ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad����� �������� ��������������� ?���:�H����¥J°@���� �"!hH#!>=�?$� � @A:&%r='�)(�?)�¥;h=�@)�E=�;>=*�EH_G+�¥:<?:#,��)�¥J�� � �®HK?��¥L-!>=�?A@ � ?A;>=.%( :�/"!��¥=10*H<JEH#�¥L[@)�2(�%rL�JEH�/"� (gH#�+��L��@A=*/3�gC'4�L*H����¥J�� � �¥H<?5!>=�?A@ � ?a;EHK@)�¥:<J76�=�/l=RHKJ*@��j=, :�� ��:86��¥J��EC9 � !:!h:;��=<6Y=<�EH_G�=FH=(�:�/"!��¥=105H<JEH#�¥L[@)�2(>/XH#!@?BAC6����2(D�FE;, :<?��)�G/"!�� � (*��@UL�E6Y=H6�� � ��H��)� � /l=X@A: ;h=«HI!h:���L�Jj:�/"�¥H���:#,>%r=.�<?a=�=KJhCMLY:<J:�)� %j=�?�@��j=5Jj=.6N/XH#!O?QP:<;r@yH#�¥Jj=.%�;�L�H�%:%��J:�lH$!>:��¥=*HI@>RHSUTV��:@��EHI@

?QP�WYX+ZQ[\?5A;W]X;ZF^ _WYXa`&RZcb6��j=�?a= _ SdTe�2�*HU!EHK?yH#/l=�@A=�?RCgfh�j=3/XH#!i? P �2�*HI���¥J�� � �¥H<?a!h=g?�@ � ?a;EHI@���:�J :�,h? AH��C��:�:�J H�� _kj[ml@�)��J:(�=�@��j="%r=*��?A=g= :�,�@)�j="/XH#!-(D�EHKJ:�<=.��, ?a:�/nJ«@A:UJ7^po CFq��H�(�:<J:�A=.r � =gJ:( =�E<@��j=a%rL�JEH#/"�2(�H#�>;>=*�EH_G+�¥:<?�:#,s?QP$�2��/ � (D�5?��2(D�j=�?Y@)�EH<J-@��EHI@3:#,s?BA;EH#��@)�j: � ���O�A:�/l=H!>:<?A@)�¥:<J:��:#,(@)�E=t%rL�JEH�/$�2(*��:#,>?5At!>=�?D���2��@�%r=*!>=�Ju%��J:� � !>:<J @��j=��:(�HK@)�¥:<J«:#,�@)�j=a!>:��¥=aR>C

fh�j=(?A=RH��A:<J�, :<?�@)�E=���J[@A=g?A='�U@��¥JH���¥J�� � �¥H<?Q!h=g?�@ � ?A;hHI@)�¥:<Ju��HK?�� �A=.�5, ?a:�/wv7=*6(@a:<JFx �/=�@)�j:%wCM9 � !�!h:;��=t6Y=«HK?a=5H#!:!���L+�¥J��yv7=*6(@a:<JFx �=/l=�@)�j:% @a:{zEJu%¨@)�j=«?a:[:<@���:#,7H,¼H#/"�G�¥L±:#,�!h:���L�Jj:�/"�¥H�� �7|QPI6��:� (D�i�EH���HU/ � ��@��G!:��=-?a:�:K@ [email protected]�aH_L�E6@)�j=H!hHK?yH#/l= @a=�?_ [NljCg}j:�?�=10jH#/"!��¥=�EF(�:<J:�)� %j=�?�@)�j=X='��!>=.(*�¥H��G�¥L{�)�G/"!��¥=H(�H���=:#,h|sP�WYX+ZC[~X���^ _ C���j=�J _ [kl(@����2�F!>:��¥L[JE:�/"�¥H����EH���Hh/ � ��@��G!:��=Y?A:�:K@�HI@5l�H<J:%�@)�E=�v�=.6(@A:<J=��@a=�?yHI@)�¥:<J, � J:(,@���:�JI� ����� /"!���L@� A WYX+ZM[\X�#�jC��:86�=�G<=g?.E+6��j=gJ _&j[\l�Er@��j=7v�=.6(@A:�JI��@a=�?yHI@)�¥:<J, � J:(,@���:�J�;h='( :�/l='�

�$P�WYX+ZQ[ X � ` _�#X [ X� `�_�#X

HKJ:%$6�=���=g=(@)�[email protected][H����¥Jl@)�j=>,¼H#/"�G�¥L�? P EK@��j=�%r=.�<?a=�=Q� � /"!:�YH���6�=�/l:IG<=�H�6(H_LC, ?A:�/_ [�ljCF�bJ°H�%�%��@)�¥:<JFEF�¥J:��@A=RH�%¨:#,Q� � ��@=�EH_G+�¥J���HI���¥:<;hH#� ��L HK@�@a?aH�(,@)�¥J��Uz�0r='%&!h:���J[@HI@7@��j=:<?��G����JFE>H8,�@a=�?�@��j=$!h=g?�@ � ?A;hHI@)�¥:<JBEE@��j=�%rL�JEH�/"� (gH#��;h=.�EH_G+��:�?�:#,�� P ;>=.(�:�/l=/ � (D�«?)�2(D�j=g?gC

}j:<?a�)�G/"!�� �2(1��@UL�EF�¥J @����2�a!EH#!>=�?'E�6Y="6�� �G��( :<Ju���2%r=�?�@��j=3��� /"!���='�U@7!>:����)��;��¥=3(�H���=6��j=�?a==?5A�WYX+Zh[�X��«HKJ:%U���\�rC�9r:X@)�j=�%rL�JEH#/"�2(*��:�,M?BAHK?a=a6Y=.�G� � J:%r=g?���@A:�:%wC���,� X �+��� E�@)�E=�:<?a;���@�:�,sX@a=�J:%��3@A:l@)�E=*HI@�@a?aH�(,@���J��$z�0r='%t!>:��¥J[@7HI@(@)�E=�:<?��G����J � J:%r=g?��@a=�?yHI@)�¥:<J :#,�?BA[C8��, � X ��\� E_@��j=(:<?a;���@�:�,uX�@A=gJ:%���@a:7��6��:� (D�$� ��H�� �A:�H<JHK@�@a?aH�(,@)�¥J��z�0r=.%{!>:��¥J[@�, :�?<?5A$��J @)�j=X9��¥=*/XHKJEJ���!:�j=�?a=<C � J�@��j=$(*��?D(1�¥= � X � [ � EE@��j=$/lH�!��2�6Y=.�G�u�[JE:863Jl@A:*;>=�(D�EH<:K@�� (��K@)�:� ����=�@��2��@)�E=��;����Y =¡*¢1£�:�,g?5A;E�6����2(D�36�� �G�>;h=<%r=1zhJj=.%;h=.��:86�C

}j:<?Q/l:���@�:#,h@)���2��!EH�!h=g?.E_@��j=����¥J�� � �¥H<?)�¥La!>=�?A@ � ?a;h='%�,¼H#/"�G�¥L=6�� � �r;h=3:�,E@)�E=h, :<?�/?QP:W]X;ZM[\X � ^ _X �

6��j=�?a=*H#�[H#�¥JK�&�¤�jCu9�:l@��j=*HI@A@A?yH�(,@���J:��z:0�='%U!h:���J[@7HK@3@)�j=�:�?)� ���¥Jt�2�3?a=*!��®H�(�=.%�;�LH�!>:��¥=�E#� � �U@7H��h��J«@��j=7v�=.6(@A:<JBx �h/l= @��j:%«=10jH#/"!��¥=*HK;>:IG<=�C

� =�?A=./XHK?���@��EHI@h6Y=C( : � �2%X=.r � H��G�¥L"6Y=.�G�g( :�J:�)� %r=g?�@��j=a(�H��A=�6��E=�?a=�@)�E=<%r=.�<?a=�=:#,�!h=g?�@ � ?a;EHI@���:�J-�2�7J j[¥��E�; � @*@)�E=K(�H���=HJy[¥�d�EH��7�A:�/l=H�)!h='(1�®H#���AL+/"/=�@A?���='�@)�EHK@>/lH��<=�@)�:� �>(gH��A=F=RH��)��=g?(@A: � J:%j=�?D�U@yHKJ:%wC

Page 3: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q �fh�j=7!j?���Ju(1� !EH#�s�<:[H#�6:�,�@)���2��!hH#!>=�?��2�7@A:K����G�=�HH� � ?aG<=�L«:#,M�A:�/l=�:�,�@)�j=�/lH<J�L

?A='( =�J[@(?A='� � ��@D��(�:<J:(�=�?aJ���J:�*@)�E=C%jL[JhH#/"� (gH#�wHKJu%H!EH<?aH�/=�@A=g?�!��®HKJE=.��, :<?�@)�j=<,¼H�/$� �¥L?sPhC�qm� � ;r@)�E=*/l==�¥J�@)���2��!EH#!>=�?�� �7@A:K�)�j:86�@)�hHI@7@��j==(gH��A=76��j=�?a=7�-[ �"� �<r � ��@A=%����=�?a=�J[@t, ?A:�/ @��j=y(gH��A={6��j=g?A=y� � �rC �bJ:%r=g=.%gEY@)���2�@��:86�=�?t%r=*��?A=g=I,¼H#/"� ��L :#,/lH�!:��@ � ?AJu��: � @�@a:a�EH_G<=h/ � (D�"/:�?A=�(�:�/"!�� � (gHI@A='%"%rL�JEH#/"�2(�H#�r;>=*�hH_G;�¥:<?�@��EHKJl@��j=(�H��A=$6��j=�?a="� � �rCgq�Jj:<@)�j=g?7� � ;r@)�j=./l=3� �F@A:I%r='�)(�?)�¥;>="�A:�/l=:#,Y@)�j=3�¥J[@A=�?a=.��@)�¥J��@A:�!>:��¥:���� (gH#�g�)!EH�(�=.��@)�hHI@7HK?��2��=�H���@��j=�� � �G�®H"��=�@��(:#,�@)�E=a/lH�!:��?QPEC

����� ����)� �V�����Q:�����LY:<J:�)� %j=�?(@)�E=C,¼H#/"�G�¥L-:�,s/XH#!u�

? P W]X;ZM[\X � ^ _X �6��j=�?a=C� � �rC�fh�j=3�;����Y $¡*¢*£Y:�,s?QP:E�%j=�Jj:<@A=.%�;�L��MWY?QP+Z E��2��%r=1zhJj=.%5@A:l;>=F@��j=7�A= @:#,Q!h:���J[@��FHI@>6����2(D�«@)�j=7,¼H#/"�G�¥L«:#,Q��@a=�?yHI@A='�3:#,�?QP",¼H#� �2�3@A:X;>=�HlJj:�?)/XH���,¼H#/"�G�¥L@��J@)�j=C��=gJ:�A=7:#,���:<J[@A=.�eC+fh�j=g?A=�HK?a=>/XHKJ�Ll:K@��j=�?�=.r � �¥GIH#�¥=�J[@h%j=1zEJ���@)�¥:<Ju�Y:�,w@)�j=�� � �G�®H��=�@gCg}j:�?F=*0rH�/"!���=�EE@��j=�� � � �®HK�A= @��2�F@)�j="(1�¥:�� � ?a=*:#,�@)�j="��=�@F:�,�?A=.!h=.�G� �¥J��K!>=�?���:%�2(!h:���J[@��*:#,�?sPuEPH<J:%gEF�¥J¨: � ?���!>=.(*�¥H���(�H���=�Eg��@=�2�*H�� �A:«@)�j=X;>: � Ju%jHK?aL :�,Y@��j=K�A= @*:#,!h:���J[@���6��j:��A=�:�?A;���@���=.��(�H�!h=3@a:"�4C�q��YH7(�:<J:�A=.r � =gJ:( =�EK@��j=�� � � �¥H7�A= @�� ��@)�j=<��=�@Y:#,!h:���J[@���:<J36����2(D�3?sP*;h=.�EH_G<='�M(D�EHK:<@)�2(�H��G�¥L�E#�)��Ju( =�HK?a;���@A?yHK?��G�¥L7(1�¥:��A=(@A:*H<J�L=!>:��¥J�@���J��W]?sP+Z�@)�E=�?a= �2��;h:<@)�XHF?A=.!h=.�G� ��J:��!>=�?���:%�2( !h:���J[@�HKJu%�H�!h:���J[@�6��j:��A=�:<?a;���@�='�)(gH#!>=.�@A:t�4C�fh�j=7( :�/$!:��=./=gJ[@(:�,P@��j=�� � �G�®H"��=�@>� ��(gH#� ��='%-@��j=��5 8£��8�U¡*¢1£�C

fh�j=�?a=�H<?A=5H J � /�;>=�?l:#,�(�?)��@)�2(�H���!>:��¥J[@��=, :<?"? P C � Jj=@( ?���@�� (gH#� !h:���J[@l:+(.( � ?D�HI@3�kEQ6����2(D�O� �H#� 6(H_L+��HIz�0r=.%O!h:���J[@gC�qe��='( :<Ju%O( ?���@�� (gH#� !h:���J[@:+(.( � ?D��HK@ @��j=:<?��G����JFE�6���� (D�"�2�M/XH#!:!h='%3%��?a=.( @)�¥L*@a:$� C;qk��@A?yH#� ���[@�, :�?)6(HK?D%=(�:�/"! � @yHI@)�¥:<J3�)�j:86>�@)�EHK@3@)�j=g?A=�HK?a=7�8� :K@)�E=�?>( ?���@)�2(�H#�g!>:��¥J[@��h, :�?>? P ���¥G<=�J«;�L�� P [ _ � ! � � C:fh�j='��=�H<?A=@)�j=#"%$)¢D¢a(�?)��@)�2(�H��g!h:���J[@D� , :<?�?sPhCjDF=.�)!���@A=�@)�E=C,¼H�(,@(@��EHI@(@)�E=�?a=�H<?A=a�8��, ?A=g=a( ?���@�� (gH#�!h:���J[@��.E�@��j=�?a= HK?a=�:<J��¥L @c6Y:i( ?���@)�2(�H#�7G_H�� � =.�3����G�=�J ;�L'&�P\[)(C�+* _ C��7:86Y=gG<=g?.E@)�j=g?A= ?A=RH#� ��LU�2�7:�J���L�:�Jj=7, ?a=�=$( ?���@�� (gH#��:<?a;���@C���¥J:(�=�E�6��j=�J��O� �F=�G�=�JFE>;>:K@)� :#,�@��j=( ?���@�� (gH#��GIH#� � =.��HK?a=$/lH�!�!>=.% @A:«@)�j=3�aH#/l=$!h:���J[@gCg���E=�J-�O� ��:%�%FE�@)�j=l:<?a;���@���:#,@)�j=@c6�:@(�?)��@)�2(�H���GIH�� � ='��;h=.�EH_G<="��L+/"/l= @a?)�2(�H��G�¥L � J:%r=g?�@)�j="/XH#!-X-,. `>XEC�fh� � �@)���2�<,¼H#/"� ��L :�,�/XH#!u�*Eg� �G��=�@��j=Hr � H�%j?aHK@)�2(=!>:��¥L�Jj:�/"�®H#�s,¼H#/"� ��L�Eg� ��H�JEHI@ � ?yH#��:<JE=0/!EHK?yH#/l= @a=�?B,¼H#/"�G�¥L�:#,�/XH�!:��C8fh�j=�!EHK?yH#/l=�@A=�?5!��®HKJj=CW�@��j= _ /�!��¥H<Jj='ZB�2�6@��j=�JH�?a=.( :�?�%:#,�@)�j=7;h=.�EH_G+��:�?�:#,�@)�E=�, ?A=g=�( ?���@)�2(�H#�E:<?a;���@'E1� � ��@YH��M�¥Jl@)�j=�(gH��A=3:#,�@)�E=�� H<J:%r=*�¥;j?a:K@��=�@gCfh�j=�?a=FH<?A=�H#�2��:$�8�I!j?a=*!>:��¥=.��HI@�@)�j=<!>:��¥J�@D��1 P [ Wc` _ Z �2! � � E+��C =�CGE? P W�1 P Z�[�lEC

��=�?a=��2�Y:<Jj=F?A=RH��A:<J36���Ll@)�j=C(�H���=��I[ �a�2���A:�%�����=�?a=�J[@ , ?A:�/ @��j=C(�H��A=�� � �jCLY:<J:�)� %j=�?(@)�E=7��='( :<Ju%K��@A=g?aHK@A=�:#,P@)�j=7(�?)��@)�2(�H��g!h:���J[@��gC��±=7( :�/$! � @A=

? �P W�� P Z�[k� � _ � ! � ^ �� � _43 � ! �6587 �:9���j=�J&� � �EF��@a, :�� ��:86>�F@��EHI@�?a�P W;� P Z<. � H�� _ . lECF4 � @76��j=�J&��[N�Ew@��j=

��='( :<Ju%y��@A=g?aHK@A=-:�,#� P ?a=.% � ( ='��@A:>= _ ^ � �?=uEB��:{?a�P W;� P Z@. � �A= H�� _ . ljCsfh�j=?A=RH��A:<J-@����2��� ���)� �<J��Gzu(�H<J[@h6�� � �6;h='( :�/=7(*��=RHK?h�¥J�@��j=�Jj=*0[@��A=.( @)�¥:<J6C

