Atlantic Research Corp. of Alexandria pushed its profits up 49 percent last year to a record $5.2 million ($2.11 a share) from $3.5 million ($1.52) in 1981.

Sales were up 26 percent to $117.8 million from $93.2 million and the backlog of orders at yearend jumped to $73 million from $56 million.

For the fourth quarter ended Dec. 31, Atlantic Research earned $1.5 million (55 cents) on sales of $32.6 million, up from $986,000 (43 cents) on sales of $25.8 million in the same period a year earlier.

Atlantic Research is one of the leading makers of solid fuel rocket motors, and has subsidiaries in the electronics research and manufacturing fields. Recently the firm completed a 600-barrel-a-day pilot plant at Fredericksburg to make a liquid fuel by mixing water and powdered coal.

Net income of Syscon Corp. climbed 15 percent to $2.76 million from $2.44 million last year and sales of the Washington computer systems firm jumped to $71.4 million from $61.1 million for the fiscal year ended Nov. 30. Earnings per share fell to $1.02 from $1.05 because of the sale of about 375,000 shares of additional stock last November.

Fourth quarter income increased to $934,000 (34 cents) from $761,000 (33 cents) while quarterly revenues jumped to $19.2 million from $15.8 million.

President Jose J. Yglesias said the fourth-quarter results show "a strong trend of business" for Syscon, which primarily serves government customers. He said annual earnings were trimmed 7 cents a share by an unfavorable arbitration ruling on a contract dispute.

Madison National Bank profits last year climbed 6.2 percent to $2.87 million ($7.00) from $2.7 million ($6.59) in 1981. Chairman K. Donald Menefee said the bank's assets at yearend were up 7.9 percent to $297.5 million from $192.3 million, and deposits climbed 10.2 percent to $177.7 million from $161.3 million.

Fourth quarter earnings for the bank, a subsidiary of James Madison Ltd., slipped to $615,000 ($1.50) from $791,000 ($1.93) in the last quarter of 1981.