Let \(f(x)\) be a logistic map (for example, \(f(x)= 2x^2- 1\)). The authors investgiate the second-order difference equation (1) \(x_{n+ 2}= g(x_n, x_{n+ 1})\), where \(g(x, y)\) is a polynomial of second degree and \(g(x, f(x))= f(f(x))\) (such a map \(f(x)\) is called an invariant of (1)). The Lyapunov exponents \(\lambda_i(x, y)\), \(i= 1,2\), are calculated for a.e. \((x, y)\) such that \(y= f(x)\).