Page 4: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

� ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

B6= @��hP�;h=Y@)�j=h(1�¥?D(1�¥=����¥G<=gJ*;�L � X � [ � _ � �2! � � C'v7:K@A=Y@)�hHI@�� PC(�:<J[@aH���Ju��H#� ��:#,j@��j=( ?���@�� (gH#�Q!h:���J[@D��H<J:%�@��j="!j?a=*!>:��¥=.�gC�qV��@A?yH#� ���[@�, :�?)6(HK?D%U( :�/$! � @aHK@)�¥:<Jy�)�j:86>�F@)�hHI@?sP$/XH#!u���hP3�8��@a: � :<J[@A:*@)�j=a��@A?yH#� ���[@�� ��Jj=C( :�JjJj='(,@)�¥J���@)�j=F@c6Y:$( ?���@�� (gH#�>GIH#� � =.�gC� =$(�H#� ��� P-@)�E=�� $ � £]���D #���*� $��*�G¢lHKJu%U��@D��� /XH#�<=*@)�j=�0$ � £ ���D 8�Q�����¢ C�fh���2���2�7JE:K@7@��j=(�H��A=7, :�?7@��j=�,¼H#/"�G�¥L{X;�=^ _ �#X �=6���@���� j[ JuEg��:5@)���2�<� ��:�Jj=�:�,�@)�j=?A=RH��A:<J:��6���L6Y==%j=gH#�P:<J��¥LU6���@���@)�E==(�H���=7�&[ JEC � Jj=7/XH_L�H#�2��:H(D�E=.(D��@)�EHK@�HKJ�L«:K@��j=�?�(*��?D(1�¥=( =�J[@a=�?a=.% HK@7@)�E= :�?)� ���¥JI� �</XH#!:!h='%{�¨@A: � :�J�@a:�H<J =*� �G� !:�A==6��j:;��==, :(1��HK?a= (�&�PhCq>�2��:uEE@��j=$��@A?yH#� ���[@�?aH_LU( :�JjJj=.( @)�¥J��tl-@A:{�8H<J:%{!hH������¥J��X@��j?A: � ��� � P W¼H��0$ � £ ���D 8�$) ��+Z��2�M/XH#!�!>=.%H�F@a: � :<J[@a:�@)�j=���@A?yH#� ���[@�?yH_L=( :�JjJj='(,@)�¥J��*=.��@��j=�? & P :�?>` & P @a:$�HKJ:%±=10�@a=�J:%�¥J��«@)�E=K( ?���@�� (gH#��� ��JE=<C59+� /"�G�®HK?��¥L�Ew=gH�(D�-��@A?yH#� ���[@�?yH_L{( :<JEJj=.( @)�¥J��{l�@A:� H<J:%3!EH��)�)��J:��@)�j?a: � ���<1 P W¼H�� $)¢�� �8�G¢�$) ��+ZM�2��/XH�!�!h='% � @a: � :<J[@A:�@��j=���@A?yH#� ���[@�G�¥Jj=a!EH��)�)��J:��@��j?A: � ����@)�E=�:<?��G����J�HKJ:%t!>=�?�!h=gJ:%�2( � �®HK?(@a:l@)�j=7(�?)��@)�2(�H��g�G�¥Jj=<C

fh�j=�/XH#!u� ? P �EH_G�=7H�%�%���@���:�JEH#�g�AL;/"/l= @a?)�¥=.�gC�B6= @��°;>=�H=!j?��G/"��@)�¥G<=a�8�����*?a:�:K@:#, � J���@UL<C+fh�j=gJH6Y=��EH_G<=3@��EHI@h?sP�W���X;Z�[�� � ?sP�WYX+Z,C;��=gJ:( =7@)�j=F:<?a;���@D��:�,g!h:���J[@��Y:#,@)�j=�, :<?�/����*XH#� �E;>=*�EH_G�=��)�AL+/"/=�@A?�� (gH#� ��L � � Ju%r=�? ��@a=�?yHI@���:�JX:#,B?QPEC+}j:�?�=*0rH�/"!���=�E� ,7?�!P W]X;Z�. �kE�@��j=�J\?�!P W��#"�X+ZXH#�2�A:¨@A=�Ju%��X@a:d� , :�?-=gH�(D�%$�CM��,=?�!P W]X;Zl@A=�Ju%��@A: H<J¨HK@�@a?aH�(,@)�¥J��{(�L+(*��=�Ew@)�E=�JO�A:{%r:�=.�7? !P W�� " X+Z,Cs��:86�=�G<=g?.E>@��j=K( L(*��='�a��J�G�:��¥G<=.%/lH_L�;>=$%�����=�?a=�J[@a%r=*!>=�J:%���J��«:<J�$�HKJu%gE���J:%j=�=.%FEh@��j=�LI/XH_L�=�G�=�J{�hH_G<=�%�����=�?a=�J[@!h=g?)�¥:%���C5v�:<JE= @)�E=*�¥=.���*E�H��G��!h:���J[@��7��L+�¥J���:<J±@)�:� ����=�@*:#,�HI@A@A?yH�(,@���J:�U(�L(1�¥=.�*H<?A=l:#,@)�j=�, :<?�/����.X'A�, :<?��A:�/l=�X'AaSIT�Cgq�Jj:<@)�j=g? �AL+/$/l=�@A?aL$�2�����¥G<=�JX;�L @)�j=<�¥J[G�:�� � @)�¥:<J&"P�W]X;Z�[ * _ �#XEC��=g?A=C6�=a�EH_G<=C?QPuW'&3PuW]X;Z)ZM[k?QP�WYX+Z,C

���j=�J � X � �2�a�®HK?��<=�E�@)�j=X@a=�?�/ _ �8X � �¥J¨@)�j=H, :<?�/ � �®Ht, :<?=? P �2��Jj=.��� �G����;��¥=�E5��:? P WYX+Z�(\X � Jj=gH<?�� C�LY:�J:�A=.r � =�J[@���L�E�@)�j=7!>:��¥J[@7HI@�� �2��H"� � !h=g?aHK@�@A?yH�( @)�¥J��$z�0r=.%!h:���J[@@, :<?I? P H<J:%k��@U� �@6�=*� �<��Jj:863J4@)�EHK@U? P � �U(�:<J8� � �[HI@a= @A:¤X ,. X;���¥J HJj=*� ����;>:<?��j:�:+%�:�,h�kE��A:K6�==�EH_G�=�HKJ��G/"/l=.%�®HI@a= ;hH��)��J :�,�HI@A@A?yH�(,@���:�J*) P HK@7�4C9+��Ju( =7? P �EH��3H$!>:��¥=�:#,P:�?�%j=�?h� HK@�l�E�@)�E=�?a=<�2�3HKJ«:�!>=�J«Jj=.�G���[;>:<?��j:�:%-:�,Ql @)�hHI@� ��/lH�!�!>=.%U�±@A: � :<J[@A:-HXJE=*� ����;h:�?)�E:[:%«:#,�� ��J*)CPEC:��,�@)�E=*=�J[@)�¥?a=*;EH��)�¥J�:#,��� � %�2� �U:���J[@�, ?a:�/ @)�:� ��Jj=.�G���[;>:<?��j:�:%lHK?a: � J:%@��j=�:<?��G����JFEK@)�j=gJ-@)�E=�?a=��2��H*:�!h=gJK�A= @HK;>: � @alX@��EHI@C� �</XH#!:!h='%{� @a: � :<J[@A:+)<PuEwHKJ:%�@)���2��=�J[@)�¥?A="�A= @C� �<%�2� �U:���J[@<, ?A:�/)<PhCBfh���2�=�A= @�� ��(�H#� �¥=.% @)�j=�£�$) ,�.- � � $+Es�)��Ju( =-HKJ�L :<?a;���@�@)�EHK@�=�G�=�J[@ � H��G�¥L =gJ[@A=�?D�)<P@/ � �U@a%r:U��:«;�LI!EH����)��J��5@)�j?a: � ����@��j=@A?yH#!�%r:�:�?gC��±=$%r=gJj:K@a=@)�j= @A?yH#!-%r:�:<?;[L0/gPEC

1��)��J��(@��j=��AL+/"/=�@A?aL�?QP�W���X+ZM[2� � ?QPuW]X;Z1E*��@5�2�B��@A?yH#� ���[@�, :�?)6(HK?D%3@a:>(D�j=.(D�F@)�hHI@)<PuE3/FP:E>H<J:% �MWY?QP+Z�H<?A=*H��G�Q��L+/"/l= @a?)�2( � J:%j=�?�X ,.4��XhC:�±=��aH_L5@��EHI@�@��j=.�A=��A= @D�!h:;�)�A=.���M�8� / , :��2%=�AL+/$/l=�@A?aL<C#�bJ"!EH<?�@�� ( � �¥H<?.E8�)��Ju( =(@)�E=>( ?���@�� (gH#�!>:��¥J�@D��HK?a=3HK?a?aH<J��<='%��L+/"/l= @A?��2(�H#� �¥L HK;>: � @�@)�j=�:�?)� ���¥JFE8��@M, :�� �¥:86>��@)�EHK@�� ,w:�Jj=3:#,>@)�j=�( ?���@�� (gH#��!>:��¥J[@���� ��='���J�) P W¼?A='��!6C E5/ P Z E<@)�j=gJ-H#� �E:#,>@)�E=�( ?���@�� (gH#�u!h:���J[@��M�G�¥=>�¥J�) P W ?a=.�)!6CGE6/ P Z C�q>�2�A::EK@��j=/lH�!7& P ��J[@a=�?D(D�EHKJ���=.��) P HKJu%8/ P H<J:%3!j?a=.�A=�?aG<='��;>:K@��-@)�E= � � � �¥H HKJ:%l@��j=C}EHI@a: ���=�@gC

}j:<?7:K@)�E=�?�(�:�/"!>:<Jj=gJ�@D��:�,�@)�j=$}EHI@a: � �A= @'Ej@)�E=��AL;/"/l= @a?ALU�)��@ � HI@���:�JI� �<��:�/l=0/6��EHI@>%� �>=g?A=gJ[@:9r=*��@)�E=�?�H"( :�/$!>:<JE=�J[@h(�:<J[@aH���Ju�����1X'A<, :<?�H����¥G<=gJtX'AC��J«@��j=a}EHI@a: ���=�@�H<J:% H#� �<;IS�="E[:�?F=.� �A=�� � (D� Ht( :�/$!>:<JE=�J[@<(�:<J[@aH���J:�FJj:�Jj=�:�,�@)�j=>���.X'A=6���@)�; j[\l�/l:%I�8��C:9�=g=8? @BAF, :<?�H$!j?a:�:#,P:�,�@���� ��,¼H�(,@RC

Page 5: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q q����� ���������+� ���<���u��������� ���

}j:<?P@)�E=�6�=*� ��/ ��@ � %�¥=.%a,¼H#/"�G�¥L�:#,:r � H�%r?yHI@)�2(M/XH�!:�� ��.W]X;Z�[\X��;^@��6���@�����H�( :�/$!:��=*0!EHK?yH#/l= @a=�?�@)�E=�?a=a� ��@��j=a6Y=.�G�g��Jj:863JU} � J:%jH#/l=gJ�@yH#��D��2(D�j:K@a:�/�L�9� Ch��,w@)�j=�:<?a;���@�:#,w@��j=7:�J��¥L�, ?a=�=�( ?���@�� (gH#��!h:���J[@ l�@A=�Ju%���@a:"�kE<@��j=�J-@)�j=�� � �G�®H7�A= @:#,� ��h�2�3HKL�HKJ[@A:�?���=�@.�

�rCh��,w@)�j=7:<?a;���@Y:#,Fla%r:�=.��Jj:K@�@a=�J:%X@A:$�kEK@��j=�Jl@��j=�� � � �¥Ha�A= @��2��H=( :<JEJj=.( @A=.%3�A= @RC�bJ�@)���2���A=.( @)�¥:<JI6�==%�2�)( � �)�7HH��� /"�G�®HK?7?A='� � ��@>, :�?�? P @��EHI@�6�==(gH#� �w@)�j=�����(�H�!h=

f�?�� (D�E:K@A:�/�L�C61�J:�G� �<=F@)�j=�,¼H#/"� ��Ll:�,5r � H�%r?yHI@�� (>/lH�!:�� � E�@��j=�?a=7=*0� ��@Y@��j?a=�=�%�����=�?2/=�J[@��6(H_L+� �X@��EHI@�@��j=H( ?���@�� (gH#��:�?A;���@<, :<?7? P (�H<J @a=�J:% @A:U�¥JzEJ���@UL<Cg��,�@)�E=H( ?���@�� (gH#�:<?a;���@�@A=�Ju%���@a:{��J�zEJ���@UL�EP@)�j=gJ°H#� ��:#,(@��j=K( ?���@)�2(�H#�YGIH#� � =.�=/ � ��@=� ��=H�¥J =*��@)�E=�?>) P E/ P E�:<? ��:�/=F:#,B!E?A=.�G/XH#��=.��:#, / P Cfh�j='��=7@)�j?a=�=<%� �>=g?A=gJ�@h!>:������¥;�� �G��@)�¥=.����=RH�%l@a:�@��j=@)�j?a=�=�%�2�U@���Ju(,@�@UL+!h='��:#, � � �G�®HC��=�@��s, :<?M?QP�@��EHI@�(�:�/"!j?�� �A=�@��j=�����(�H�!h=hfP?)�2(D�j:K@a:�/ L<C�����������! WYfh�j="���)(gH#!>=3f�?�� (D�E:K@A:�/�L�Z,C�#�� �5� ��¡*¢7�\�w�O ��-U£%$� 8£<£%$�¢ � $'&1� £]¡ ��"£($:¢ "%$)¢D¢ �0$ � £ ���D 8� ��#��:£]¡�� ">?QP{£ ¢ �-3£����*)�+,$�¢ .-/ )10 " �B�¢ � "3£%$�¢0� $ � £ ���� #�328 8���:¢1¡"���Y¢ ¡H�� )<P54>£($:¢ ��W]?sPZ3�2¡H 76� B:£��A$"¡*¢*£< ��-

?sP � �MW]?sP+Z��2¡>  ���¢98�¡ ��-�¢ -=¡:$+� "D£ �B"�#�y¡ ��;<&%�8� ¡')>=h£%$�¢ $@?��2¡*¢@4B£%$�¢�� $)¢1�(;3 'A;¢�/FP�2¡ -8�2¡(B �8��u£ "%$6��; )<P.)

C )10 "��B�¢ ��"�£%$�¢ � $ � £]���D 8��2� #���:¢1¡>���Y¢ ¡>��>/ P j[.) P 4F£($:¢ ��W]? P Z>�2¡� "6M �u£8� $h¡*¢1£ ��"¡ �(; ���G¢0�*� ��¡*¢,- �*� $@2�¢1¡EDGF*�: �¡ ���1� $ �1�G¢1¡IHJ4� 8� ���%�� �D¢ :£ $ ���H K&%�#�£>£($:¢ �A$ �LA���.)�+,$+�2¡�D �¡*¢>- �.¢1¡ �#£ � �,�1� $M?�$:¢ t�I[ �)

N )10 " ���¢<� ">£%$�¢�� $ � £]���D #��28 8���:¢1¡����Y¢1¡���I  � $)¢*�(;3 'A;¢���"#) P £%$� #£��2¡ -8� O�¢ $)¢ :£ "%$2�P;/ P 4M£%$�¢ >�MWY? P Za�2¡= Q#��Y¢0$ �����¡:R#���*� $@2�¢S)

0 "C£%$�¢ �0$ � £ ���D 8��� $'&1� £]¡��¢'2�¢ $=¢1¡ �D ,��¢�£����T4M£($:¢ ��MW]?sPZ7�2¡7 7�%��3�¢,�1£ ¢,-$¡*¢1£U)9�=�G�=�?yH#�:� � �G�®H"��=�@��h� �G� � ��@A?yHI@)�¥J��@)�:� �(@a?)�2(D�j:<@A:�/ L-HKJ:%@%r?yH�63JK, ?a:�/ @)�j=C,¼H�/$� �¥L

6��j=�?a=C�I[ =XHK?a=��¥J:(1� � %r='%U��JU}s�G� � ?a= � C � CqV#u�Y¢ $ ����:¡:R8� �*� $@2�¢$�2�FH�G<=g?AL@�¥J�@a=�?a=.��@)�¥J��-@a:�!>:��¥:���� (gH#�B�)!EH�(�=<C�4�LU%j=1zEJ���@)�¥:<JBE

