Abstract :
We consider an isolated difference resonance of the form (2p)¿1 - (2q)¿2 = n + ¿ where (2p) and (2q) are positive integers with (2p)+(2q)>2, n is 0 or an integer and ¿¿¿<<1. With action-angle variables (Ik, ak), the driving term of this resonance in the Hamiltonian takes the form D·(2I1)P(2I2)q cos(¿), ¿ =(2p)a1-(2q)a2 +const. Unlike sum resonances, two action variables I1 and I2, which are proportional to emittances in two directions, are bounded and any definition of resonance width will involve the concept of an "acceptable" growth in I1 or I2. We propose a definition such that inside the resonance width, an initial condition of large I2 and very small I1 will lead to an order of magnitude growth in I1. With this definition, the width is indefinite for (2p)=1. An arbitrarily small I1 can grow to a sizable fraction of (p/q)I2 for any value of ¿¿¿. For (2p)=2, the width is proportional to D·(I2)q. One cannot have resonances for (2p)>2 according to this definition, but there is a threshold value of initial I1 above which I1 will grow by a large factor if ¿¿¿ and the invariant quantity I1+(p/q)I2 satisfy a certain relation which will be given analytically. We thus propose a definition involving one parameter for (2p)=2 and two for (2p)>2. The picture is clearly symmetric in two directions: if the initial I2 is very small and I1 large, one simply uses (2q) in place of (2p) to classify the resonances.
