Why did the Dutch invade Portuguese America (Brazil)?

Question

I heard that the French and the Dutch both invaded Portuguese (Brazil) because the Portuguese never repayed them respectivly in the promised surger cane.

Answer 1

Sugar.
"The second and more durable attempt started in the Summer of 1629. This time the objective was Pernambuco, the best sugar colony in Brazil."
"The Dutch were, at this time, the masters of the Atlantic Ocean and Recife was, like Batavia in the East, the capital of the WIC (West Indische Compagnie) empire."
"The history of the Jews in Brazil is quite long and complex as it stretches over many centuries. Jews settled early in Brazil, especially when it was under Dutch rule, setting up a synagogue in Recife - the first synagogue in the Americas - as early as 1636. Most of these Jews had fled Spain and Portugal to the religious freedom of the Netherlands during the re-establishment of the Inquisition in first Portugal, Spain, and again Portugal. Amsterdam and a few other Dutch towns soon had small Jewish communities. However, Jews were barred from almost all guild trades and faced limited opportunities. The community soon had more members than it could support."
The French had a different agenda.
"The French ports of Normandy, especially Rouen and Dieppe, where were, in 16th century, a flourishing textile industry became the principal trading competing with Portugal in Brazil. Owing to the presence, on the Brazilian coast, of vast forests of "Pau Brasil" (which was used in the process of cloth colouring), the French soon started to trade with the Indians."

Answer 2

I think the Dutch were helping the Dutch West India company (not to be confused with the Dutch East
India co.) eliminate trade competition.