H�9+�¥=�?�!���Ju���+��( � ?aG<=3�2� HI!:�¥H<JEHK?=�A= @�@)�EHK@��2�7�j:�/=g:�/l:<?�!����2(@A: @)�j=t6Y=.�G� / ��Jj:863J9+��=g?)!:��J:�)�+�a(�HK?�!>= @@, ?aH�(,@aH��eCh4 � @�Hd9+�¥=�?�!��¥J:�)�;�a( � ?AG�=��EH���HKJ H���@a=�?aJEHI@���G�=�@A:�!h: /��:����2(�H#� (D�EHK?yH�( @A=g?)�XWgHK@)�¥:<J 96H<J[L-!��®HKJEH<?��A= @ @)�EHK@��2��(�:�/"!EH�( @.EQ( :�JjJj=.( @A='%gE5�¥:(�H��G�¥L( :<JEJj=.( @A=.%FE6Jj:86��j=g?A=H%r=gJ:�A=�E�H<J:%&�EH���@)�j=H!E?A:�!h=g?�@UL�@��EHI@HKJ�L @c6�:I( :�/"!��¥=*/l=gJ /@aHK?aLy%j:�/XH#�¥J:��H<?A=X;>: � J:%r=.%¨;�L&%�2� �U:���J[@��)�G/"!��¥=t(*��:;��='%&( � ?aG<='�a� �=��Jj:863J±@A: ;>=�j:�/l=�:�/l:<?�!����2(@A:�@)�E=t9+�¥=�?�!���Ju���+��(gHK?�!h=�@0? � @�ATC���:�?A=g:IG<=�?'Eg� � (D�°HI��=�@=� ��H � J�� /G<=�?D�aH#�s!��¥H<JEHK?a�A= @C��J @)�E="�A=�Ju��=@)�hHI@a��@7(�:<J[@aH���J:��H@�j:�/l=g:�/l:<?�!����2(=(�:�!�L�:#,(HKJ�L( :�/"!EH�(,@'E�( :<JEJj=.( @A=.%FEj:<Jj= /�%��G/l=�Ju���¥:<JEH��g� � ;u��=�@�:#,�@��j=a!��®HKJj=�C

� =Y?A=./lH<?)�7@)�EHK@�@��j=��A=.(�:<J:%a!hHK?A@�:#,r@��j=Y���)(gH#!>=�f�?��2(D�j:K@a:�/�L�6�H��gzE?D��@B!j?a:IG<=.%;[L �{( � � �G�¥=�J�? ��� A�CF�7=H���j:86�=.% @)�hHI@�@)�:� ��?a=.� � ��@*:<J��¥L{�j:�� %:�a� ,h� � ��HKJ:% _ �2�� �[Z (*��=gJ�@���LO(*��:;��=5@A:-ljCQ�bJ:%r=g=.%gE�@)�E=@(�?)��@)�2(�H��YGIH#� � ='��(gHKJ°Jj=gG<=g?�� ��=@�¥J /FPy6��j=�J�I[ �<, :<? � _ � ��/XH��G�����¥J:(�=�@)�E=�� /lH��<=�:�,4&�P�@A=�Ju%���@a: � �?=�H�� _ . ljC����E=�JK� � ��E@)�j=a� /XH#�<=�:�, & P @a=�J:%:��� H�� _ . ljC

�bJt}s�G� � ?a= � C �E;6�=��)�j:86 @)�j= _ !��®HKJj=<�¥J-@)�E=<(gH��A=��{[ =ECfh�j=�: � @A=�?�?a=*���¥:<JK��J@)���2�>� /XH#�<=7(�:<J:�)�2�U@D�3:#, _ /�GIH#� � ='�h, :<?�6���� (D� �MWY? P Zh�2��HtL�H<J�@a:<?>�A= @FH<J:%U�2�>(�H��G�¥=.%

Page 6: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

� ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

�K�r§*´ m�� o � �h��§�´ m�� m �

�g�r§*´���m��]p o��� }���} ��} s<Og��^X­ � �]Z���Qb^aSUQ*�®Og`����¦ §r©g���R��Zc�F§ ´ m�� o �<f�����������Z]Q ��»��gVISUOg` Qb^aS���Z]�§*´ m�� mgmRm � f�����������Z]Q��7»��gVISUOg`�Qb^aS�O��>�yZc`U�y�]^yQ�����V���Zc�w§*´ m��]p ogq��bf �!��������Z]Q��Fs<Z]^a`U��Z]V[Qb¬KZ� � `U\_^R�

@)�j=�6� B:£��A$�� �:�*�+¡,Cufh�j=$( =gJ�@a?aH��B%�2���U�2�3@��j=#"*�$"U�� � ¢ - �P;" #��&��J{6��:� (D� �MWY?QPZ� �3HHL�HKJ[@a:<?>�A= @3:#,���� /"!��¥=C(*��:;��='%t( � ?aG<=.�gCfh�j=�?a=*���¥:<J�;h=�@c6Y=g=�J«@)�E=.�A=F@c6�:3��=�@��h�2�(�H#� �¥=.%l@��j=��6�B3�¢ �*£ ¢ -B�¢1¡D¡�� � �1�+¡�H���@)�j=�� � �G�®H��A= @D�YH<?A=�H#� 6�H_L��( :<JEJj=.( @A=.%H6��j=�J _�G�¥=.����J«@)�:� �7?A=.���¥:<J6C:fh�E=�:K@)�E=�?�%�� �)�����J�@��j==(�:<JjJj='(,@a=.%rJj='�)�>��:( � �>( :�?A?a=.�)!h:�J:%5@A:#u�Y¢ $ �����¡:R#��$�#�G¢1¡7��JU6����2(D�5@��j=7( :�?A?a=.�)!>:<J:%�¥J���� � �G�®H"��=�@��3HK?a=79+�¥=�?�!���Ju���+�F( � ?AG�=.�gC

����% ����h��&H�'&H�5�#�����;� ���<��$:�����>���� ��

�bJ @����2�3��='(,@)�¥:<Jd6�=@!E?A:IG+�2%r=�H ?a: � ���p�)�<=�@�(D�n:#,F@)�j=I!j?a:�:#,�:#,7@��j= ����(�H�!h=IfP?)� /(D�j:K@a:�/�LHfh�E=�:<?a=*/�C

Page 7: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q �

��� }���}��E}�� ��^3����`�����^aSU^a`�������V[^(����^yV¡l´ �<�

}5�¥?D�U@K�¥= @-&�P°;>= :�Jj= :�,�@)�j=-( ?���@�� (gH#�7GIH�� � ='�-:�,=?sPnHKJu% H���� � /l=>&�P S )CPhC9�: (�&�P«;h:<@)�-�G�¥="��J�)<PhC6B�= @�(�� ;h=@c6�:@�AL+/$/l=�@A?�� (lHKJu%y%�2�Y�U:��¥J�@7( � ?AG�=.����L+�¥J����J%)CPnHKJ:%\( :<JEJj=.( @)�¥J�� (�&�P£@A:�� C(B6=�@�� � ;>= @��j=-�#� !j?A=.�G/XH��<=.�-:#, (��\, :�?;U[ �

9 9 9 �#��C � Jj=3(D�j=.(D��F=gH��)� ��L�@��EHI@�=gH�(D��� � �G�¥=.�<�¥J )CPuEw=10�@A=gJ:%���, ?A:�/ l5@A:

��E<H<J:%"( :�J�@yH#�¥J:��HC�)��J�����=>(�?)��@)�2(�H��!h:���J[@RC���=gJ:( = )CP*H<J:% /FP�HK?a=(Jj:K@�%�2�Y�U:��¥J�@RC�B�= @� ;>=lH5Jj=*� ����;>:<?��j:�:%�:#,h�8@��EHI@7( :�J[@aH#�¥J:�<(��¨H<J:%{�2�a�8� / , :��2%��AL+/$/l=�@A?�� (<C��±=/lH_L$(D�j:�:��A= ����:*@)�EHK@�� /l=�= @D��=RH�(D��� � ��J�H7���¥J�����=�H<?�(<C<B6=�@�� [�&3P�W��7ZQ6��j=�?a=6Y=a/XH_L5H��)� � /l=�@)�EHK@�� HKJ:%�� HK?a=7%�2� �U:���J[@RC:fh�j=�J�;>:K@���� H<J:%��~�G�¥=a�¥J ) P H��%r:*@��j=�� � C9�: �MW]? P Z�� �¥=.����JX@��j=C( :�/"!��¥=*/l=gJ�@�:#,w@��j= � J:��:�J-:#,��$E��3E�HKJu%@)�j=�� � E6���� (D�K� �YH=( :��G�¥=.( @)�¥:<JX:�,F�#�I��� /"!���L3(�:<JjJj='(,@a=.%H�A= @D�*E��

�9 9 9 � � � C;�T@�, :��G�¥:86>��=RH��)�G�¥L

@)�EHK@�=RH�(D� :�,�@)�j=�� � �2�</lH�!�!>=.% � J���GIH���=gJ�@���L�:�J�@a:�H5?A=.���¥:<J�@)�hHI@a( :<J[@yH#�¥J:�FH��G��:#,@)�j=�� " C:fh�j=gJU��@aH<J:%jH<?�%5@A='(D�jJ��2r � =.�h, ?a:�/�(�:�/"!��¥=10U%rL�JEH#/"�2(*���)�j:864@��EHI@�@��j=7�A= @:#,Q!h:���J[@���6��j:��A=�:<?a;���@��3?a=*/XH#�¥J@, :<?7H��G�6@)� /==��J�@��j= � J��¥:<J�:�,P@��j=�� " �2��HKL�HKJ[@a:<?��=�@ HKJ:%±@��j=5?a=.��@A?��2(,@)�¥:<J¨:�,�?QP @A:�@���� �$�A= @=�2�=(�:<J8� � �[HI@a=l@A:�@��j=5:�Jj=0/ ���2%r='%-�)���G,�@/lH�!�:<JI�8�y�AL+/�;>:��2��C�fh�:� ��L�H<J�@a:<?>�A= @��2���MWY?QP;Z C

}j:<?�(�H��A=��E�6Y=��EH_G�=3;�L H���� � /"!r@)�¥:<J@)�hHI@#)CP�H<J:% /FP*HK?a=>%�2� �U:���J[@�:�!h=gJ3%� �)��HKJ:%°H��G�Y:#,3@��j=t(�?)��@)�2(�H���!>:��¥J�@D�=� �¥=K�¥J�? 7 �P W�/FP�Z C5�±=t(*�¥H��G/ @��EHI@"? 7 �P W�/FP�Z=� � HKJ:�!>=�J HKJEJ � � � ���*CufP:H�A=�=�@)�:� �7Jj:K@a=�@)�EHK@�? 7 �P W�/FP�Z>(�HKJEJj:K@�;>=�H3( :��G�¥=.( @)�¥:<J�:#,M�8�%� �Y�U:��¥J[@ %�2�)�+�.E;, :<?'E��¥Jt� � (D�5H�(�H��A=�E[=gH�(D�X:#,6@��j=C%� �)���6�: � � %l;>=�/XH�!�!h='%5HK@��¥=gH���@� @A: � :<J[@A:8/FPEC:fh�j=g?A=*, :<?a=F@��j=�?a=a6�: � �2%5;>=�HK@>��=RH���@ =��y!E?A=.�G/XH#��=.��:#,�HKJ�Lt!h:���J[@��J/ P Eu( :�J[@A?yH�%�2(,@���J��@��j=a,¼H�( @�@)�hHI@3@)�E==%r=*��?A=g=�:#,�? P �2���8��C:�7=�J:(�=�HI@���=RH���@3@c6Y::#,�@)�j=3!j?a=*� /XH#��=.��:#,�/ P / � ��@7�¥J�@a=�?D��='(,@RCB4 � @�@)�j=gJFE6;�Ly�AL+/"/=�@A?aL�E6H��G��:�,Y@��j=.�A=!j?A=.�G/XH��<=.�5/ � �U@��¥J�@a=�?D��='(,@���:F@)�E= !E?A=.�G/XH#��=Y:�, / P � ��H<�)��J�����=(:�!h=gJ�( :�JjJj='(,@A='%��A= @RC4 � @7@)�E=�J�@��j=9>��=./lH<JjJ / � � ?�6���@:Wa, :�?)/ � �¥HH�G/"!�� ��='��@��EHI@��m/ � ��@F;>= HKJ H<JjJ � � � �@)�EHK@�( :<J[@yH#�¥J:�3H#� �w:#,P@)�j=7(�?)��@)�2(�H��g!h:���J[@��gC

Page 8: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

� ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

v�:86 �~� ��='�C��J @)�j=XH<JjJ � �¥H<?�?A=.���¥:<J @��EHI@7�A=*!EH<?aHK@A='� )CP�HKJu% /FP�HKJ:%-%�¥G+� %r='�@)���2��?A=.���¥:<J{�¥J[@A:-@c6�:t(*��:;��='%{� � ;EHKJEJ � �G� E � ! � HKJ:%��������,C�fh�j=gJy?QP�@aH��<=.�F=gH�(D� :#,@)�j='��=-H<JjJ � �G��H���HKJ&� @A: � ( :IG�=�?���J���:<J[@A:«@��j=X=�J[@)�¥?a=H( :�/"!��¥=*/l=gJ�@�:�, )CP�� /FPECfh��� �h� /"!�� ��='��@)�EHK@3@)�j=g?A=�HK?a=�H$!EH���?(:�,s� � ;EHKJEJ � �G��@��EHI@�HK?a=</lH�!�!>=.%«:<J[@A:��7Er:�Jj=:#,>6����2(D�O� ��='�=�¥J°=gH�(D� :#,�� !G� H<J:%¨@��j=�:<@)�j=g?=�¥J �������,C�LY:�J[@)�¥J � ��J��uE�:�Jj=t�)�j:86>�@)�EHK@�? 7 �P W�/FP+Zh( :�J:�)� ��@��3:�,�� �07 � :�!h=gJFE:%�2�Y�U:��¥J�@<� � ;hHKJjJ � � �]EE=gH�(D�«:#,Q6����2(D�I�G�¥=.�>��J@)�j=$}EHI@a: � �A= @RC�fh�E=�( :�/"!���=./l=�J[@7:#,�@��j= � J���:�J :#,�H��G�P:#,�@��j=.�A= H<JjJ � �G� W�@a:��<=�@)�j=g?6���@��7) P H<J:%8/ P Z��2��@��j=�J5@)�j= � � � �®H=�A= @�6��:� (D�K(gHKJ-;>=��)�j:863JX@a:�;>=FH"L�HKJ[@A:�?��A= @:#,���� /"!���=a(*��:;��='%t( � ?AG�=.�gC

�bJ:(1�2%r=�J[@yH#� ��L�EE@��j= H<;h:IG�=�HK?�� � /l=gJ�@���J:%�� (gHI@A='��6���L�@)���2�C(�H���==(gHKJjJE:K@��hH#!�!>=�J6��j=�JI��[��rCu9��J:(�=�� ! � HKJ:% � ���� HK?a=�=gH�(D�@/XH#!�!>=.%�H���� @A: � (�:IG<=g?)�¥J�����:�,�@��j=HKJjJ � � � �MT@`� ) P � / P � E#��@�, :�� �¥:86>��@)�EHK@�@)�j=�/l:% � � � ��:#,�=gH�(D�:#,>@)�E=.�A=>� � ;hHKJjJ � � �� �3=*0jH�(,@���LK�EH#�G,�:�,P@��j=7/l:% � � � �3:�,P@��j=�;�� ���<=g?�HKJjJ � � � �gC:fh���2��� /$!:�G�¥=.�(@��EHI@3@��j=�?a=� ��Jj: ?A:�:�/V�¥JKT-`� )CP�� /FP � , :�?�H<J[Ll:<@)�j=g? �A= @D�*E[=*� �G/"�¥JEHI@���J:��@��j=�!>:����)��;�� � ��@ULl:#,@)�j=�=*0+�2��@A=�Ju( =�:#, � HKJ:%t, � ?A@)�E=�?�!j?a=*� /XH#�<='�(:#, /gPECu9�:�&�PK(�HKJEJj:K@�� ��=a�¥J�/gP36��j=�J�I[ �rC

