Condition for production of circulating proton beam with intensity greater than space charge limit

Transverse e-p instability in the proton rings could be damped by increase a beam density and the rate of secondary particles generation above a threshold level, corresponded a decrease of the unstable wavelength λ below a transverse beam size h( increase a beam density nb and ion density ni above a threshold level: nb + ni> β2 /re h2.; re =e2/mc2 ).In high current proton rings is possible to reach this 'Island of stability' by fast. concentrated charge exchange injection without painting and enhanced generation of secondary plasma as it was demonstrated in small scale PSR at BINP. After damping of e-p instability has been accumulated a circulating proton beam with intensity, corresponding a tune shift Δv=0.85x6 in the ring with v=0.85. Experimental observations of a transverse instability of proton beams in different accelerators and storage rings will be discussed. The methods of instability damping will be considered. These experimental dates could be useful for verification of computer simulation tools developed for investigation of space charge and instabilities in realistic conditions