The last two chapters provide means how to grapple with some real systems with variables and parameters that keep on changing. Although these could be mathematized, there exist approaches that require few more notions, types, definitions, theorems etc. There are many for theorizing, in some more depth, ideas on bifurcation, and catastrophes. We begin with a resume of what we have learnt in the earlier chapters. We take bifurcation first, then catastrophes and after that, interweaving one with the other to the extent they facilitate both in understanding of and pushing further concepts. But at every stage, we reckon non-linearity whose effect is to bend the solution branches, which we have already met in the example -u″ (x) = λ ∥u∥2u(x), 0 < x < π. We can call this as a mildly non-linear problem, in the sense, that ∥u∥2u does not reflect a genuine non-linearity (or pseudo non-linearity) so well represented by, say, f(xu)u where f(u) is nothing but a vector, which turns the vector to a new direction in the space.

Resumé on Bifurcations

Let us recall that our objective has been not merely to solve the system of equations in variables x = (x1, x2, …, xn)∈ Rn of the form

but also to investigate the behaviour of solution of (1) as the real scalar parameter λ varies; F = (F1, F2, …, Fm) is a non-linear mapping with the values in Rm.

Poincare–Bendixson theorem

Any text on nonlinear systems can hardly afford, not to refer to the celebrated result called ‘Poincare–Bendixson Theorem’. It was first established by the French mathematician Henri Poincare, in 1880 and later on, simplified by the Swiss mathematician Ivan Bendixson in 1901. It is known to be an indispensably vital result that asserts the existence of a periodic solution for a general class of non-linear systems. The theorem has been stated earlier without a formal proof, in connection with periodic orbits, which, of course, focus its primary concern. The ideas in its proof are relatively difficult and so, the proof is not set forth on the first exposure to this theorem. Perhaps, it will now be fairly comfortable to have glimpses of ways and ideas leading to the proof of this theorem. The proofs are achieved apparently in different ways but they do depend on concepts and results treated in the early part of this book. Poincare-Bendixson theorem (PB) has, on the face of it, different versions, as shown below in different sections.

Revisiting concepts treated earlier

We look upon the PB theorem as a basic wherewithal even for understanding planar dynamical systems, which we have already met. For this, we need to bank upon some properties of the limiting behaviour of orbits at the level of abstract topological dynamics, followed by an analysis of the flow near non-equilibrium points of a dynamical system.

ARUNACHAL

Demography

Arunachal Pradesh is the largest and least developed hill state of the region. It covers the mountain ranges of the eastern Himalayas and is located to the north, east and south-east of Assam. Its neighbours besides Assam include Tibet (China) to its north and east, Myanmar to its east, Bhutan to its west and Nagaland to its south-west. Its 2001 census population is 10,91,117.

It has well over 100 ethnic groups of which 20 are considered to be major tribes. The demographic profile of its 20 main tribes is:

• Adis (Padam, Minyong, Pasi, Pangi etc) 27%, (inhabitants of Siang and Dibang)

• Nissi/Dafla 22%, (inhabitants of Subansiri and South Kameng)

• Wangchoos 8%, (inhabitants of Tirap ethnic lineage akin to Nagas)

• Monpa 8%, (inhabitants of North Kameng and a few other areas on Tibet border)

• Mishmi 7%, (inhabitants of Lohit lineage probably a mix of Bodo and Naga

• Nocte 7%, (inhabitants of Lohit lineage somewhat akin to Nagas)

• Tagins 6%, (inhabitants of South Kameng and parts of Subansiri)

• Tangsas 4%, (inhabitants of Lohit)

• Apatani 4%, (inhabitants of Subansiri)

• Miris 2%, (inhabitants of Northern Foothills of Brahmaputra valley)

• Sulungs 1%. (again inhabitants of the foothills)

• Khamptis 1%, (inhabitants of Changlang Lineage a mix of Naga and Bodo)

• Others (includes migrant Chakmas, Tibetans, Nepalese, Bangladeshis) 3–4%. (Tibetan refugees who fled on Chinese occupation of Tibet and Chakma migrants who fled Bangladesh have been settled on the plains of Lohit and Changlang. […]

MEGHALAYA

Demography

Meghalaya is a hill state with Assam to its north and east and Bangladesh to its south and west. Its population is 23,06,069 as per the 2001 census.

