
homoclinic kaksha

gatishil pranalinchya abhyasat, homoclinic kaksha haah tappyachya jagetoon jaanara eka marg aahe joou kathichya samtol bindula swatashi jodato. adhik tantotantpane, eka samalingi kaksha sthir bahugunit aanee samtol asthir bahugunitchya chhedanbindumadhye asate. hii eka visham-clinic kaksha aahe-konatyahi done samtol bindunmadhil eka marg-jyamadhye antim bindu eka aanee samaan aahet.
samanya vibhedak samikaranadvare varnan kelelya satat gatishil pranalicha vichar qaraa.
samaja sootra _ 2 varr samtol aahe, tarr samadhan sootra _ 3 hii samalingi kaksha aahe jarr
jarr tappyachya jaagechi teen kinwa adhik parimane asateel, tarr kathichya binduchya asthir anekpatanchya topologycha vichar karane mahatvache aahe. aakadevari done prakarane darshavate. pratham, jevha sthir manifold heey topologicaldrushtya eka cylinder asate aanee dusare mhanje, jevha asthir manifold heey topologicaldrushtya mobius patti asate; yaa prakaranat homoclinic kakshela "twisted" mhanatat.
swatantra gatishil pranali.
samalingi kaksha aanee samalingi bindu punaravrutti kelelya karyansathi tyach prakare paribhashit kele jaatat, jase sthir sanch aanee pranalichya kahee nishchit bindu kinwa niyatkalik binduchya asthir sanchache chhedanbindu. 
swatantra gatishil pranalincha vichar kartana aaplyala homoclinic kakshachi kalpana dekhil aahe. asha paristhitit, jarr sootra 5 heey bahugunit sootra 6 chey bhinnarup asel, tarr aapan mhanato qee sootra 7 haah eka samalingi bindu aahe, jarr tyacha bhootkal aanee bhavishyakal samaan asel-adhik visheshtah, jarr ekhada nishchit (kinwa niyatkalik) bindu astitvat asel tarr. 
sootra _ 8 asey qee
malmatta.
eka samalingi binduche astitva heey tyanchya anant sankhyeche astitva soochit karate.
heey tyachya vyakhyevarun yeteh sthir aanee asthir sanchache chhedanbindu. donhi sanch vyakhyenusar aparivartaniy aahet, jyacha arth assa aahe qee homoclinic binduchi pudhe punaravrutti sthir aanee asthir asha donhi sanchavar aahe. n vela punaravrutti karoon, nakasha sthir sanchadvare samtol binduparyant pohochato, parantu pratyek punaravruttimadhye too asthir manifoldver dekhil asato, joou haah gunadharm darshavito.
haah gunadharm soochit karato qee samalingi binduchya astitvamule guntaguntichi gatishilata udbhavate. kharokhar, smelane (1967) dakhavun dile qee heey bindu ghodyachya nalyachya nakashakade netat, jase qee gatishilata, jee arajakateshi sambandhit aahe.
pratikatmak gatishilata.
markov vibhajan vaparoon, sanketik gatishilatechya tantracha vapar karoon hyperbolic pranalichya dirghakalin vartanacha abhyas keela jaau shakato. yaa prakaranat, homoclinic kakshache visheshtah sope aanee spasht pratinidhitva asate. samaja sootra _ 10 haah "m" chinhancha maryadit sanch aahe. x" yaa binduchi gatishilata nantar chinhanchya dvi-anant stringdware darshavili jaate. 
pranalicha aavartak bindu haah fakt aksharancha aavarti kram asato. nantar heteroclinic kaksha mhanje done bhinn niyatkalik kakshanche sanyojan. asey lihile jaau shakate
jethe sootra _ 13 haah lambichya 'k' chinhancha kram aahe, (arthat, sootra _ 14), aanee sootra _ 15 haah chinhancha aanakhi eka kram aahe, lambicha 'm' (tyachpramane, sootra _ 16). sanketan sootra _ 17 heey fakt "pii" chee anant vela punaravrutti darshavate. asha prakare, heteroclinic kaksha heey eka niyatkalik kakshetoon dusya kakshet sankraman mhanoon samajale jaau shakate. yaulat, samalingi kaksha asha prakare lihili jaau shakate 
madhyavarti anukram sootra _ 19 rikt naslyas aanee, arthat, "pii" naslyas, anyatha, kaksha fakt sootra _ 17 asel.