� =YJj:86i%r=.��( ?���;>=�@��j=MzEJhH#�+(�H��A=�:�,[@��j=Y���)(gH#!>=�fP?)�2(D�j:K@a:�/ L�6��j=g?A=�@��j=h( ?���@�� (gH#�G_H�� � =.�H�hH_G<= :�?A;���@��-@)�EHK@�=gG<=�J[@ � H#� ��L =.��(�H�!h=�@��j?a: � ��� @)�j= @A?yH#!\%r:�:�?.E�; � @�@��j=( ?���@�� (gH#��GIH#� � ='�X@)�E=*/3��=.��G�=.�H%j: Jj:K@K� �¥={��J�/FPEC��bJ @���� �t(�H���=�@��j= � � �G�®Hi�A= @K�2�-H9+��=g?)!:��J:�)�+��( � ?AG�=<C�fP:��)�j:86 @)���2�*E 6�=�Jj=�='% @A:°G�=�?�� , L£@)�j=�zEG<=I!E?A:�!h=g?�@���='�X@)�hHI@(D�EHK?yH�( @A=�?��XW�=�Hi9+�¥=�?�!��¥J:�)�;�>( � ?aG<=<C��T@l@ � ?AJu�X: � @l@��EHI@3, : � ?-HKJ:%nH-�EH#�G,�:#,7@��j=.�A=!j?A=.!h=g?�@���='�FH<?A=*@A?���G+�®H#�6@a:t�)�j:86�Cu}s�¥?���@�H<J:%{�A=.(�:<J:%gE��)��J:(�==6�=�HK?a=�H��)� � /"��J:�-@)�hHI@@)�j=U( ?���@�� (gH#�Y:<?a;���@=�G�=�J[@ � H��G�¥L¨=�J[@A=g?��*@)�j=«;EH����¥J°:#,<�kEs6Y=t�EH_G<=X@��EHI@@)�E=�� � �G�®H��=�@"�2�$���¥G<=gJ°;�L�TV`��Q? 7 �P W�) P Z,C�fh�EHK@"�2�.E �MW]? P Z�� �3T 6���@��d(�: � J[@yHK;��¥L&/lH<J�L%� �Y�U:��¥J[@.E���� /"!���LU( :�JjJj='(,@A='%gEE:�!h=gJ{��=�@��7?A=./:IG�=.%wC��7=�J:(�= �MWY? P Z��2��( :�/"!EH�( @7HKJ:%( :<JEJj=.( @A=.%6CFfh����?D%gEF�)��Ju( =-�MWY? P Z j[~T<EM��@7� �C��Jj:863J @)�EHK@ �MW]? P Z<(�H<JjJj:<@a(�:<J[@aH���JHKJ�Ll:�!>=�JK�A= @D�*E�A: �MWY?QP+ZM�2��JE:86��j=�?a=�%r=gJ:�A=<C+}j: � ?�@��FE���¥J:( =7@)�j=C( ?���@)�2(�H#�>:<?a;���@���H#� �@A=�Ju%l@A:$� HKJu%"�j=�J:(�=�%r:*JE:K@�� �¥=>��J5:<?�H�(*( � / � �®HI@a=�:<J �MWY?QP+Z E;��@�, :�� ��:86>��@)�EHK@ ?QP� �a��L+!>=�?a;h:��G�2(:<J ��W]?sP+Z�H<J:%&�U@yHKJ:%jH<?�%±HK?�� � /=gJ[@��C���E:86 @)�hHI@ �MWY?QP+Z<� �a�¥:(�H��G�¥L( :<JEJj=.( @A=.%¤W]��=g= ? � � A Z,CF��=gJ:( = �MW]?sPZ�, � �Gz:� � �*@)�j="zh?���@a, : � ?�:�,Y@��j=K(�:<J:%��@)�¥:<Ju��@A:;h=*H39+�¥=�?�!��¥J:���+�F( � ?AG�=<C

f�:KzEJ��2���y!j?A:IG+�¥J��X@��EHI@ �MWY?QP;Z��2�FHU9+�¥=�?�!��¥J:���+�Q( � ?aG<=�6Y= Jj=�='%�@a:@�)�j:86"@)�hHI@@)�j=*;>: � J:%jHK?��¥=.�(:#,�)CP-H���6Y=.�G�PH���H��G�6:�,P@��j=7!j?a=*� /XH#��=.�(:#,�)CP5H<?A=a�)�G/"!��¥==(1�¥:��A=.%( � ?aG<='��H<J:% @)�EHK@�@)�j='��=3;>: � J:%jHK?aL=( � ?AG�=.��HK?a=h!EH#�¥?�6�� �A= %�� �Y�U:��¥J[@gC�v7:86aE#���¥J:( = )CPC�2�H7��� /"!��¥L"(�:<JjJE=.(,@a=.%H(�:�/"!>:<Jj=gJ�@Y:#,w@)�E=�}EHK@A: � ��=�@.E;��@�, :�� ��:86>��@)�EHK@Y@��j=7;>: � Ju%jHK?aL:#, ) P �2���¥:(�H��G�¥L$(�:<JjJE=.(,@a=.%wC;��:86�=�G�=�?'EI@)���2�Y;h: � J:%jH<?AL$/XH_L��EH_G�=>!��¥J:(D�H!>:��¥J[@���H����J¨@��j=H(�H���=X:�,hr � H�%r?aHK@)�2( � � �G�®HI��=�@��7� � (D� H���@)�j=X;EH���� � � (gH�:<?*DF: � H�%rL�x ��?yHK;E;���@RCfh��� � �2�Y@)�j=�:<J:��LlJj:�J /���@aH<J:%jH<?�%3!>:<?A@)�¥:<J5:#,6@��j=C!j?A:�:�,UC;�7:86Y=gG<=�?'E�H���� � /"�¥J��*@��j=.�A=;h: � J:%jH<?)�¥=.�lHK?a=@��J:%j=�=.%¤�)�G/"!��¥=I(1�¥:��A=.%p( � ?aG<='�*E�@)�E=�L�(gHKJjJj:<@3��J[@A=g?��A=.( @;>=.(gH � �A=HKJ�L$� � (D�3�¥J[@A=�?D�A=.(,@���:�J3!h:���J[@ 6Y: � � %lJE=.( ='�)�aHK?�� ��L ;h=�H7( ?���@�� (gH#�u!h:���J[@gC+fh���2�M, :��G�¥:86>�;h='(�H � ��=�E1, :<?5��:�/=����G���a!h:86�=�? $�E.? "P @yH#�<='�w;>:K@)��:�,�@)�j='��=�( � ?aG<=.�w@A:3@��j=�;>: � Ju%jHK?aL:#, )CPuE#��:7@)�j=(:<?a;���@�:�,>H<J[LC�¥J[@A=�?D�A=.(,@���:�J=!>:��¥J[@Q/ � ��@s!EH��)�P@��j?A: � ���H�(�?)��@)�2(�H��+!h:���J[@���¥J:( =�? "P � �Q�¥:(�H#� �¥L=�3@a: � @)�E=�?a=<C8}j:�?P@��j=h!j?a:[:�,E@��EHI@�@)�E=.�A=Y;>: � Ju%jHK?���='��HK?a= �)�G/"!��¥=(1�¥:��A=.%@( � ?aG<='�*E+6�=�?A=*, =�?�@A: ? @�ATC

Page 9: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q �

����� � �� � �+���<� �s���c������& ���;� ���<��-� � ���h��)���� �F� �q��$(�HKJ ;>=t�A=�=gJi��Jp}5� � � ?A= � C ��E6@��j=�?a=5HK?a=5HI�®HK?��<=XJ � / ;h=g? :�,�9+��=g?)!:��J:�)�+���j:���='���J�@��j==!EH<?aH�/l= @A=g?>!��®HKJj='�h, :<?7@)�j='��==/lH�!:�gC:�±=��aH_L-@)�hHI@�� � (D� H"�j:���=7�hH���¢1¡ �D  �:¢£]�(;3¢�@�G,cE+, :<?3=gH�(D� _ �¥J5@��j=a�j:���=�E�@��j=a( ?���@�� (gH#�w:�?A;���@�� �®HKJ:%K��J)CPlHI@���@A=�?yHI@���:�J��Cqw!hHK?yH#/l= @a=�? _ � �a(gH#� ��='% @��j=3( =gJ�@a=�?�:�,�@)�E=39��=g?)!��¥J:�)�+�s�E:��¥=$�G,�@)�j=l:<?a;���@�:#,�@��j=( ?���@�� (gH#�F!>:��¥J[@���:�,M?sP�H#� �F�®HKJ:%�:�J�@)�j=7!>:��¥J[@FHI@a� ?yHI@)�E=�?3@��EHKJ�@a=�J:%«@a:@� C�fh�j=, :�� ��:86��¥J��l?A='� � ��@��2�h!j?A:IG�=.%K��J ? � A�E ? �� ATC�����������! �� +,$�¢ $)¢a�2¡a "� :� F*�:¢��D¢ u£�¢ $�� "a¢� 6� $ #u�Y¢ $ �����¡:R#� $�8� ¢ ) " �A$�¢6�P2�¢ $ 4M£%$�¢ $)¢ A$)¢�¢��+ 5�*£]� �<W �M` � Z1W �8�BZ�� 7�� #u�Y¢ $ �����¡:R#�[$ �8�G¢1¡ ?�� £($$¢1¡ �D ,��¢h£ �(;3¢�K��"£%$�¢ �: A$) P;"¢*£ ¢0$���G B�¢S)

fh�j=�!E?A:�:#,�:�,�@)�:� ��?a=.� � ��@ � �A=.�<r � H��)�2( :<J�, :<?�/lH��s� � ?��<=g?AL�@A='(D�jJ��2r � =.�F@A:U���E:86@)�EHK@�@)�E=�?a=>� ��H � J:� r � =<( =�J[@a=�?�:�,6=gH�(D�39��=g?)!��¥J:�)�+�u�j:���=�C��<�¥G<=gJl@)���2�*E�@)�E=7=.r � HK@)�¥:<J, :<?�@��j=K( =gJ[@A=�?D��:#,�@)�E=H�j:��¥=.�.E6JEH�/l=*�¥Ly? � 7 �P W;�*P;Z7[Nl:EF� ��=gH���� ��Ly��=g=�J±@A:�?a=.% � ( =@A:�HK!h:���L�Jj:�/"�¥H���=.r � HK@)�¥:<J :�,�%r=*��?A=g=KW �I` � Z1W �8�BZ�� 7�� E�HKJ:%{�A:-@��j=�?a:�:K@D��:#,�@)���2�=.r � HI@���:�J«H<?A=�H��G�F%�2��@)�¥J:(,@RC

q���HKJ�=10jH�/$!:��= :#,�@)�E= HK;>:IG<=�( : � J�@�:#,�9+�¥=�?�!��¥J:�)�;�5�j:��¥=.�.E�6��j=�J{�i[��X@��j=�?a=HK?a=K�{9+�¥=�?�!���Ju���+���E:��¥=.�a�¥J @)�E=K!EH<?aH�/=�@A=g?a!��¥H<Jj=H6���@)�°='�)(gH#!>=X@)� /l=���E � �@�j:���='�6���@��X=.��(�H#!>=(@)� /l=#=:E =����C9+�¥=�?�!��¥J:�)�;�:�j:��¥=.��6���@)�X='�)(gH#!>=(@)� /l= @�E<H<J:% � ��l �� ��� � ��j:��¥=.� 6���@)�5='�)(gH#!>=7@��G/l= � �EC;q��G�w:�,6@)�E=<!hHK?yH#/l= @a=�?D��, ?a:�/ @)�:� �h�®HK?��<=�(�:�� ��='(,@���:�J5:#,�j:��¥=.��@)� � ���EH_G<=�� � � �®H7��=�@���@)�EHK@(H<?A=��j:�/l=�:�/l:<?�!����2(#E��A:�@��j=FJhHI@ � ?aH��ur � =.��@)�¥:<JK� �:9HK?a=F@��j=7%rL�JEH�/$�2(*�(:�J5@��j=.�A=�� � �G�®H"��=�@���@��j=7�AH�/l=!

fh�j=�HKJ:�)6Y=g?�@A:l@)�:� �>r � ='�U@���:�Jt�2�h���¥G<=gJt�¥J ? "BA 9

�����������! �� W�����(�H#!>=afh�G/l=�LY:�J8� � ��H�( L�Z,C$#5¢1£? P W]X;ZM[\X � ^ _X � R;�FJ@?&%gW]X;Z�[\X � ^ �

X �?�$�¢ $)¢ _  B - �\  $)¢ ��  $) �;3¢1£ ¢ $D¡C£%$� #£�� �Y¢7�� #u�Y¢ $ �����¡:R#��$�#�G¢1¡9)/ )10 " _  ��- �����Y¢h��"£%$�¢�¡* �;3¢ #��Y¢0$ �����¡:R8�5$�#�G¢94�£%$�¢ a?sPK � - ? %   $)¢h£�� � �8� � A#���D #� � �

�%��'B �KA; 8£ ¢ ���£%$�¢*� $=�;����Y $¡*¢*£Y¡('C )10 " _  B - �O���Y¢<�� #u�Y¢ $ ����:¡:R8� $�#�G¢ ¡1?�� £($+-8� O�¢0$)¢ :£M¢1¡ �� ,��¢<£ �(;3¢ ¡ 4B£%$�¢ $?QPI ��-

? %   $)¢ �8£�£8�,��#� � A#���D 8� � ���%�� B �KA� #£�¢ ��y£($:¢1� $7�;����Y 3¡*¢1£]¡('N )M#u� �6��8¡1¢ _  ��-��m A$)¢��D¢ u£�¢ $D¡���"7-8� O�¢ $)¢ :£ #u�Y¢ $ ����:¡ R#� $ �8�G¢1¡K£%$� #£�$: �28¢@£($:¢

¡* �;3¢3¢ ¡ �D  �:¢$£]�(;3¢S)&#5¢1£�) &D¢" �� $ �(;�� £ �(28¢3W �U` � Z+* � $2� �8£#� "=� u� £'�K) +,$:¢ I? P ��-"?&%¤ A$)¢3£8�,��#� �:A����D 8� � � �6�B'B �KA; 8£ ¢ ��&£%$�¢*� $3���� �Y t¡1¢*£]¡=� "$ B ->�B:� �@� " 44"0�A$¡ ��;3¢7��u£�¢IA;¢0$ ;4, �{[�)s��� _ 4 � $, �{[�)s��� _ )+,$�¢ $)¢;"0� $)¢94�� " _  ��- �m  $)¢ �: A$� �;3¢1£ ¢0$D¡H� �8��!Ay��2-#� O�¢0$�¢ u£ #��Y¢0$ �����¡:R8��$�#�G¢ ¡

?�$�8¡1¢a¢1¡ �D  �:¢<£]�(;3¢ ¡�  $)¢<£%$�¢�¡* �;3¢@4s£($:¢ "?sPI B -<? %  A$)¢�£�� � �8� � A����� #� � ���%��'B �[8A� #£�¢@��U£($:¢1� $<�;����Y �¡*¢1£]¡>� "< B - �B:� ��� "�£%$�¢��D¢ :£ ¢0$D¡ � "�£%$�¢1¡*¢1#u�Y¢ $ ����:¡ R#� $�#�G¢ ¡$� P2�¢�£%$�¢$ K&%��28¢ � $2�,��¢ $ £'�K)

Page 10: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

pym ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

fh�j=y!E?A:�:#, :#,*@��j=�zh?���@@!hHK?A@«:�,*@)���2�«@)�E=�:<?a=*/ , :��G�¥:86>��;[Lkr � H����2( :�J, :<?�/XH#�� � ?��<=g?AL�@a=.(D�jJ:� r � ='��C�fh�E=>��='( :<Ju%�!hHK?A@s, :��G�¥:86>�s, ?a:�/ @��j=h,¼H�( @�@)�EHK@YHKJ�L=(�:<J8� � ��H�(�L;h=�@c6Y=g=�J=?QP�HKJ:%7? % / � �U@P@aH#��=���)CP3@a:���) % E�� /gP�@A:�� / % E_HKJ:%�@)�j= $ ��� !j?a=*� /XH#��=.�:#,�� /FP@a:@)�j==(�:<?a?A='��!>:<Ju%��J:�=!j?a=*� /XH#�<='�(:#,�� / % C:4 � @�@��j=a!j?a=*� /XH#�<='��:#,�/gP-HKJ:%/ % (�:<J[@aH���J��¥J��l@��j=�(�?)��@)�2(�H��B!>:��¥J�@D�7H<?A==�)!>=.(1�®H#� 9j@��j=*�¥?7;>: � J:%jHK?��¥=.�7H<?A=a/XH�!�!h='%��@A: � :�J�@a: @��j=*�¥?"�G/XH#��=.�.E�HKJ:%°@)�j='��=�HK?a=-@��j=�:<J��¥L&!j?A=.�G/XH��<=.� :#,�� /FP±HKJu%�� / %@)�EHK@��EH_G<=�@���� �M!j?A:�!h=g?�@UL�C#��=gJ:( =�EK@c6�:C� � (D�3( :�J8� � �[HI@A=�/XH�!:�Q/ � ��@��EH_G�=�@)�j=>�aH#/l==.��(�H#!>=�@��G/l='��C}s��JhH#� ��L�E�, :�?�!hHK?A@�@)�E?A=g=�E+��@�� �[Z ( =.��@A:"( :�J:�)� %r=g?�@��j=</lH�!:��@)�EHK@(H<?A=@)�j=a( =gJ[@A=�?D�(:#,P@)�j=a(�:<?a?A='��!>:<J:%���J����j:��¥=.�gC�4 � @(@)�j='��=C/XH#!u�3HK?a=7��( ?���@�� (gH#� ��L"zEJ:��@a=:���Jl@��j=>��=gJ:�A=(@)�EHK@�H#� �j:#,�@)�E=>( ?���@�� (gH#�j:�?A;���@���=gG<=gJ�@ � H#� �¥L=�®HKJ:%:<J@��j=hz�0r=.%"!>:��¥J[@�HI@� C�4YLUfh� � ?D�U@a:<JFx �>fh�j=g:<?a=*/ ? � A�E:�¥J�@)�E=*:<?���=gJ[@aHI@���:�JU!j?A='��=g?AG+�¥J��3(�H���=�Er@c6Y:H� � (D�/lH�!:�>(gHKJ�;>==( :�J8� � ��HK@A='%�;�L«H ���:<;�� � ��@A?yHKJ:��, :<?�/XHI@)�¥:<J�C�4 � @�� � (D��H3(�:<J8� � ��H�(�L/ � ��@ @��j=�J£@aH��<=U� @a:&� WY�)��Ju( ={� �2� @)�j=«:<J��¥Li� � !h=g?aHK@�@A?yH�( @)�¥J��{z:0�='%i!>:��¥J�@ ZHKJ:%@l*@a:3ltW]���¥J:( =al$�2��@��j=�:<J:��L3!j?a=*� /XH#��=F:�,Q��Z C��T@h, :�� ��:86>�Y@��EHI@(@)�E=C(�:<J8� � ��H�(�L/ � ��@F;>=�:�,�@��j=7, :�?)/ X ,. )5XhCufh�j=gJFE�( :�/"!EH<?)�¥J��H(�:[= Z (1�¥=�J[@�����J�@��j=�(�:<J8� � ��H�(�L=.r � HI@���:�J

