Summary: We have studied the order of appearance of oscillation modes associated with a piecewise linear map (PLM) obtained from the Oregonator model excited by an external periodic force \((A\cos(\omega\tau))\). It is shown that the bifurcation of the transition from \(m\)-peak oscillation (\(m\)-PO) to \((m+1)\)-PO occurs when the value of a shift parameter in the PLM varies from large to small. The region of appearance of the oscillation mode \(\pi_{\tau, s}(m, m+1)\) consisting of \(r\) times of \(m\)-PO and \(s\) times of \((m+1)\)-PO is obtained from analytical treatment, where \(r, s\) and \(m\) are the positive integers, respectively. The series of the winding number of the oscillation mode \(\pi_{\tau,s}(m,m+1)\) form a part of successive Farey series, and the Farey series can be decomposed into the part of series of unstable oscillation mode and that of stable oscillation mode. The bifurcation diagram of the order of appearance \(\pi_{\tau,s}(m,m+1)\) of the oscillation mode in the PLM has a self similar hierarchy. Analyzing theoretically the order of appearance of the oscillation mode \(\pi_{\tau, s}(m,m+1)\) and \(m\)-PO in the PLM, we can explain the order of appearance of the oscillation modes in the parameter space \((A,\omega)\) for the Oregonator excited by \(A\cos(\omega\tau)\).