Its demographic profile is:

• Khasis/Jaintias/Garos 68%,

• Dimasa/Hmar/Kuki/Karbi/Lakher/Hajong/Mon/Naga 10%,

• Mizo 4%,

• Others (including Bangladeshi and Nepalese migrants) about 18%.

The Khasis, Jaintias and the Karbis are of Austric origin whereas the Garos, Dimasa and Hajongs are of Bodo origin.

History

Historically, Meghalaya was excluded from the Mughal, Ahom and Burmese invasions on account of its inaccessibility. The British annexed it in 1835 and decided to set up their headquarters at Shillong, the capital of Meghalaya, for control of the north-east (undivided Assam), on account of its salubrious climate and relatively easier access. They therefore set up a more effective administration, while allowing limited autonomy. This area was not therefore given a segregated ‘Excluded Area’ status as was done for Nagaland and Mizoram and was amalgamated into the Assam administration. The British set about civilising the tribals through conversion to Christianity. As they were not segregated and there was greater interaction with other communities unlike the other hill states, they progressively accepted their merger with the Indian Union more gracefully, whilst maintaining their culture modified by Christian and western influence. Political Awakening Independence brought political awakening; demands for a separate state started from 1949. In 1952 the Khasis and Mizos led the demand for a separate tribal identity as they felt the Assam government was neglecting them.

KEY ISSUES AND STEPS TOWARDS RESOLUTION

Necessity for Review of Political Structure

The north-east has more than 500 different ethnic groups, many of whom feel neglected in some way or the other. Since it is a nation's chief responsibility to ensure the security and well-being of its people, it is the government's duty to ensure a socio-political structure that meets the aspirations of all. Provisions for this have already been made in the Sixth Schedule of the Constitution wherein it is stipulated that adequate measures may be instituted where necessary, to apply a suitable political structure based on existing tribal and local structures in the north-east with a view to protect the values and culture of the people of the region. We have however unfortunately faltered by foisting an alien and unworkable political model on the north-east.

Balkanisation – A Mistake

Firstly, we must understand it is next to impossible to keep balkanising the existing states of the nation to meet demands of the numerous ethnic groups. The redrawing of state boundaries on so called ethnic lines, in the manner as has been done repeatedly, has been a major error as it has given further impetus to splintering the already fragile pluralistic civil society structure. It contributed to the creation of extremist groups like the ULFA, NDFB, NSCN, a number of Manipuri and Arunachal insurgent organisations and an upsurge in insurgency in the north-east.

Nonlinear studies, by and large, have emerged in a variety of disciplines such as physical, chemical, biological, social and technological sciences, and in particular, differential equations. As trends go, there is every likelihood that an undergraduate curriculum in areas of mathematics, science, technology and even social sciences should start reflecting a qualitative approach, mainly through differential equations. Hence, this book is intended not only for students of mathematics, but also for a wider readership.

The genesis of this book can be described essentially as an outgrowth of my postgraduate teaching. I am especially confident of the content, thanks to the classroom notes my students assiduously took down over the years and later passed on to me to structure this volume.

I must confess, I have had the privilege of sitting through the valuable lectures of leading personalities, such as Prof Rene Thom, Delegue de l'Academie des Sciences, France, Prof E C Zeeman, Warwick University, UK, Prof D J Chillingworth, University of Southampton, UK and Prof Michael A B Deakin, University of Monash, Australia, on catastrophes and bifurcations, at the Jadavpur University, Kolkata. I must also mention about my own exposure and interactions with the faculty at ICTP, Trieste, Italy, under the leadership Prof E C Zeeman, on a course on ‘dynamical system’.