)s?QPuW]X;ZM[\? % W )sX;Z���j:86>��@��EHI@ ) � 7 � [ � HKJ:%��¤[ )s� _ CF�bJ±@��j=H(�H���=l:#,3HKJ¨:�?)�¥=�J[@aHK@)�¥:<J ?A=gG<=�?D�)��J��( :<J#� � ��H�( L�E���@<�2��=gH��AL«@A:@(D�j=.(D�«@)�hHI@a?sP@�2�C( :<J#� � ��HK@A=*@a:@? P G+�¥HtX�,. X:E��A:�@)���2����¥G<=.�(H��G�6:�,�@��j=a!h:;�)�)�¥;���=7(�:<J8� � ��HI@a=<(�=�J[@A=g?��(:�,�9��=g?)!��¥J:�)�+�g�j:��¥=.�gC

fh��� ��?A='� � ��@ H��G�¥:86>� � �*@A:{����G�=5HI!j?a=.(*� �A=H( : � J[@�:#,3@��j=-J � / ;h=g?�:#,�%�����=�?a=�J[@( :<J#� � ��H�( LU(1�®H�����='�3:#,�='�)(gH#!>=�@)� /l=$9+�¥=�?�!��¥J:���+�Q( � ?aG<='�*EE;h='(�H � ��= :<J��¥L5@��j:��A=7�j:���='�@)�EHK@FHK?a=a��L+/"/l= @A?��2( � J:%r=g?�?A:<@aHK@)�¥:<J�;�Lt� � (*(�=.�����¥G<=a��r � HK?a=.�(:#,�H"!j?��G/"��@)�¥G<=HWY�@`� Z�* � ?a:�:K@3:#, � J���@UL5:<?3;�LK(�:�/"!��¥=10t(�:<J8� � ��HI@���:�Jt�hH_G<=F@)�j=7�aH#/l=a%rL�JEH#/"�2(*�gC�����������! W]v � / ;h=g?�:�,5LY:<J8� � �[H�(�L3L��®H�����='��C Z�+,$�¢�:�[;<&�¢ $�� "�£�� � �8� � A����� #� �%��'B �[8A� 6� ���1�G 8¡D¡1¢1¡�� "�¢1¡ �� ,��¢$£ �(;3¢�#��Y¢0$ �����¡:R#� �*� $@2�¢$�;����Y K¡*¢1£]¡�?�� £%$y¢1¡ �D  �:¢$£]�(;3¢ ��2¡A#�(28¢ & �/ )=W]�#�BZ � 7 �K� "h���2¡�� -5- 'C )=W]�#�BZ � 7 ���#�>^p� � 7H� "h���2¡=¢'2�¢ .)

}j:<?F=10jH#/"!��¥=�E�6��j=gJ��O[ ��E�6�==�EH_G�==��=g=�J�@��EHI@�@)�E=�?a= HK?a=�=*0jH�(,@���L =����H9+�¥=�?2/!���Ju���+���j:���='�Q��J@���� ��,¼H#/"�G�¥L=6���@)�l='�)(gH#!>=Y@��G/l= @:EK; � @�@��j=�?a=3HK?a=�:�J��¥L�� � @<%�����=�?a=�J[@( :<J#� � ��H�( L{(1�®H����A=.��:#,h� � (D�-/XH#!:�gCB9+� /"�G�®HK?���L�E�@)�j=g?A=lHK?a= � ��l �� ��� � �"9+��=g?)!:��J:�)�+��j:��¥=.�a6���@)�£=.��(�H#!>=l@)� /l= � ��; � @�:�J���L�@�l = @5@ .��� @H%� �>=g?A=gJ�@$( :<J#� � ��H�( Ly(*�¥H��)�A=.�.E��:U(1�¥=gH<?)�¥L�@)�j=g?A="�2��Ht��?A=RHI@�G_H<?)�¥= @UL�:�, %� ��=�?a=�J[@7%rL�JEH#/"�2(�H#��;>=*�EH_G+�¥:<?D�7:�J @��j=.�A==.��(�H#!>=�@)� /l==9+��=g?)!:��J:�)�+�F( � ?aG<=<� � � �¥H"�A= @��gC

fh�j=*?a=gH��A:<J@, :�?�@)�E=�%� �>=g?A=gJ[@FJ � /�;>=�?F:#,�(�:<J8� � �[H�(�Lt(1�®H����A=.�>6��E=�JI�i�2��=�G�=�JHKJ:%�:%�%�( :�/='�F, ?a:�/�@��j=�,¼H�(,@�@)�[email protected]�6��j=gJ=�H�2��:%�%FEg@��j=�?a=YH<?A=YJj:<9��=g?)!��¥J:�)�+���j:���='�@)�EHK@�/l=�=�@7@)�E=�?a=gH#��H80�2�=W¼H<J:%{�A:H�EH_G�=�Jj:K( :�/"!���=$( :�J8� � ��HK@A=7�E:��¥=.�DZ,Cuq���:�J��-@��j=?A=RH#��H#0+�2�3@��j=�?a=7�2��:�J���L�H3!EH���?7:#, � HKJu%r=*�¥;j?a:K@��A= @D�7HKJu%«@)�j= �{( � � �G�¥=�J{%r:�/XH#�¥J6Cq���HI(�:<J:�A=.r � =gJ:( =�Ew@)�E=�?a=-HK?a=-H#� 6�H_L��=10jH�( @)�¥L��y` � %� ��=�?a=�J[@�9+�¥=�?�!���Ju���+�M�j:���='�6���@��K( :�J8� � ��HK@A=>%rL�JEH�/"� (.��C+9�=�=�}s� � � ?a= � C �<, :�?YHa!��2(,@ � ?a=�:#,�@)�E=>!EHK?yH#/l=�@A=�?�!��¥H<Jj=6��j=�Jt�I[��jC;���j=�Jt�{� �Y=�G�=�JFE�@)�E=<�)��@ � HI@���:�JH�2��G�=�?aL"%�����=�?a=�J[@.��@)�E=�?a=��2��H��G6(H_L��H

Page 11: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q pRp

L�HKJ[@A:�?3Jj=.(D�+�®H�( =*H���:�J��@)�j=�Jj=*�[HI@)�¥G<=�?a=gH���H80�2�aW /l:<?a=*HK;>: � @7@)���2�>�¥J{9�=.( @)�¥:<J � Z C9�=�=C}s�G� � ?A='� � C �*HKJu% � C , :�?h!�� ( @ � ?a=.��:�,6@)�j=C!EH<?aH�/l= @A=g?�!:�¥H<Jj=.��6��j=�J@�{[ =lHKJ:%�I[ �rC

����� ���:��������� �Q������� �[��������� ��2��B����� � �����93=.(gH#� �E@��EHI@'E;6��E=�JK� � �E;� , � _ � � � � �5Z (*��=gJ[@)�¥LH��/XH#� � E &�P$� ��='���¥J0/FPHKJu%��MWY?QP+ZM�2�HaL�H<J[@A:<?Q�A= @�:#,���� /"!���=�(*��:;��='%=( � ?AG�=.�gC�fh�E=�=gJ[@)�¥?A=(?a=*���¥:<J=��J @)�j=�!hHK?yH#/l= @a=�?s!��¥H<Jj=, :<?�6��:� (D�l@���� �Y:(*( � ?��M� ��(�H#� �¥=.%@)�E=��{( � � � ��=gJK%r:�/XH���JFE�� C��T@ � ���[JE:863J ? �BAE@)�hHI@� � ��H<J :�!>=�JBEF��� /"!���L{(�:<JjJE=.(,@a=.% ?A=.���¥:<J @��EHI@C�2�F;>: � J:%r=.% ;�L Ht�)� /$!:��=3(1�¥:��A=.%( � ?aG<=�C �bJ @)���2�I�A=.(,@���:�J\6�=-%j=.��( ?���;>=-�A:�/l=±:#,*@��j=¨?A=./XHK?��_H<;��¥=y��@A? � (,@ � ?A= @)�hHI@� � ?a?a: � J:%:�� �¥J @)�j=@!EH<?aH�/=�@A=g?7!��®HKJj=�CQ9+�¥J:(�=5@��j=�?a=K�2��JE: �{( � � � ��=gJ�%r:�/XH���J6��j=�JU�{[k�Ej@)�j=7��@A? � (,@ � ?a=<6�=a%r=.��( ?��¥;h=�;>=*�¥:86\�2��HK;:�A=�J[@��¥J�@����2�>(�H���=�C

�bJp}s�G� � ?a= � C �y6Y=U%�2��!:�¥H_L¨@��j=U!EHK?yH#/l= @a=�?$!��®HKJj=t, :<?"� [ � @A:��<= @��j=�?$6���@)�@c6Y:-� � (.( ='�)�)��G�=H/XH#��J�� z�(�HI@���:�J:��H<?A: � J:%� CQv�:<@A=-@��EHI@@)�E=�?a=�H#!:!h=RHK?D��@A:±;>=�H( :�� �¥=.(,@���:�J�:�,�(1�¥:��A=.%$( � ?AG�=.�Q� � ?A?a: � Ju%��J:����� @��EHI@�!EH��)�P@)�E?A: � ���$/l:<?a=�H<J:%=/l:�?A=9+��=g?)!:��J:�)�+�h�j:��¥=.�lH�� @��j=.�A=I( � ?aG<='� H�!�!j?a:�H�(D� ��� C�L��¥:��A=�?$��Ju��!>=.( @)�¥:<J¤��=g=*/3� @A:��J:%�� (gHI@A=�@��EHI@3@��j=.�A=a( � ?aG<='�3H#�2��:$!EH��)��@��j?A: � ���U�)/lH��G�5( :�!:��='�(:#,�� HKJ:%r=.��;E?A:<@��A= @D�H�� 6Y=.�G��C�fh���2� � �h�¥J:%r=g=.%5@A? � =�EjH���@��j=<, :�� ��:86��¥J�� ?A='� � ��@�6�H����)�j:863Jt��J�? ��A�HKJ:% ? � A�C�����������! W¼9��¥J�����H<?A: � J:%�@��j= �{( � � �G�¥=�J&%r:�/lH���J�Z,C � �A$3¢� 6� $I�¤� �[4h£($:¢ "*�98"U�� � ¢ �- ��;3 8��-�2¡C¡ � $ $2�8� �-�¢,-E& �3��� :� £ ¢1� �<;3 � �"¡ �(; ���G¢>�*� �8¡1¢,-+�1� $@28¢1¡�� " "0�A$$"[ � �

9 9 9$� �2'��!At£%$�¢ � $2�,��¢ $ £ �H£($: 8£U-

/ )��� 5� $ �*� $@28¢�� " ¡ � $ $2�#� -�¡��  8¡ ?�¢*� �B �¡�� "�� � 4� ��-3£%$�¢�� "  5�,�1�[;$��G #£�¢���£($:¢ &%�#� -�  $ ����"C£%$�¢ "*��"I�� �G¢ - ��;3 8��i �¡ $ . �*)

C ) +,$�¢8�*� $@2�¢�� " ;3¢�¢*£]¡C£%$�¢8�D¢ :£ ¢0$D¡�� "�� �" #u�Y¢ $ �����¡:R#�>$ �8�G¢1¡ 4�¢D 6� $ ?�� £%$y¢1¡ �D  �:¢£]�(;3¢ $C^p��4�?�$:¢0$�¢

� �" [ WY�t`i��Zc� " 7 � ^ �9N ) +,$�¢ �1� $@28¢�� "  8� ¡ ��� 8¡�¡*¢1¡@£%$ $6�8�KAJ$�� �" �D¢ u£�¢ $D¡>��" &D K& � "� � -�¢1�X&0$2�#£$¡1¢*£]¡ 4

 ��-a£%$�¢1¡1¢ "� � -�¢1�X&0$2�#£s¡1¢*£]¡h ��- #��Y¢0$ �����¡:R8�,$�8� ¢ ¡� 8� £�¢ $ � #£�¢� �¡�£($:¢ �: A$) P;"¢*£ ¢0$?��� -8¡� A$2�#� -�� " )4YL Hy(�=�J[@A=g?X:#,�H±;EH<;�L'� HKJu%r=*�¥;j?a:K@3��=�@.EM6�=t/l=RHKJ @)�j=I!EH<?aH�/l= @A=g?3%r?aH�63J

, ?A:�/ @)�E=K/XH#�¥Ji(�HK?D%�¥:��2%±:#,(@)�j=�� HKJ:%j=*�¥;j?A:<@=��=�@7, :<?=6���� (D�¨@��j=K(�:<?a?A='��!>:<J:%���J��HI@�@a?aH�(,@���J��t( L(1�¥=7�2�FH�(,@ � H#� ��LU� � !h=g?aHK@�@A?yH�( @)�¥J��:Eu�eC =<C E>:<Jj=*:�,�@)�j=$( ?���@�� (gH#�5!h:���J[@��F:#,?sP$�2��!>=�?��¥:+%�� (<C;�T@�@ � ?AJ:�Y: � @Y@��EHI@3H#� �h:�,w@)�j='��=�;EH<;[L � H<J:%r=.��;j?a:K@h��=�@���HK?a=7; � ?)�¥=.%��J5@)�j=C��=gJ:�A=�@)�hHI@�@)�j=gL"%j: Jj:<@�@a: � (D�X@��j=�: � @a=�?�;h: � J:%jH<?ALl:#,�@)�j=C( :�JjJj=.( @A='%rJj=.�����:( � �a��J¨@��j=3!EH<?aH�/=�@A=g?a!��¥H<Jj=<C5fh�j=�Ji��@7� �7�[JE:863J @��EHI@HKJ�L{!EH<?aH�/=�@A=g?7%r?aH�63J, ?A:�/1@��j=@/lH���Jp(�HK?D%�¥:��2% :#,3@����2� � HKJ:%r=.��;E?A:<@���=�@$�hH��H � � �G�®H��A= @ @)�EHK@$�2�H#�2��:H�9+�¥=�?�!���Ju���+��( � ?aG<=<C�fh�j=.�A=y9+�¥=�?�!��¥J:���+�<( � ?aG<='�5HK?a={%rL�JEH�/$�2(�H��G�¥LnG<=�?aLp%�����=�?a=�J[@, ?A:�/ @)�E=�=.��(�H#!>=�@)� /l=�9+�¥=�?�!��¥J:���+�B( � ?aG<=.�h!E?A:% � (�=.%«;�LX@)�E=�����(�H�!h=afP?)�2(D�j:K@a:�/ L�E���¥J:( =�=gH�(D�3�EH��(H<J5HI@A@A?yH�(,@���J:��( L(*��=�6��j:��A=7;>: � Ju%jHK?aL3( � ?aG<='��HK?a=�%�� �Y�U:��¥J[@h�)�G/"!��¥=(1�¥:��A=.%3( � ?AG�=.��@��EHI@(HK?a=>��J�GIHK?��®HKJ[@ � J:%j=�?��A:�/l=>��@A=g?aHK@A=�:�,B? P C+}j:�?YHKJX='�)(gH#!>=3@)� /l=9+��=g?)!:��J:�)�+�;( � ?aG<=�Eg@)�E=�?a=��2��:<J��¥L�:<Jj=���J�GIHK?��¥H<J[@P;>: � J:%EHK?aLC( � ?aG<=�ERJEH�/=.��L ��) P C8}E:<?

