Given the complex Ikeda mapping En+1 = A + BEn exp i C 1

Given the complex Ikeda mapping En+1 = A + BEn exp



i

φ −

C 1 + |En|2



, where A, B, and C are constants, show that the steady-state solution, say, En+1 = En = ES, satisfies the equation cos C

1 + |ES|2 − φ

=

1 2B 1 + B2 −

A2

|ES|2

.