Problems have been set in the book with a view to engage students' creativity as well as exercise their mental prowess.

In the preceding chapters, we had an overview of the region and all the states, including major problem areas that need to be addressed. In this chapter, I will familiarise the readers with certain little-known facts related to the role of servicemen (SM) and ex-servicemen (ESM) of the armed forces, Para Military Forces (PMF) and Central Police Organisations (CPOs) in nation-building in their respective regions.

DEMOGRAPHY OF SERVICEMEN/EX-SERVICEMEN

Ukhrul District of Manipur

This district, the heartland of the NSCN (I&M), has a 2001 census population of 1,40,000. Of this, about 7500 are ESM with another 500 SM from the Army. In addition, there are about 1000–1500 SM and ESM of the Assam Rifles and other PMF/CPOs, totalling to about 9000–10000. When one considers the SM/ESM as a family unit (father, mother, 3 or more children), we have a population of about 45,000 or about 30% of the total population of the district. All these are people for whom the parent organisations – the central and state governments – are duty-bound to look after and provide for in terms of pensions, healthcare, priority for education, canteen services, payment rations and for problem resolution. When one expands the family sizes to individuals that the SM/ESM can directly influence, i.e. his brothers and sisters and their children, the family unit size increases to about 10 or about 90,000 to 1 lakh – 65–70% of the total population of Ukhrul District.

The geography and history of a region naturally shape the contours of its people, their ethnic groupings, culture, traditions, value system and way of life. They also indicate the patterns of migration of ethnic groups and reasons for the same. In a nutshell, as would be seen from the forthcoming analysis, it is a region's geography and history that largely contribute to the problems and prospects of its people and their destiny.

The Lie of the Land

Geographically, the north-east consists of the states of Assam, Nagaland, Manipur, Mizoram, Tripura, Arunachal, Sikkim and though not officially accepted yet as part of it – North Bengal. It is definitely an integral part of the region covering an area of about 300,000 sq km, with a population of about 46 million. The north-east is connected to the rest of India over a narrow stretch of plains through North Bengal and Assam known as the Siliguri Corridor (also called the Chicken's Neck by some), which leads into the Brahmaputra river valley plains. The Brahmaputra valley plains are about 70 to 90 km wide and 900 km long. West of Guwahati the valley widens substantially to merge with Bangladesh and the Gangetic plains.

The Himalayan mountains of Nepal, Sikkim, North Bengal, Bhutan, Arunachal and Tibet bound the Brahmaputra and the Gangetic valley plains to the north, the eastern hills link India and Myanmar to the east, and the Meghalaya-Karbi Anglong plateau, the Barak River valley and the plains of Bangladesh to the south.

In this chapter, I propose to give readers an overview of the effects of our foreign policy, power politics and events in the neighbouring countries, as they have had a profound effect on developments in the north-east.

Effects of Partition of India

The effects of the British policy of ‘Divide and Rule’ with reference to the Indian sub-continent's Hindus and Muslims are well-known and documented. Their consequent decision to partition the sub-continent into India, Pakistan and Myanmar and the way they drew up boundaries to do so, was perhaps one of the greatest blunders they committed throughout their colonial and imperial history. It resulted in anguish and massacres which had never been seen on the subcontinent, created millions of refugees fleeing to safety across artificially created boundaries, turmoil for people who had been split across these borders, and unrelenting hatred and distrust between India and Pakistan which continues till today. This is the backdrop to the foreign policies of both India and Pakistan since that fateful period of Partition and Independence.

The idea of creation of a homeland for the sub-continent's Muslims germinated in Bengal, arising out of the first Partition of Bengal by the British in 1919. After this, the so-called ‘Muslim Homeland’ was created through partition of the sub-continent by the British in 1947, in the form of East and West Pakistan. Whoever could have imagined that Pakistan would get balkanised in the fashion that occurred in 1971?