15. Given $f(x)=x^{2}+c$, and $f(f(x))=f\left(x^{2}+1\right)$.
(1) Let $g(x)=f(f(x))$, find the analytical expression of the function $g(x)$;
(2) Let $\varphi(x)=g(x)-\lambda f(x)$, try to find the value of the real number $\lambda$ such that $\varphi(x)$ is a decreasing function on $(-\infty,-1]$ and an increasing function on $[-1,0)$.
15. (1) $f(f(x))=f\left(x^{2}+c\right)=\left(x^{2}+c\right)^{2}+c, f\left(x^{2}+1\right)=\left(x^{2}+1\right)^{2}+c$, we get $c=1$, $g(x)=x^{4}+2 x^{2}+2$.
(2) $\varphi(x)=g(x)-\lambda f(x)=x^{4}+(2-\lambda) x^{2}+(2-\lambda)$, set $x_{1}1+1+2-\lambda=4-\lambda$, to make $\varphi(x)$ decreasing on $(-\infty, -1]$, it is only necessary that $4-\lambda \geqslant 0$, i.e., $\lambda \leqslant 4$. Similarly, it can be proved that when $\lambda \geqslant 4$, $\varphi(x)$ is increasing on $(-1,0)$. Therefore, $\lambda=4$.