Gaddafi is a rare family name in Libya. The most famous is Colonel Mu'ammar al-Gaddafi who seized power in Libya in a military coup on 1st September 1969 and formed a Revolutionary Command Council to rule the country. On March 2nd 1977 Libya changed its name to the Socialist People's Libyan Arab Jamahiriya and declared a "State of the Masses" with a novel form of direct participatory democracy that is outlined in The Green Book, attributed to the authorship of Mu'ammar Gaddafi. On April 15th 1986 Libya was attacked by the British and USA via aerial bombing, that killed hundreds of Libyans in Tripoli and Benghazi, included the adopted baby daughter of Mu'ammar Gaddafi. On June 11th 1988 a Great Green Charter of Human Rights and Freedoms was proclaimed and for a while human rights made some advancements before Gaddafi and his 6 sons again consolidated effective direct rule over the Libyan people. Responsibility for the PanAm Flight 103 bombing over the Scotish town of Lockerbie killing hundreds in the air and on the ground was accepted by Gaddafi as part of a deal for establishing ties with Britain and USA after which an excellent relationship between the Gaddafi family and the USA and Britain ensued for many years right up until March 2011 when British covert forces entered Libya and assisted in turning demonstrations there into a rebel uprising against the regime of Gaddafi. The African Union (AU) has ruled out any foreign intervention, but France along with Britain and USA have passed resolutions in the UN Security Council that violate African sovereignty and give support to rebel forces in Libya.
Gaddafi has only one spelling in Arabic but in English and other languages a great many ways of spelling, including but not limited to: Gadhafi, Gadhdhafi, Qadhafi, Qathafi, Khadafi, Khaddafi, Gadafy, Gaddafy, etc.