Chaos, catas­tro­phe and math­e­mat­ics

The Asian Age - - Oped - Jayant V. Nar­likar

History of math­e­mat­ics tells us that its role as an aid to de­scrib­ing nat­u­ral phe­nom­ena re­ally took off with Isaac New­ton. Amongst his many con­tri­bu­tions to maths and sci­ence was the new branch of math­e­mat­ics known to­day as cal­cu­lus. His deriva­tion of the laws of plan­e­tary mo­tion from his laws of mo­tion and grav­i­ta­tion was greatly fa­cil­i­tated by this new math­e­mat­i­cal tech­nique. As if that was not enough, New­ton made fun­da­men­tal dis­cov­er­ies like demon­strat­ing how sun­light is made up of light of seven colours, how to cal­cu­late the speed of sound in a medium, etc, be­sides leav­ing be­hind home­work for his fol­low­ers by iden­ti­fy­ing im­por­tant queries to be an­swer by them. For ex­am­ple, his query as to whether the force of grav­i­ta­tion acts on light also, was prop­erly an­swered only in the last cen­tury.

A gen­eral tru­ism that postNew­to­nian sci­en­tists came to be­lieve in was the con­ti­nu­ity of cause and ef­fect. If we throw a ball up in the air it at­tains a max­i­mum height be­fore com­ing down. What will hap­pen if the ball thrower puts in slight ex­tra en­ergy in his throw? The be­lief sub­stan­ti­ated by ob­ser­va­tions is that the ball will fur­ther rise to a slight ex­tra height. A man bor­rows money from the bank at a cer­tain in­ter­est spec­i­fied by the bank. What will hap­pen if that in­ter­est rate is slightly raised by the bank? Whether he likes it or not, the bor­rower will have to pay the bank a slightly more ex­tra in­ter­est.

The ef­fec­tive way in which New­ton’s meth­ods han­dled these and many other phe­nom­ena led sci­en­tists and math­e­ma­ti­cians to the be­lief that the New­to­nian frame­work can ac­count for all di­verse sit­u­a­tions. Even lay per­sons out­side sci­ence and maths came to feel that these in­puts from New­ton are suf­fi­cient to un­der­stand nat­u­ral phe­nom­ena. A poet like Alexan­der Pope un­der­scored his ad­mi­ra­tion for New­ton in his po­etic trib­ute when he wrote: “Na­ture and na­ture’s laws lay hid in night: God said let New­ton be and there was light.”

But in the 20th cen­tury new branches of math­e­mat­ics be­gan to sur­face and it be­came clear that in many sit­u­a­tions New­ton did not have the last word. For ex­am­ple, New­to­nian meth­ods dealt with sit­u­a­tions de­scribed above, which are of a “con­tin­u­ous” na­ture. But na­ture may have other types of sit­u­a­tions to which New­to­nian ideas can­not ap­ply. Imag­ine the plight of a moun­tain climber who is scal­ing a steep hill. As he goes up, he sud­denly dis­cov­ers that the next stage in­volves ne­go­ti­at­ing a precipice. Or, think of a dog­fight in which two dogs fight for a ter­ri­tory. Af­ter a nearly equal en­counter, one dog de­cides that enough is enough and runs away leav­ing the ter­ri­to­rial au­thor­ity to its ri­val. Are there math­e­mat­i­cal mod­els, which help us un­der­stand bet­ter what is go­ing on in such cases?

French math­e­ma­ti­cian Renè Thom showed how this can be done. There are phe­nom­ena char­ac­terised by abrupt changes which oc­cur in a dis­con­tin­u­ous mode. Such crit­i­cal changes, Thom ar­gued, may be char­ac­terised by the ad­jec­tive “cat­a­strophic”. These can­not be de­scribed by New­to­nian meth­ods but are amenable to the ideas and mod­els of Thom’s catas­tro­phe the­ory.