Page 12: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p,o ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

��� }���}��E} � �[^���� `��g��^aSU^a`��[���gV�^��¥Og`�SU��^��®�g��Z]�cd*���E¦5§r©g�����gV[�*Qb^a\_^a`��g�r�*���RV�Z]¯[�,��SUZ]OgV[Qy�� ��^��a^yVISb`��g�(�<Z]Qb¬±Z]Q SU��^«i�,i � �]�]^yVn�<OR�*��Z�V���� � ��^�QbZ]������^-�y�]OgQb^,� � � `U\_^aQ Z]V¨SU�[^� ��^yOg`U^a� ���y� � � � ��� SU^(ORV�SU�[^3�rO � V�����`bd*Og��� �

H=!j?a:�:#,P@)�EHK@h!hHK?yH#/l= @a=�?D�M, ?a:�/ @��j=a/XH#�¥Jt(�H<?�%���:�� %���:�,P; � ?��¥=.%-;hHK;�L � H<J:%�¥=�;j?a:K@��=�@��3H#�2�A:L+�¥=*�2%I9+��=g?)!:��J:�)�+�F( � ?aG<='�*E�A=�=8? � ATC

��=�?a= H<?A=$��:�/l=�:#,�@)�j="�2%r=gH����¥J�G<:��¥G<='%U��J @)�E=�!j?a:�:#,�:�,�@)�j=?���J��;�FH<?A: � J:% �@)�j=g:<?a=*/�C����j=�J _ �aHI@)�2��zE=.� � _ �+� � 7�� � ! 3 � 7 � 5 EI:<Jj=h(D�j='(D�+��=RH��)�G�¥L�@)�EHK@ � &�P �� � �*P � C9�:�@)�j=>(�?)��@)�2(�H���(1�¥?�(*��= �hPa� �¥=.���¥J@)�j=3=*0[@a=�?���:�?�:#,���@D���G/XH��<=�E_@��j=>( ?���@)�2(�H#�:�G�¥Jj=<C�fh�j=�J@)�j=g?A=�� ��HH!E?A=.�G/XH#��=*:#,#�hPuE�� �P EE@��EHI@C� ��='�7: � @��)� %j=�:#, � P«H<J:%{�2��/XH�!�!h='%{�¨@a: �:<J[@A:�hPECFfh�E=�J @)�j=g?A=$� ��H<J : � @A=g?C!j?A=.�G/XH��<=:#,�� �P E���P E�@��EHI@a�2�C/lH�!�!>=.%y�°@A: �@A:�� �P E�HKJu%O��:�, :<?A@)�6Cs�±=-@)� � �=zEJ:% HKJO�¥JzEJ:��@a=@(�:�� ��='(,@)�¥:<J :�,�(1�¥:��A=.%O( � ?aG<=.�� "P/:IG+�¥J���: � @c6(HK?D%I, ?a:�/ @)�j=H(�?)��@)�2(�H��M(1�¥?�(*��="�¥J @��j=H%rL�JEH�/$�2(�H���!��®HKJj=-H<J:%��hH_G;�¥J��@)�j=3!j?a:�!>=�?A@UL�@��EHI@*=RH�(D��� "P ( :�J[@aH#�¥J:��=*0rH�(,@���L{� " � �8�p!h:���J[@D��@)�EHK@ HK?a="/XH#!�!>=.%;[L3? "P @a:�:�Jj=�:#,w@��j=C( ?���@�� (gH#�:!>:��¥J[@�����J+� P H<J:%l@)�E=��aH#/l=7J � / ;h=g?Y:�,F!h:���J[@���@)�hHI@HK?a=<�)�G/"� �¥H<?)�¥LH/XH#!:!h='%-@a:lH$!j?a=*!>:��¥=C��J� P C�v�:86 ? P @yH#�<='�Y@)�j=a�¥J[@A=�?��¥:<?(:#,�� P @A:

Page 13: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q p �@)�j=K(�:�/"!��¥=*/l=�J[@�:�,(@)�j=@( ?���@�� (gH#�M� ��JE=5H��*H<Ji�n@A: � (�:IG<=�?��¥J��U/lH�!6Cs9r: :<Jj=H(gHKJ( :<Ju���2%r=�?�@)�j=@/XH#!��sW _ ZH[ ?QP�W &�P+Z,CQfh���2�$� �H{/XH#!°@)�EHK@ @yH#�<='�*@��j=K!hHK?yH#/l= @a=�?!��¥H<Jj=@A:�@��j=H%rL�JEH#/"�2(�H���!��¥H<Jj=<C � JE="(gHKJ&���E:86 @)�hHI@�@)�:� �a/XH�!y�2� � J��¥GIH#�¥=�J[@*:<J=gH�(D�£:#,�@��j=t�&` � �A=.( @A:�?�����J£@)�j=@!EH<?aH�/=�@A=g?=!��®HKJj=«;>: � J:%j=.%°;�L¨@��j=@��@A?yH#� ���[@?aH_L�(@��j?A: � ����@��j= ����!��¥Jj='���X:�,�@)�j=��@` � �¥H<?)��=�� HKJ:%j=*�¥;j?A:<@��A= @�����L+/"/l= @a?)�2(�H��G�¥LHK?a?aH<J��<='%�HK?a: � J:%*@��j=�:�?)� ���¥J6C ��:�?A=g:IG<=�?'E��X@yH#�<='��=gH�(D�"� � (D�"��='(,@a:<?�:�J�@a:aT /"�¥J � �H�!hH#�¥?3:#,Q�EH#�G, / � �¥Jj=.�gC

v�:86\(�:<J:�)� %j=�?3H=!hHK?A@)�2( � �®HK? $ ��� !j?a=*� /lH��<=F:#,P:�Jj=�:#,6@��j=a( ?���@�� (gH#�g!>:��¥J[@�� ��L+�¥J����J � P @)�hHI@�� �¥=.�>�¥J � "P CFL�H#� �6@)�:� ��!h:���J[@�� P C�fh�E=�J�� P GIH<?)�¥=.�FHKJEH���L[@)�2(�H��G�¥LK6���@)� _H�� _ ?yHKJ���=.��:IG�=�?*=RH�(D� :#,3@��j=t�A=.( @A:�?��gC59�:y6Y=K(gHKJO(�:<J:�)� %j=�?*@��j=�H<JEH#�¥L[@)�2(3/XH#!��W _ Z [���7 � W�� P Z Cufh���2�>/XH�!�@yH#��=.�(@��j=7��='(,@a:<?>�¥J�@)�j==!EHK?yH#/l= @a=�?�!��®HKJE=�@A:3��@��A=*�G,UCfh�j=�J:<Jj=�(gHKJ��)�j:86 � ���¥J���@)�j=>9(D�+6(HK? WYB�=*/"/XH�@��EHI@���EH���H � J��2r � = z�0r=.%�!h:���J[@��J�@)���2���A=.(,@a:<?RCfh��� � z:0�='%H!>:��¥J[@h�2�3H�!EHK?yH#/l= @a=�? _� , :<?h6����2(D���sW _ ZM[�� P�� E�eC =<C E?sP � W &�P � Z"�®HKJ:%:�X:<Jn@)�E={���¥G<=gJ $ �'� !E?A=.�G/XH#��=�:#,�Hi( ?���@�� (gH#�>!>:��¥J[@gC�fh�E=�J =*��@)�j=g?@)���2�>( ?���@)�2(�H#�g!>:��¥J[@�:<?���@��3JE=*��HK@)�¥G<=<� ��z�0r=.%«;�LK? "�� �P � C:fh���2�h!j?a:+% � ( ='��H"( =�J[@a=�?3:#,H�;EHK;�L � HKJu%r=*�¥;j?a:K@��A= @7, :�?�=RH�(D� :#,(@��j=K����G�=�JO( ?���@)�2(�H#��!h:���J[@�� :<J �6"P W /l:% � �¥:HKJ-� %r=gJ[@)�Gzu(�HK@)�¥:<J H�� _ 6���Ju%���HK?a: � Ju%�@)�E=:�?)� ���¥JuZ C��±=3��� /"�G�®HK?��¥LI�<= @7(�=�J[@A=g?���:#,9+��=g?)!:��J:�)�+���j:��¥=.��;�L3��=�@�@���J����gP ;h=�H�!j?A=.�G/XH��<=7:�,�H�!j?a=*!>:��¥=��¥J:��@A=gH�%-:�,PH"( ?���@�� (gH#�!h:���J[@gC

����� � ������M�� �B��� �Q� � �fh�j=�?a=��2��H<Jj:K@��j=�?Q�U@a? � (,@ � ?A=�(gH#� ��='% 6M �u£8� $#�¢ � R#�G 5�D¢1¡�@)�hHI@P:(.( � ?D�F�¥J*;>:K@)��@)�j= %rL+/JEH#/"�2(�H#�wH<J:%3!EH<?aH�/=�@A=g?�!��®HKJE=.��, :<?Y@)�j='��=</XH#!:�gCq L�HKJ[@A:�?�Jj='(D�;�®H�(�=��2�(H=!��®HKJhHK?��=�@7@)�hHI@��2���j:�/l=�:�/l:<?�!����2(�@A:-@��j=7, :��G�¥:86��¥J��H��=�@gCFLY:<J:�)�2%r=�?7@)�j=3L�HKJ[@A:�?�/"� %:%��= /@)���¥?�%:���A= @��¥L+��J:�7��Jl@��j= � J���@ ��J[@a=�?aG_H��eC[93=*!:�¥H�( =�=RH�(D�l?A=./l:IG<=.%:�!h=gJH��J[@a=�?aG_H���6���@)�H7(1�¥?�( � �®HK?�:�!h=gJK%�2���$6��j:��A=�%�®H#/l=�@A=�?�� ��@)�j=<�AH�/l=7H���@)�j=���=gJ��K@��X:#,w@)�E=�?A=./l:IG<=.%��J[@A=g?AGIH���H<J:%{6��E:��A= ;>: � Ju%jHK?aL�@A: � (D�j=.�F@��j=�@c6�:«=�J:%!>:��¥J[@���:�,�@)�j=l?a=*/l:IG<='%I�¥J /@A=�?aGIH#��Cfh�j= � J��¥:<J5:�,�@��j==L�HKJ[@A:�? �A= @�6���@)��@)�j='��=C( : � J[@aHK;:��L3/XHKJ�LX:�!h=gJ@%�2�)�+� �2�H3L�H<J[@A:<?3JE=.(D�+�¥H�( =�C

f�:��A=�=�L�HKJ[@A:�?�Jj='(D�+�¥H�( =.�F��J�@)�j='��=�,¼H#/"�G� �¥=.�P:#,�/XH#!:�.E*6�=�?a=.��@A?��2(,@F, :�?s�)�G/"!�� �2(1��@UL@A:@)�j=7(gH��A=��{[k�@W @)�j: � ����@)�j=7�aH#/l=<!j?a:[:�,56�:<?����/l:<?a=�:<?h�¥=.���YG�=�?a;EHI@��G/�6��j=�J� � ��Z,CEB�= @�@��j=a, : � ?�( ?���@�� (gH#�g!>:��¥J[@���:�,Q?QPX;>==%r=gJj:K@a=.%«;�L�� A

9 9 9 � � 6��j=g?A=�@��j=

� � GIHK?aLlH<JEH#�¥L[@)�2(�H��G�¥L�6���@)� _ HKJ:%gE+6��j=gJ _ S�� � E � AaS�� � H<J:%l@)�E=F:<@)�j=g? (�?)��@�(gH#�!h:���J[@���HK?a=�H<?A?yHKJ���=.%�HK?a: � J:%«@)�j= :<?��G����JI�¥Jy(�: � J[@a=�?D(1�¥:+(D�+6��2��=C,¼H����:��:�J6Cu}E:<?�=gH�(D�_ �¥J�@)�j==(�:<JjJE=.(,@a=.%rJE=.������:( � �*E:6Y=a/XH_LK!:� (D��� � � ��:@��EHI@�@��j==(*��?D(1�¥=�:#,�?yH�%� � ��i� ��='�=�¥J )CPjC�B�= @���P ;h=�@)�j=@!j?a=*� /XH#�<=5:#,3@)�:� �"(1�¥?D(1�¥=-@)�hHI@�� �¥=.�$��J )<P±HKJu%O�¥= @�.P¨;>=�@)�j={(�:<?a?A='��!>:<J:%���J��-!j?a=*� /XH#�<=@�¥J2/FPECMfh�j=�J�(�:<J:�)� %j=�?@)�j=U, :��G�¥:86��¥J��±@c6Y:?A=.���¥:<J:� � A HKJ:%��

�C�fh�j=>�A= @ � A �2��@)�j=���/XH#� �¥=�?�:�,h@��j=3@c6�:�?A=.���¥:<J:��;>: � J:%j=.%l;�L�@��j=

( ?���@�� (gH#�E?yH_L��@)�j?a: � ��� � A H<J:% � � HKJ:%@��j=�( � ?AG�=.��� P H<J:% � P C�B�= @���

[ ` � A C+fh�j=�J? P /XH#!:�F;>:K@)� � A HKJ:% �

�� J:��GIH#�¥=�J[@���L�:IG<=g?�� A ���

�Cg9�@aHKJu%jHK?D%«@A='(D�jJ��2r � =.��, ?A:�/

( :�/"!��¥=10I%rL�JEH�/$�2(*��@)�j=gJ��)�j:86�@)�EHK@7@)�E==��=�@F:�,Q!>:��¥J[@���6��j:��A=*:�?A;���@��7?A=./XH#�¥Jt, :�?H#� �E@��G/l=��¥J�� A ���

��2�YH"L�HKJ[@A:�?��A= @.E.� � ��@(H��M�¥JK!EH<?�@ � :#,w@��j=1���)(gH#!>=�fP?)�2(D�j:K@a:�/ L<C

Page 14: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p � ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

v�:86�@)�j=g?A=�� ��H"z�0r='%{!>:��¥J�@<��J�� AH<J:% H3!E?A=.�G/XH#��=*:#,�@����2��z:0�='%I!>:��¥J�@<��J���HKJ:% :<Jj=H(D�E=.(D���=gH��)� ��L�@��EHI@*;>:K@��¨:#,�@��j=.�A=3!>:��¥J�@D�C�G�¥=3�¥J ��)CPjCs9+�G/"� �¥H<?)�¥L�E�@��j=�?a=

HK?a=�@c6�:t!E?A=.�G/XH#��=.�F:#,Y@)�j="!j?a=1z�0r='%{!>:��¥J�@a�¥J � /FPE6H<J:%y!j?A= / !E?A=.�G/XH#��=.�7:#,�@��j=.�A=!h:���J[@��C�¥J @c6Y:@!j?a=*� /lH��<='�7:�, � /FPrC5LY:<J[@���J � �¥J��@��J @)���2�C/XHKJjJE=�?'Eg6Y=$/XH_L H�%��U:���JH#� �h:�,>@��j=.�A=FH�!�!j?a:�!j?��®HI@A=�!E?A=.�G/XH#��=.��:#,</FP�@a:*@)�E=0��=gJ:%!>:��¥J[@�� �a��J-@)�j=aL�HKJ[@A:�?��A= @HKJ:%5@)�j=�?a=.� � ��@)�¥J��3��=�@>� �7H3L�H<J[@A:<?(Jj='(D�+�¥H�( =<C

f�:U��=g=HK@<�¥=gH��U@�Ht!h:�?�@���:�J :#,�@)�j=KL�HKJ[@a:<?�Jj=.(D�+�®H�( =$�¥J @)�E=$!hHK?yH#/l= @a=�?<!��¥H<Jj=6��j=�JU�{[k�E�(�:<J:�)�2%r=�?(@��j=�?A=.���¥:<J����¥JU!EHK?yH#/l= @a=�?h!��®HKJj=�@��EHI@>�2��@)�j=a!>:<?A@)�¥:<J«:#,@)�j=>%�2�)�*:#,�?aH�%� � ����(�=�J[@A=g?A='% HK@�@)�j=3:�?)� ���¥J*@��EHI@M� �¥=.�Q�¥J @��j=��¥=1,�@M�EH�� ,�!��¥H<Jj=<C;9+��Ju( =@)�j=7�A=.(�:<J:%t��@a=�?yHI@a=�:#,�@��j==(�?)��@)�2(�H��g!h:���J[@>�2�h���¥G<=�J«;�L-@��j=a/XH#!��sW _ Z�[ = _ ^ � �A=:E��@$, :�� ��:86>��@)�hHI@ �5W��KZ�� �lHy�EH#�G,�%�2�)� @��EHI@3��@A?�� ( @)�¥LO( :<J[@yH#�¥J:��� CQv7:86aEB, :�?l=gH�(D�_ S��IEs6�=t(�H<JO( :�J:�U@a? � (,@*@��j=UL�H<J�@a:<?=�A= @$��J°@)�j=@%rL�JEH#/"�2(�H#��!��®HKJj=5H���H<;h:IG�=<C} � ?A@)�j=g?)/l:�?A=�Eu�G, _ S��IE>@)�E=�J @��j="��=�@��

��2�F=RH��)�G�¥LI��=g=�J�@a:�;>="( :<J[@yH#�¥Jj=.%I�¥J @��j=

%� �)��� , :<?�=RH�(D� _ S��{EB!j?a:IG+� %r='% @)�j=-: � @A=g?�?yH�%� � � � � �jC�WYfh�E=@L�HKJ[@A:�?=�A= @( :<Ju�U@a? � ( @)�¥:<J@�j:��2%��(=gG<=�J@�G, _ �2�3Jj:<@���J«@)�E=7( :<JEJj=.( @A=.%jJj=.���h��:( � ��C Z

v�:86 z�0 H<J[LU!>:��¥J[@CX8P@�¥J @)�E=3L�H<J�@a:<?C��=�@gCg��=g?A="X8Pt� ��HH!>:��¥J[@a6���@�� Ht���¥G<=�J��@���Jj=g?aH<?AL3�¥J � A � �

