The Portuguese expelled the Dutch from Northeastern Brazil in 1649 in the 2nd Battle of Guararapes. After that the Dutch went to their recently founded colony of New Amsterdam, that later became NYC. What if the Dutch had defeated the Portuguese and remained in Brazil?
Second Guararapes is too late for the Dutch to have any hope of controlling Pernambuco. They already suffered a huge defeat one year early, in the first battle, and lost much of their artillery and guns for the Portuguese/Brazilians. In the second Guararapes their situation was even worse. The Dutch troops were suffering from famine (as the Brazilian "guerrillas" cut their supply lines).
Also, the commanders' sttuborness in fighting using European methods didn't work against an enemy that fought using "guerrilla" in a territory that they did have the advantage of knowing well. The Dutch needed to wait an entire day in position of battle under the summer's sun, while the Brazilians simply waited for the night to come to make incursions against them. The Dutch also lacked leadership, as most of the commanders had been killed in the first battle. We can't forget that most of the Dutch troops were mercenary soldiers, that were fighting for their salaries, while the Portuguese were fighting for their lands. During the initial attacks many of the Company's soldiers simply deserted, while others, who were French and Italian Catholics, decided to join the Portuguese.
If you want a Dutch Brazil, you need an earlier POD, during the Dutch conquest, with a different policy that really gives the land to loyal Protestan settlers. Without it a Dutch Brazil is pretty much doomed, as their permanence in the colony would always depends on the will of the landowners.