One may ap­ply the catas­tro­phe the­ory to some of the cel­e­brated bat­tles like Water­loo or Panipat, or Sin­hagad, where un­ex­pected de­vel­op­ments changed the out­come. A bat­tle like Panipat left be­hind last­ing changes and it is an in­ter­est­ing ex­er­cise to spec­u­late what would have hap­pened if the out­come of the bat­tle had been dif­fer­ent. In the cam­paign to win the fort of Sin­hagad, the at­tack­ing leader Tanaji was killed un­ex­pect­edly. But his un­cle She­lar mama stepped in his place and re­vived the morale of the in­vad­ing group to en­sure vic­tory. If at the Panipat, the Marathas had won con­clu­sively in­stead of los­ing dis­as­trously, would sub­se­quent history of the sub­con­ti­nent have been dif­fer­ent? And what would have been the po­lit­i­cal shape of Europe, had Napoleon re­ceived the timely re­in­force­ment at Water­loo? Rel­a­tively mi­nor hap­pen­ings in a bat­tle­field could lead to a dif­fer­ence be­tween vic­tory and de­feat.

In a sim­i­lar vein one can men­tion an­other branch of math­e­mat­ics called “chaos”. At first sight this name sounds strange. How can math­e­mat­ics which, in the lay mind is as­so­ci­ated with neat­ness and or­der, have any as­so­ci­a­tion with chaos? The an­swer is that the chaos it­self may be of a kind that is gen­er­ated by math­e­mat­ics! The fol­low­ing ex­am­ple will il­lus­trate this point. Take the ra­tio of cir­cum­fer­ence of a cir­cle to its di­am­e­ter, nor­mally de­noted by the Greek let­ter Pi. Now con­sider gen­er­at­ing a se­quence of numbers se­lected by tak­ing ev­ery third mem­ber in the dec­i­mal ex­pan­sion of Pi. The dec­i­mal ex­pan­sion pro­ceeds as 3.141596… with the dots warn­ing us that the ex­pan­sion is end­less. Thus, our se­quence of ev­ery third mem­ber will be­gin 3, 1, 6, etc. If you ask an ex­pert what is the next num­ber he will cal­cu­late and tell you. How­ever, he can­not give a rule, which will tell us what is the num­ber at a pre- spec­i­fied or­der in this se­quence. To any­one not know­ing where these numbers are from, the se­quence above would look chaotic, with no ap­par­ent or­der.

Like catas­tro­phe, chaos talks of numbers or events with no ap­par­ent rule gov­ern­ing them, although some un­der­ly­ing math­e­mat­i­cal method of their deriva­tion ex­ists. An ex­ag­ger­ated ex­am­ple of an event in me­te­o­rol­ogy is of­ten given: a but­ter­fly flap­ping its wings in Brazil de­vel­ops weather con­di­tions that lead to a tor­nado in Texas! In this case there may be an un­der­ly­ing rea­son but un­til our weather fore­cast­ing de­vel­ops as an ex­act sci­ence, the con­nec­tion is qual­i­ta­tive at best.

Chaos as a sub­ject, as a part of math­e­mat­ics tries to probe ap­par­ent chaotic con­di­tions so as to find the un­der­ly­ing rea­son. From a steadily and gen­tly flow­ing river to a Ni­a­gara type wa­ter­fall, where does chaos en­ter the scene and how does maths cope with the grow­ing chaotic changes that we see?

Such con­sid­er­a­tions show that be­yond New­ton there are ter­ri­to­ries for math­e­mat­ics to con­quer. Catas­tro­phe and chaos pose chal­lenges to them and tempt them with glimpses of new vis­tas where even New­ton feared to tread.

The ef­fec­tive way in which New­ton’s meth­ods han­dled var­i­ous phe­nom­ena led sci­en­tists and math­e­ma­ti­cians to the be­lief that the New­to­nian frame­work can ac­count for all di­verse sit­u­a­tions

The writer, a renowned as­tro­physi­cist, is pro­fes­sor emer­i­tus at In­ter- Uni­ver­sity Cen­tre for Astron­omy and As­tro­physics, Pune Uni­ver­sity Cam­pus. He was Cam­bridge Uni­ver­sity’s Se­nior Wran­gler in Maths in 1959.

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