�E���:"X8PGIHK?���='��H<JEH#�¥L[@)�2(�H��G�¥L"6���@�� _ C�fh� � � 6Y=C�EH_G�=�@c6�:�/XH�!:�

@)�EHK@ HK?a=3%r=1zEJE=.%¨:<J��IE6@)�j=H/XH�! � HKJu%-X8PECBfh�j=H/XH�! ���2�a��J�G�=�?A@)�¥;���=�EB�A:«:�Jj=/lH_Li(D�j='(D� =RH��)�G�¥L @)�EHK@ @��j=K/XH#! _ ,. ��7 � WYX8P;Z�@yH#��=.������Ju���2%r=K��@��A=*�G,UC�4�L @��j=9(D�+6�H<?:W�B6=./$/XH:Er@)���2��/XH#!�@)�E=�JI�EH��3H � J�� r � =Cz:0�='%@!h:���J[@>�¥J�� C:fh���2��!h:���J[@>�2�@)�j= � J:� r � =3!EH<?aH�/l= @A=g?C, :<?a6����2(D�±@)�j=H�A=.(�:<J:%-��@A=g?aHK@A=l:#,�@)�j=H(�?)��@)�2(�H���!h:���J[@D�*H#� ��¥H<J:%5:�J«@)�j=a����G�=�JK!>:��¥J[@>��J«@��j=�L�HKJ[@A:�?���=�@gC:�7=�J:(�=<6�=a�<=�@3:<Jj=a� � (D�t!hHK?yH#/l= @a=�?( :<?a?a=.�)!h:�J:%�¥J��«@A:�=gH�(D�&!h:���J[@=�¥J¨@)�j=UL�HKJ[@A:�?7��=�@.EPH<J:% @)�j='��=K!h:���J[@���@)�j=gJ&, :<?�/@)�j=aL�HKJ[@A:�?���=�@�!>:<?A@)�¥:<JX:�,>@��j=CL�H<J[@A:<?�Jj=.(D�+�®H�( =���JX@��j=�!hHK?yH#/l= @a=�?M!��®HKJE=<C�fP:7��= @@)�j=�:<@)�j=g?�!EHK?A@3:#,�@��j=�Jj='(D�;�®H�(�=�E+6�=�H�%��U:��¥J�@)�j=*H��)�A:(1�®HI@A='%t9+�¥=�?�!��¥J:�)�;�g�E:��¥=.�s� � ��@H��h6�=C%�� %@�¥J�@��j=7%rL�JEH�/$�2(�H��g!��®HKJj=a(�H���=�C

� Jj=h/lH_L =10�@A=gJ:% @)���2��JE=.(D�+�¥H�( =�@A:a�¥J:(*� � %j=�!EHK?yH#/l= @a=�?D�5, :<?M6����2(D�@)�j=�( ?���@�� (gH#�:<?a;���@D�F�®HKJ:%*:<J�@��j=�!>:<?A@)�¥:<J�:�,r@)�j=>L�HKJ[@a:<?5�A= @s��J � A�C89�=g= ? � A;, :<?5%j= @aH��G�2�gC'}s� � � ?a= � C =%� �)!��®H_L��@)�j=a!>:<?A@)�¥:<J�:#,�@)�E=�L�HKJ[@A:�?3Jj=.(D�+�®H�( =a(�:<?a?A='��!>:<J:%���J�� @A:3!hHK?yH#/l= @a=�?D��, :�?6���� (D�«@��j=7( ?���@�� (gH#�w:�?A;���@�� �®HKJ:%«:�Jt!>:��¥J[@����¥J��

�C

���� ����� � � �+� �� �q���/l=�J[@���:�Jj=.%«HK;>:IG<=�E�@��j=7�)��@ � HI@���:�Jt�¥J«@)�j=a(�H���=C�{[k���2�3G<=g?ALH%� ��=�?a=�J[@gC�±=�Jj:��:�J��<=g?��EH_G�=FH@��( � � � �¥=�Jt%j:�/XH#�¥J6C�9�HI@��j=�?'E<@��j=�, :�� ��:86��¥J��*?a=.� � ��@��2� �)�j:863J3�¥J ? � A 9�����������! �� #u� �5� ��¡*¢@�¥[���)E+,$�¢ 54$�� ¢'28¢0$ � �¢*�LAS$&6� $:$� �:- � "{£%$�¢ �A$ �LA��� ��£($:¢ �: A$) P;"¢*£ ¢0$ ���G ��¢@4=£%$�¢ $)¢- A$)¢���� :� £ ¢1� � ;3 � � -8�2¡(B �8��u£ � �:¢ \¡1¢*£]¡ � � 4#;k[� � �

9 9 94#�%��u£� #��:��!A �: A$) P;"¢*£ ¢0$D¡1$� P2.���At£($:¢ "0�#� � ��?����A � $2�,��¢ $ £]�Y¢1¡'-

/ )10 " _ S�� � 45£%$�¢ K£%$�¢<�;����Y a¡*¢*£ ��"M?QP@�2¡� �#u�Y¢ $ �����¡:R#� �*� $@2�¢94F¡ �=£($: 8£Q� " _ S�� � ��- �&S�� " 4M£($:¢=�;����Y 3¡*¢1£]¡<� ">?QPy B -�? %  A$�¢ $ ��;3¢6�P;�� $ �5$��� 'C )����£Q� " $ j[ ;4B£($:¢1;" ,�u¡�?QPI ��-�? %  A$�¢ �8£Q£�� � �8� � A#���D #� � ���6�B'B �KA; 8£ ¢���@£($:¢1� $

$)¢1¡'�:¢,�1£ �(28¢��;����Y "¡1¢*£]¡9)

Page 15: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q p q

��� }���}��>} k �rO�`bSUZ�OgV«O���SU�[^�»��gVISUO�`�V[^y��¬K�����y^�Z]V-SU�[^F����`�����^aSU^a`3�����gV[^7�®O�`3SU�[^F�®�g��Z]�cd����¦X§r©g��� ����O�SU^�SU�[^�����`U�R^�s<Z]^a`U��Z]V[Qb¬KZ[��OR�]^��g�]ORV���SU��^�V[^y�R��SUZ]\_^�`U^,�g�[���<Z]Q����gV[¬R^,�F�Id7S���OQb�*���]��^A`�s<Z]^a`U��Z]V[Qb¬KZ>��OR�]^yQ�����Z����-��`U^gfrZ]V-S � `UVhf[^,�g�����r��V[¬_^,�l�Id�S���O�Qb�*���]��^A`�s<Z]^a`U��Z]V[Qb¬KZ��OR�]^yQyf[^aSU�R�

�bJ@}s� � � ?a= � C 6�=a%�2��!��®H_LX@��j=C!EHK?yH#/l= @a=�?h!��®HKJj=C, :�?(@)�j=7(gH��A=��{[ �*@a:���= @)�E=�?6���@�� H{/XH��<J��Gzu(�HK@)�¥:<J6Csfh�j=@�¥H<?)��=K( =gJ�@a?aH���?a=*����:�Ji� � Jj:K@XH>�{( � � � ��=gJp%r:�/XH#�¥JF�?aHK@)�j=g?���@>�2�3H39+��=g?)!:��J:�)�+�g�j:��¥=*HKJu%K��@�%r:�=.�(Jj:<@�( :�J�@yH#�¥J5@��j=�:<?�� ���¥J6C

��� }���}E}� �[^P����`�����^aSU^a`w�����gV[^��®O�`wSU�[^6���g��Z]�cd7���g¦�§r©g�����gV[�F���*�g�gV[Zc¯��,� SUZ]ORV7�y^yVISU^a`U^,�� S�SU�[^(O�`UZ��gZ]Vh�

Page 16: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p � ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

� =C����= @D(D�-@)�E=<!E?A:�:#,P:#,6@���� �gC�±=C�)�EH��G�g�)�j:86 @)�EHK@(@)�j=g?A=�HK?a=��¥JzEJ���@A=.��LK/lH<J�L:�!>=�J3�¥J�@a=�?aGIH#�2����J ��7p��J-H<J[L Jj=*� ����;>:<?��j:�:%�:�,>@��j=�:�?)� ���¥J"��J3!EH<?aH�/l= @A=g?��)!EH�(�=���J6���� (D� @)�E=H( ?���@�� (gH#��:<?a;���@�=gG<=gJ�@ � H#� �¥L =.��(�H�!h='��Cgfh�j='��=H9+�¥=�?�!��¥J:���+���j:���='��Jj=�='% JE:K@�G�¥=*H#�¥:<J�� ��7��:6Y=a(D�j:�:��A=F@)���2�Q� � ��@3@A:H�)�G/"!�� �G, L-@)�E=a!j?A:�:�,UC93=.(gH#� �w@)�[email protected]�6��j=�JI�y[ ��Ej@��j=a, : � ?�( ?���@)�2(�H#�B!h:���J[@��FHKJ:%@, : � ?�!E?A=.!h:���='�3:#,M?QP

H#� �s� �¥=:�J @��j=3( ?���@�� (gH#��(1�¥?�(*��=:#,�?aH�%� � � � _ � �2! 3( =�J[@a=�?a=.% HK@�@)�j=l:<?�� ���¥J6C�q�� �A::E�@��j=��='( :<Ju%K� /XH#�<=�:�,�H#� �6:�,�@)�E=7( ?���@�� (gH#�g!>:��¥J�@D���2�h���¥G<=�J«;�L

? �P W��*P+ZM[ = _ ^ �=

HKJ:%&�A: _ ,. ?a�P W�� P ZC� ��HKJ HKJEH���L[@�� ($, � Ju(,@)�¥:<J¨:�, _ @)�EHK@=� ��HU�j:�/l=�:�/l:<?�!����2��/�C��,�` � � � @ � _ � l�E�@)�E=�J±:<Jj="(D�j=.(D���=gH���� �¥L�@)�EHK@�@��j=3( ?���@�� (gH#��(1�¥?D(1�¥=��2�C/XH#!�!>=.%�U@a?)�2(,@���Lt��J:�)�2%r=7��@��A=*�G,UC:�T@�, :��G�¥:86>��@)�hHI@��MWY?QP+Zh�2�7H3( :�JjJj='(,@A='%I��=�@7HKJ:%)CP-H<J:%�/FPHK?a=�%�2� �U:���J[@gC;�bJH!EH<?�@�� ( � �®HK?'EK@)�E=<�A=.(�:<J:%"� /XH#�<=7:#,w@)�E=<(�?)��@)�2(�H��:!>:��¥J�@ �¥H<J:%��Y:<JX@��j=?A=RH#�6H80�2�3HKJ:%t� ��='�h��J«@)�E=7( :�/"!��¥=*/l=gJ�@�:�, )CP"�¥J���C

� =�(*�¥H��G/ @)�EHK@6@)�E=�?a=�� �PHKJC��Ju( ?a=gH��)�¥J�����='r � =�J:(�= _ � _ � 9 9 9��J ��7y6���@�� _ � . l

HKJ:%�? �P�� W;�*P � ZI[ l ; � @U? !P W;�.P � Z � l&, :<?�H#� ��� � ;EC f�:p��=g=�@)���2�.E�Jj:<@A= @��EHI@'E���¥J:( =3?a�P W;� P Za[ = _ ^ � �?=uE�@)���2�ar � HKJ[@)��@UL{�¥J:( ?a=gH���='�</:�Jj:K@a:<J��2(�H��G�¥L�@A:86(HK?D% � �?=H�� _ . l 7�C v�:86�@)�j= :<?a;���@�:#, � �A= ?A=./lH���Ju�3��J � � , :<?�H��G����@a=�?yHI@���:�J:�X:#,7? AHKJ:%&%r='( ?a=gH��A=.�C/l:�Jj:K@a:<J��2(�H��G�¥L @a:{ljCB�7=�J:(�=�EB����G�=�Ji�iEg, :�? _ � �[Z (*��=gJ�@���L-��/XH��G� E? �P W;� P Z�H�� �A:U�G�¥=.�a�¥J ���~, :<?$��� ;��~� HKJ:%-/l:<?a=�:IG�=�?�@)�:� �7zEJ���@A=K�A=.r � =gJ:( =3�2�%r=.(�?A=RH��)��J��hC;v7:86\� � !�!>:��A= � � ) � lEC+� =<�EH_G<=<?��sW�Z � ?��BW �Z�, :<?3H#� ��{S�� � Cq>�2��:uEs? �� W;� � Z � ? �� W�� � Z �n� �A=ECB�7=�J:(�=@? �� W�� � Z � ? �� W;� � ZC, :�?H��G��;{, :�?�6��:� (D�? �� W;���uZ S�� ��C:fh�j=�?a=.� � ��@(@)�j=gJt, :��G�¥:86>�(;�LK( :�J[@)�¥J � ��@UL-:�,Q?QP"6���@)��?a=.�)!h='(,@3@A: _ Cfh� � �H6�={�EH_G<=���JzhJ���@a=*�¥L¤/lH<J�L¤9+�¥=�?�!��¥J:�)�;�<�j:���='�H�¥J @)�j=y!EHK?yH#/l=�@A=�?K!��¥H<Jj=( :<J�G�=�?����¥J��-@a:Ul@�¥J ��7�CB9+�¥J:( =l@��j=X=.��(�H�!h= @)� /l=.��:�,�@)�E=.�A=39��=g?)!��¥J:�)�+���j:��¥=.��H<?A=H#� ��%�����=�?a=�J[@.E��@h, :�� ��:86>�Y@��EHI@�HKJ�Ll@c6�:�!hHK?yH#/l= @a=�?D��%r?yH�63J3, ?a:�/�%�� , =g?A=gJ�@��j:���='� ��J@)���2�>( :��G�¥=.( @)�¥:<J@�EH_G<=�Jj:�J /�(�:<J8� � ��HI@a=a%rL�JEH#/"�2(*�aW¼H������E:863Jt�¥J{9�=.( @)�¥:<J Z,C

v�:K@a=3@)�hHI@ _ � [ ` � � � @jCB1����¥J��*@��j=�!E?A=gG;�¥: � ��:<;:�A=�?aGIHI@���:�JFE#6�=>/XH_L=zEJu%X:�!>=�J��J[@A=g?AGIH�� � � � HK;>: � @ _ � , :�? ;�[ � �

9 9 9�EH_G+��J���@��j=�!E?A:�!h=g?�@UL @)�[email protected]�� , _ S � � EK@)�E=�J

? �P W;� P Z S+/ P ErHKJ:%t��:�? � � �P W�� P ZhS�) P Cfh�j=�?a=1, :�?A=�E;? �P W�� P Z . � H����>. �kErHKJ:%@)�j=����)(gH#!>=CfP?�� (D�j:<@A:�/�Ll@)�E=�JI���E:86>��@��EHI@<�MW]? P Z �2��H39+��=g?)!:��J:�)�+�F( � ?aG<=�C

����� � ������ � �g�� �y����� �F����)���w�+�w�5�#���k���c� �k�2� �fh�j=�?a=>�2��HKJE:K@)�E=�?�6(H_L*@��EHI@�@)�j=<(�H��A=��{[\�a%�����=�?D���)�EHK?�!��¥L7, ?a:�/ @)�j=<(�H���=>� � �jC��=�?a=X@��j=t�)��@ � HI@���:�Ji��J�G�:��¥G<=.��@)�j=@�U@a? � ( @ � ?a=X:#,3@��j=�� � � �¥H���=�@��*:�,>? P 6��j=�J _ �2�(1�¥:��A=(@A:=lEC�q���6Y=��aH�6 HK;>:IG<=�E86��E=�J3� � ��H<J:% � _ � � �M�)/XH#� �E=gJj: � ���FE _ �G�¥=.�M��JX@��j=�{( � � �G�¥=�JH%r:�/XH#�¥JlH<J:%"��:�@��j=�( :�?A?a=.�)!>:<J:%�¥J���� � �G�®Ha��=�@���H<?A=3H��G6(H_L��L�H<J[@A:<?M�A= @D�:#,u(�:<J:(�=�J[@A?�� (���� /"!��¥=h(*��:;��='%7( � ?AG�=.�gC'4 � @�6��j=gJ=�{[k�ER@��j= � � � �®H��A= @���GIHK?aLa6�� � %�¥L��:#,�@A=gJFE�; � @(Jj:K@(H#� 6�H_L�.EK@��j=�LXHK?a=�9+�¥=�?�!��¥J:�)�;��( � ?AG�=.�gC;}E:<?�=*0jH#/"!��¥=�E;��J@}5� � � ?A= � C EJj:K@a=3@)�EHK@�@��j=�?a=>�2��H7(�:�!�L�:�,w@)�j=�� HKJ:%r=.��;E?A:<@���=�@�6��j:;��=���@aH��G� ��H�( @ � H��G�¥L =*0�@A=�Ju%��@A:-@��j=:<?��G����J6Cu���j=�JE=�G<=g? _ �2��(D�E:��A=�JI�¥J @)�:� �C�A= @.E �MWY?QP+Z�(�:<J[@aH���J:����/XH��G�s!��¥=.( ='�

Page 17: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q p �

§*´��3p © � §*´��3p ©Ip �

§*´���m�� mKp §*´���m�� mRmKp��� }���}��E} sKZ]^a`U�[Z]V[Qb¬KZ�� � `U\_^ ­ � �]Z���Qb^aSUQF�®Og`*\g��`UZ]O � Q�V�^y�_� SUZ�\R^\g�g� � ^yQ�Og��§ Z�V�SU��^�,��Qb^¡X´noK��kY�]�>O��6SU�[^yQb^�Qb^aSUQY��`U^��[Og��^yOR��O�`U�[��Z��gf[� � S�SU�[^��Kd�V[�g��Z]�,���h�r^y�[�,\<Z]O�`�OgVl^,�g���XZ]Q\R^a`bd ��Zc³j^a`U^yVIS,�

Page 18: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p � ·�OR�r^A`bS�¸6�[¹�^y\R��V[^ad

