7/23/2019 5901 Problem Solution Essay Tianyu Wang

http://slidepdf.com/reader/full/5901-problem-solution-essay-tianyu-wang 1/3

 

The hyperbolic set inner the Henon Map

 Tianyu Wang

Course 5901, Ohio State University

7/23/2019 5901 Problem Solution Essay Tianyu Wang

http://slidepdf.com/reader/full/5901-problem-solution-essay-tianyu-wang 2/3

Suary an! Coents "or Sterling an! Meiss#s paper 

  $n the paper %Coputing perio!ic orbits using the anti&integrable liit', ()Sterling an! *)()Meiss+199-

coe up .ith a proo" o" the e/plicit paraeter#s continuous one&to&one e/tension boun!ary o" the Henon Map by

applying the etho! o" %$&iit')

  $n the "irst section, the authors cite the !e"inition an! a!aptive properties o" the classic Henon Map .ith t.o

real variables an! $&iit +.hich stan!s "or the anti&integrable liit- "ro S)*)ubury#s+1995- an!

2)S)Mac3ay#s+1994- papers) ase! on the eleentary bac3grou! an! other 3no.n "acts an! prove! theores

relate! to the trans"oration bet.een or!inary Henon paraeters an! the sybolic representation o" the orbit at

the respecte! $&liit, the author gives a thorough theoretical proo" o" the range o" the paraeters .here the non&

.an!ering orbits o" the Henon aps are continuously e/tensive .hich has been sho.e! in nuerical calculation

 by M)*)(avis+1991-) T.o ost essential steps o" the proo" consist o" the construction o" a pair o" nuber

se6uences .hich serve as the upper an! lo.er boun!ary o" the agnitu!e o" the orbit, as .ell as the application

o" copresse! apping theore to help !eterine the e/istense an! the uni6ueness inner certain range) $n the

last part o" his paper, the author re"orulates iha an! Wen7el#s etho! on e/aining the siilar syste .ith

an uni6ue attractive "i/e! point an! a3es a coparison bet.een !i""erent etho!s, provi!ing the nuerical

 boun!ary .ith ore insight an! intuitive otivation)

2egar!ing the ascen!ency o" author#s result over other analogous papers, .hich is re"lecte! in the e""iciency

o" capability to e/ten! to all the boun!e! orbit o" the Henon ap, together .ith ho. eleentary an! intuitive the

etho! the author utili7e! to solve the proble is, $n!ee! $ personally thin3 highly o" the value o" this paper)

7/23/2019 5901 Problem Solution Essay Tianyu Wang

http://slidepdf.com/reader/full/5901-problem-solution-essay-tianyu-wang 3/3

2e"erence

S)*)ubury)+1995- Physica D, 86, 48)

2)S)Mac3ay, *)()Meiss)+1994- Nonlinearity 5, 189)

M)*)(avis, 2)S)Mac3ay, )Sannai +1991- Physica D, 52, 11