How many different Chalukya dynasties were there?

 There was the one original Chalukya dynasty

1. Chalukyas of Badami (Vatapi), Karnataka (543 -753)

There are at least 4 Chalukya collateral lines of this parent:

1. Chalukyas of Vengi, Andhra (624–1189) - Eastern Chalukyas
2. Chalukyas of Lata (लाट), Gujarat (620–1070)
3. Chalukayas of Kalyani, Karnataka (973–1189) - Western Chalukyas
4. Chalukyas of Vemulvada, Telangana (650–968)

Not related to these, but sharing the same mythical origin are

1. Chaulukyas of Patan, Gujarat (960–1243) - Solankis

But again related to the Western Chalukayas are

1. Senas of Nabadwip, Bengal (1070–1230)
2. Karnata dynasty of Mithila and Nepal (1094- 1326) List of Kings of Mithila

From their capital along the banks of the 200-mile long tributary of Krishna river called Malaprabha (above), Chalukyas rose as rulers when their overlords, the Kadambas, lost power. They rose to become one of the most illustrious dynasties of medieval India that not only governed a good chunk of India but also patronized art, literature and religious thought.

How do we know all the Chalukya families are related? Their own inscriptions say so. Their seals all have the traditional varaha emblem (0r the boar incarnation of Vishnu) below. Their common gotra Haritiputras of Manavyasagotra is mentioned many times. Their mythical origin from Chuluka or handful of Brahma, and so on, is shared by most of them.

A few of their achievements are:

1. One of the greatest rulers of Dakshinapatha or the area South of Narmada. They (Pulakeshi II) halted Harsha’s aggression to the South. Chinese XuanZang makes abundant mention of them in his travels.
2. Interestingly, they were an important part of the coalition that checked both Arab aggression into Gujarat in the 8th century and later Ghaznis also into Gujarat in the 11th century
3. Landing of Parsis (Zoroastrians) into Sanjan, Gujarat happened most likely in 7-8th century when Chalukyas were ruling Navsari, etc according to this text Full text of "Parsis of ancient India"
4. The mathematician Bhaskara was born and flourished in the reign of the Western Chalukyas
5. Mitakshara - the Hindu civil law for much of India - was codified in their rule
6. Their reigns saw the development of Kannada and, later, Telugu languages while they continued to patronize Sanskrit, and even Prakrit.
7. Basava - who sought for the complete abolition of castes in India - was born towards the end of Western Chalukyas, indicating religious ferment in their rule
8. Their early reign saw experimentation with Indian styles of architecture like below from Pattadkal: both the Nagara and Dravida style side-by-side