@)�EHK@XH<?A=@�j:�/=g:�/l:<?�!����2(X@a:±@)�j=I( :�?A?a=.�)!>:<J:%�¥J�� � � � �®Hy�A= @", ?a:�/0@��j=Ir � H�%r?yHI@�� (,¼H#/"�G�¥LpX;�a^ �5@a:���= @)�E=�?36���@)���¥JzEJ���@A=.��Ld/XHKJ�L�!j?a=*� /XH#��=.�l:#, ��)<P\W¼H��"6Y=.�G��H��:K@)�E=�?3; � ?��¥=.%@!h:���J[@D��Z C

�bJO}5� � � ?A= � C @:E6Jj:K@a=l@)�EHK@H���@)�j=K!EHK?yH#/l= @a=�?*H�!�!j?a:�H�(D�E=.�al�E6@)�j=K�)�XW�=-:�,�@��j=!j?A=.�G/XH��<=.��:#,�@)�j=�@A?yH#!H%r:�:<?���=g=*/3��@a:=%r=.(�?A=RH��A=�HKJ:%@��j=�� � � �¥H7�A= @ ��=g=*/3��@a:7�<?a:86���®HK?��<=�?RC �=fh���2�h� �>��J:%j=�=.%�@��j=7(�H���=�E+, :<?��¥J ? =BAB��@>6(H��h���E:863J 9�����������! ���� $:¢ U�O[m�[4h 8¡ _ . l54 �MW]?sP+Z8�%���2�¢ $�A;¢ ¡$£8�t£%$�¢��*� ��¡*¢,-@� :� £ -#�2¡:R��p£%$�¢��= #�+¡ - �A$%O�£8�,��#� � AB�K)�= p£%$�¢ �8£%$�¢ $�$: � -�41?�$:¢ &� � ��4>£%$+�2¡K�2¡� �8£C£($:¢�� 8¡*¢@4� 8¡"£%$�¢ �B #£8�#�-¡1¢*£� #� ?� ���¡7�%��u£� #���¡H �p B3u��G A$��%��; ��B�¢ :£<£($: 8£ �6�B:£  8��:¡ �$2�#� -� B3u����+¡ � ""¡ ��;3¢ �+¢ - ?���-8£%$ "0� $K 8� � _ � �8��!A����¡ ��;3¢+-8�2¡:R- K&%�8�£<£($:¢�A$ �LA���.)

fh��� �MzE?D�U@�!hHK?A@�:#,�@)�:� �Y?A='� � ��@��2���A:�/l=.6��EHI@�� � ?�!j?�� �)��J:�:E��)��Ju( =>��@��2�M6�=*� �u��Jj:863J@)�EHK@ � � �G�®HK�A= @D��(gHKJ�Jj=gG<=g?�( :�J[@aH#�¥J :�!h=gJ���=�@�� � J��¥=.����@��j= � � � �¥HK��=�@��2��@��j=�=gJ�@���?a=9���=./XHKJjJ��)!��j=g?A=�Cs9�:��j=�?a=H6�=3zEJ:% � � � �¥H���=�@��7��= @A@)�¥J�� H<?A;���@A?yHK?�� ��L�(*��:;��=-@A: @��j=� J���@<%�2����H�� _ . ljC � ,M(�: � ?D��=�E�6��j=gJ _ [ l�EE@)�j=g?A=7�2�7H<J���� /"!���:;���¥:<J �XHKJ:%�@��j=� � � �®H$�A= @>�2�(=.r � H���@A:l@��j= � J���@�(1�¥?�(*��=�ErJj:K@3@��j= � J���@�%�2����C

fh�j=*?a=gH��A:<JU6���L�@)�j='��= � � � �®HH��=�@���( :<J�G�=�?��<=�@A:5@)�j=$(1�¥:��A=.% � J���@C%�� �)���¥�2��H��, :�� ��:86>�gC:9 � !�!>:��A=�@��EHI@�@����2��� ��Jj:K@�@��j==(gH��A=<C�fh�j=gJFE:����G�=�J�HKJ�Lt� �[Z (1�¥=�J[@)�¥Lt�)/XH#� �� � l:E�6�=>/XH_L7zhJ:%-H7�A=.r � =gJ:( =�:�,F!EHK?yH#/l=�@A=�?D� _ � . l*HKJ:%XH<Jj:K@��j=�? ��='r � =�J:(�=�:#,!h:���J[@���X � S��V� � (D�«@)�EHK@<�MWY?QP � Z�+) �� W]X � Z�[ �=, :<?3=RH�(D�8;ECu��=�?a=�) �� W]X � Z � �(@��j=%� �)��:#,E?yH�%�� � �Q� � HK;>: � @QX � C89+��Ju( =��\� �Q(�:�/"!EH�( @.Eg@)�j=g?A=�� ��H�� � ;u��='r � =�J:(�=�:#,r@��j=hX �@)�EHK@�( :<J�G�=�?��<=.��@A:7�A:�/l=(Jj:�J5W�=g?A:C!>:��¥J�@�X S��*Cu}E:<?��¥JzEJ���@A=.��L�/lH<J�L=!EH<?aH�/=�@A=g?����J�@)�j=7(�:<?a?A='��!>:<Ju%��J:��� � ;:�A=.r � =�Ju( =�E6�=�@)�j=gJt�hH_G<=���W]?sP � Z��7) WYX ZM[��jC���=�Ju( =6Y=�/lH_LXH��)� � /l=FHK@�@)�E=F: � @��A= @Y@)�EHK@ 6�=FH<?A=<%r=gH��G�¥J��=6���@)�«H=� � ;:�A=.r � =gJ:( = _ � . l� � (D��@)�hHI@<�MWY? P�� Z�+) WYX Z�[��rC

v�:86m(�:<J:�)� %j=�?�@)�E="(*��?D(1�¥=l:#,�?aH�%� � � � X � (�=�J[@A=g?A='% HK@�@��j=X:<?�� ���¥J6Cgfh��� �a(*��?D(1�¥=/=g= @D�0) WYX Z"��J4H<J4HK?D( � :�,C�¥=�J:�K@)���KChL��j:�:��A= $d��: @��EHI@U� " �p� ����Ch9+��Ju( =_ � . l�Eg6�=$/XH_L�(D�E:[:;��= ;U�®HK?��<==�JE: � ���&�A:«@)�EHK@ � ? !P � WYX+Z�`pX;��� � �2��G<=g?AL{�)/XH#� �, :<? � � � � $gE�!E?A:IG+�2%r=.%{Xt� ��='�F: � @��)�2%r=�@��j="(1�¥?�(*��= :#,�?yH�%� � � � X � ���H( =gJ[@A=�?a=.% HI@@)�j=�:<?��G����J6C5�bJO!EH<?�@�� ( � �¥H<?.EB��@�, :�� ��:86>�*@)�EHK@$? "P � W �FZa�2� H{( � ?aG<=H6��j:;��=�HK?�� � /l=�J[@��J:(�?A=RH��A=.��;�LlHI@��¥=gH��U@�����E���C =�CGE�@)�E=�( � ?aG<=�? "P � W �FZ�63?yH#!u��HI@��¥=gH��U@Y:<J:(�=�HK?a: � J:%@��j=:<?��G����J6CFq��*HU( :�J:��='r � =�J:(�=�E�@��j=K( � ?AG�="? "P�� W �FZ</ � �U@7/l=�=�@�@)�j=tL�H<J[@A:<?�JE=.(D�+�¥H�( =��J @)�j=U%rL�JEH�/$�2(�H��h!��¥H<Jj=<C�4 � @l@)�E=«;>: � J:%jHK?aL¨:#,F@)���2�Jj=.(D�+�®H�( =t�G�¥=.�"�¥J �MW]? P�� Z C��=�Ju( = �MW]? P�� Z>/ � ��@C��J[@A=g?��A=.( @�@)�:� ��;>: � J:%EHK?aL<Cg9+��Ju( =*@��j=�� � � �®HK�A= @��2�F;EH�(D�+6(HK?D%��J�GIHK?��¥H<J[@.Es��@", :�� �¥:86>��@��EHI@ �MWY?QP � Z�/ � ��@"��J[@A=g?��A=.( @ ) W]X Z,CMfh�:� � @)�E=�JnL;�¥=*�2%��lH( :<J[@a?aH�%� ( @)�¥:<JFErH<J:%U��:@)�E=�?A='� � ��@h, :�� ��:86>�gC

� �!&6�B�B��� � �

pR�3NP����V[����� `��Ef ¶ �]fr¹(^y\g�gV�^adIf<·7��¸��]fj¸EOIOR¬rfr¹��ri���fjs<^,��� f ¶ ��fr�gV��-sK�����[Zc`UO�f����jsKZ]^a`bº��Z]V[Qb¬KZ�» � `U\_^�­ � �]Z���sK^aSUQ6�gV[��s<Z]V�� � ��� ` ¶ ^A`bS � `U��� SUZ]ORV[Q6O��h»�OR���[�]^a� ¶ Og�]d<V�OR��Z��g�]Qy���� �"!$#&% ')(+*-,.!/�1032/45#7680&452&97% ':2<;>=-0&?1@A,�9B?DC � �¼ogmRmRq �Afhp,m:� � ºUp,mRqRqI�

Page 19: y gQb^yQ ¡X´°o V ¡5¢±oK - BUmath.bu.edu/people/bob/papers/grebogi.pdf · y gQb^yQ ¡X´°o V ¡5¢±oK µ7¶ ^y gQb^7 R  aOg`b`U^yQb rOgV ^aV[ y^3SUO ·

p2sKZ�V�� � ��� ` ¶ ^a`bS � `U�[��SUZ]ORV�QYO���»�OR������^A�k�V����]dKSUZ]�3¹�d<V�����Z]�,�g�hsKd<QTSU^y��Q p �

oK�(¹�^y\R��V[^adIf<·���¸�� � �[^�i� i � �]��^aVX¹(OR�*��Z�VE�[s���SU^a���]ZcSU^7il�gV[��^y�]�<`UOgS�s<^aSUQ���V��lsKZ]^a`bº��Z]V[Qb¬KZ��YOR�]^yQy��� !/4��D!/��9 2<;�� ,:!&9�,�@ �10)2&45#7680&452/9B% 'D?U� � � �¼o�mRm � �Af�p � �gºUp(�Rm<�

�<�(¹�^y\R��V[^adIf_·7�<¸6��sKSb` � �AS � `U^�Og�>SU��^(i�,i � �]�]^yV¹(Og�*�gZ]V*Z]V�SU��^ ¶ ��`�����^aSU^a` ¶ ���gV�^yQ�¥Og`Y·�� SUZ]ORV�����il���[Qy����45#/9 ,�4-@ 2 ��2/@ *-,�9 2&@ % ':2�,A� ��� �¼o�mRmRq �Afjo � �,ºTo � qI�

���(¹�^y\R��V[^adIfK·���¸��[�gV[���(��`UZ ²�O�frk���­ � �]Z���s<^ASUQ�»�OgVK\_^a`U�RZ]V���SUOFSU��^(W�V�Z]S�¹(Z]Qb¬j�����.!��'1,:,.#/% 4��/? !�� @ *�,���� = C � � � �¼ogmgm � �Af � � paº�� � � �

qK�(¹�^y\R��V[^adIf�·��_¸6�I��V���¸EOIOR¬rf_¹��_i��IN � `UZ]^,�*s<Z]^a`U��Z]V[Qb¬KZ�» � `U\_^�­ � ��Z��7s<^ASUQy��6 % ?�'D�:,�@A,2&45#�� !/4-@ % 4��!��? 680&452/9B% ':2<;>=�0/?:@A,�9B?�� � � �¼ogmgm_q �Afhp,m �_q ºUpym:� � �

� �(¹�^y\R��V[^adIf�·7�E¸6��fE¸EOIOR¬rfE¹��hi��E�gV[�5WY��Z]V[Qb¬IdIf�¹�� � ��^��6Qb�,�g�r^ � `UZ]�A��OgSUOR�7dl�®O�`sKZ]V[� � ����`U�cd ¶ ^a`bS � `U�r^,��·�� SUZ]ORV����Eil�g��Qy� �145#/% 2&452"!>4-%$#/,���?:% @ 0B9 2/@ *-,�9 2/@ % 'D?&%�!���'�4 2<; � � �¼ogmgm_q �Afro � � º�o � qK�

�K�(¹�^y\R��V[^adIf�·��r¸��r�gV��lil��`UOgSbS��Kfhsj� � �[^�i� i � �]�]^yV-¹(OR�*��Z]Vh��·�Z]V[�gQ�k�`UO � V[� SU�[^NPO � V[�[� `bdI� (-� 2/4-?�2"'D@ % !/4-? !��8@ *-,(�&� =5C � �) �¼o�mRm � �Af �RoRqIpaº��_o � �<�

� �(¹�^y\R��V[^adIf[·7��¸��j��V�� ¶ Z��]��`UZ��lf+*��ri��j¹�d<V�����Z��7»�����QbQbZ]¯[�,��SUZ]OgV�Og���6Qb�,�g�r^ � Z���^sKZ]^a`U�[Z]V[Qb¬KZ>» � `U\_^(­ � �]Z��*s<^aSUQy� � O����[�r^,� `,�

�<�(¹�O � �g�<dIf<k��[��V��,� � �[��� `��Ef<­���k ¶ `UOIO��6Og� � � � `UQTSUORV�- Q � OR�rOR�]Og�RZ]�,�g�w»�����`����aSU^a`UZcº. ��SUZ]ORVXO��w·�� SUZ]ORV�����¤ � V[�aSUZ]OgV[Qy��� 'D@ 2�C���2/@ *�C ��/�� �Tp � � � �AfEo � ��º�o � �K�

pym<�(¹�O � �g�<dIf[k7���gV���� � ������`��jf�­<��wS � ��^�¹�d<V[�g��Z10 � ^3�<^yQ ¶ Og�cd�V32OR��^yQY»�Og���[�]^a�<^yQy�¶ � `bSUZ]^��Af4�5�6�; C��2&@ *-C5687 9���?�2/0 � � :<;�� �Tp � � ���A� ¶ ��`bSUZ]^Y�b�Af��=�6�; C���2/@ *�C5687 9 ��?�2/0� �:<; � �Tp � � q �A�

pgpR��i�,i � �]�]^yVhfY»Y��k � SUOg��Og`U�[��Z]Qb��Q�Og�7·�� SUZ]ORV�����il�g��Qy�(> !/; !/9 !/�@?�*�% '����45'D@ % !/4-?2&45#A��!$#� ; %��CBPOg���(pR��il� SU�h��s<�yZ���·�^yQy���eV[QTS,� ¶ � ����� �4; ��s<��`UZ]V��R^a`,f ��^a� ��O�`U¬�Tp(� � � �A�

p,oK��iZ]�]V[O�`,fj­�� 680&452/9B% 'D? % 4A9�45,D� !/9&?5; ,FEHG�2/�1% 2I6�; ,A� ¶ `UZ]V[�a^aSUORVlWYV[Z]\_^a`UQbZcSed ¶ `U^yQbQyf�¼o�mRm � �A�

p%�<��iZ]�]V[O�`,fE­��j��V�� � ��V-¸h^yZ��jkKJTs<Z]^a`U�[Z]V�Qb¬<Zw»���`U�r^aS<L��gQ�­ � �]Z�� s<^aS,��kY�[�r^aV���Zc� ¤ Z]V��^aOR��^aSb`bd*��V��*¹�d<V�����Z��aQPO��NM � �g�<`�� SUZ��Y·�� SUZ]ORV����Eil�g��Qy� ��EO?�,��1% 9�,�4�@AC���2/@ *-C ��Tp(�!� ���Af� � º � �K�

p ����iO�`UORQU�y���Kf3sr�]f ��Z]Qb�[Z]� � `��<f ���]f � ��V[Z]� � �A��Z�fYi���f��gV�� WY^,�[�Kf � �&>8!<; !/9 !&�P?�*�% '6 0/4 2/97% 'D?��r»��g�F��`UZ����g^(W�V�Z�\R^a`UQbZcS�d ¶ `U^aQbQ �¼o�mRmRm �A�

p,qK�(·�OI^yQb���hf ¶ � ~ V*»�����S � `U^RQjOgV[^yQ6�®O�`6SU�[^�¤�����Z��cdTS�� � ����´ ���<¦�§r©g�����g��V 680&452/9B% 'D?!&4�@ *�,VU % ,�9 2/4-4 =�?�*-,��:,Af�� � `UOR�r^,��V-il��SU��^y�*��SUZ]�,���wsKOI�yZ]^aS�d �¼ogmgm � �Afhp oIpaºUp �gm<�

p � ��ª¨�Kd<� � `UVhf��F� � � � OR�rOg��Og�RZ]�,���E»�����`����aSU^a`UZ . � SUZ�OgV�Og�rSU��^�s<Z]^a`U��Z�V�Qb¬KZ[» � `U\_^R�I���45#�C��2/@ *-C � �Tp(�_q � �Af �Rogm�º��Ro ���

p �K� �PORV��R���[^yV���f$�F� ~ V3SU��^�­ � ��Z��3sK^aSUQ�O��+M � �g�<`�� SUZ���·�� SUZ�OgV����[il���[Qy�+� !/9&?5; ,FE�G�2/�1%W�2�6�; ,�?�� ��� �Tp � �_o �Af>p,�<paºUp,� �K