To me it seems an oddity that there are not at least three independent countries, say, the North, the Northeast, and the South. How has this enormous country kept together in the face of so many centrifugal forces?
The regions of Brazil sure tried to separate themselves. The revolts were simply crushed by the central government. Here are some of these attempts:
I will focus on the bolded ones to try to answer your question.
#1. Confederation of the Equator (1824)
After declaring independence from Portugal in 1822, Pedro I, the first emperor of Brazil, squeezed the provinces for higher taxes in order to maintain a growing central government - the capital Rio de Janeiro had grown from 50 thousand inhabitants in 1808 to 110 thousand in 1818. This growth had started with the transfer of the Portuguese court in 1807, fleeing from Napoleon. The transfer brought 15 thousand people from Lisbon to Rio, invigorated the economy of the city and caused the population boom.
The tranfer of the Portuguese court to Rio de Janeiro in 1807 was crucial to the power of Brazil's central government. It made all provinces connected to Rio, so when the independence happened in 1822, not much changed administratively. Contrast this to the Spanish colonies, where after independence there was no central government to look to.
The provinces had to maintain this growing central government, and obviously weren't happy about this. In 1824, a revolt broke out in the northeast: the Confederation of the Equator. The provinces of Ceará, Paraíba and Rio Grande do Norte formed a republic against the absolutist centralization of Pedro I that had been formalized in the Constitution of 1824. The revolt lasted 79 days. Inspired by the ideals of Spanish ex-colonies, the newly formed confederation abolished slave trade. This caused a split between the elites that supported the revolt: many of the landowners depended upon slavery. The Empire organized their forces with the English admiral Thomas Cochrane (1775–1860) leading by sea with English mercenaries, while the land forces were led by Colonel Francisco de Lima e Silva, father of Luís Alves de Lima e Silva (the Duke of Caxias, who we will see in the next revolt). The militia of the northeast were unable to cope with the strength of the imperial forces and were defeated.
A key person in both the Confederation of the Equator and the Pernambucan revolt was Joaquim da Silva Rabelo, known as Frei Caneca (literally "Friar Mug" - his father made mugs), a republican priest who edited a political newspaper in the first revolt and was a leader in the second. His newspaper, Typhis Pernambucano, used Greek mythology to incite republican ideals. He was executed by the end of the Confederation. He wrote poetry against the Empire, such as:
O servil acaba inglório
Da existência a curta idade;
Mas não morre o liberal,
Vive toda a eternidade.
My translation:
The servile ends up inglorious
From existence to brief age;
But the liberal doesn't die,
Lives through all eternity
(From the book Obras políticas e literárias de Frei Joaquim do Amor Divino Caneca. Org. Antonio Joaquim de Mello. 2ª .ed. 1979, PE)
#2. Ragamuffin War (1835 – 1845)
The Ragamuffin War was the longest-lived separatist movement in Brazil, lasting 10 years. The province of Rio Grande do Sul, in the south, produced beef jerky and competed with the production of Argentina and Uruguay. Brazil imported this foreign produce (used mostly to feed slaves) at a cheaper price than the national product, so the local elite of Rio Grande do Sul wanted the central government to take protectionist measures that favoured national production. These requests were not answered and so a revolt began. It didn't begin as a separatist movement, as initially they just wanted competitive beef jerky prices, but republican sentiment grew over the years and a republic was formed.
This revolt had the notable participation of the Italian Giuseppe Garibaldi (1802-1887), future leader of the Italian Unification, who sided with the rebels. The Empire had another British admiral leading by sea - John Pascoe Grenfell (1800-1869). But it was the Baron of Caxias who won in the end. Luís Alves de Lima e Silva was assigned by the emperor himself, who told him:
"End this revolution, as you have ended the others."
(Source: Morais, Eugênio Vilhena de (2003). O Duque de Ferro: novos aspectos da figura de Caxias)
The emperor Pedro II was 16 at the time, and was referring to the Balaiada Revolt (1838-1841) and the liberal rebellions of 1842.
Lima e Silva saw an opportunity when Uruguayan forces invaded the south of Brazil. He offered to help against the foreign invaders if the revolt ended, and so the rebellion came to an end in 1845. Unlike in previous revolts, the leaders were not executed and instead remained as officers of the Imperial Army. They were needed to fight foreign threats in this province that bordered Argentina and Uruguay. For this and other victories, Lima e Silva ended up promoted to Duke, the highest rank of Brazilian nobility and the only one in Brazilian history.
The point is that all of these revolts were either crushed or defeated in some form by a stronger, richer and more powerful central government. The "ragamuffins" got this name because, compared to the pretty imperial uniforms, they were wearing rags.