Study On Beam Halo-Chaos Controlled Using Wavelet Function And Generalized Synchronization Of Hyperchaotic Circuit

In this paper, we study two works on the basis of giving summary of the development in controlling chaos and chaos synchronization.In the first part, we study the problem of controlling "beam halo-chaos" for high intensity ion beam in period-focusing channel .Some dynamic equations and correlative physical concerts of describing high intensity ion beam in period-focusing channel are introduced first , then two kinds of wavelet functions are selected ,whose terms written as follows respectively, to build nonlinear feedback control functions.( b)2aaJfah(4=e a(2)The beam root-mean-square radius (rr,.,,s) is taken as feedback variable ,the general form of our control function G is presented as follows:(3)two types of wavelet feedback control functions can be achieved by mean of the equation (I) and the equation (2). G is added into the right of ion radial self-electric force equation .soFr =-qV^r.s--G(4)Using multi-particle code (particle-in-cell(PIC)code)for simulating the motion of the ion beam in the period-focusing channel controlled or not .Wè employ some statistical physical quantities to describe the control effects .Firstlv. we study continpous control with the control function corresponded to the equation (1); secondly. we explore the multi-periodical intervals control method with the control function corresponded to the equation (2) .The simulation results demonstrate that five different kinds of primary ion beams can be controlled well by selecting control parameters of continuous control .In the condition of multi-periodical intervals control ,the beam with the Kapchiskij-Vladimirskij(K-V)distribution is taken as a typical example ,good control result can also be achieved by selecting the proper control parameters and the proper number of control multi-periodical intervals.Both the continuous control and the multi-periodical intervals feedback control can reduce the energy of radial motion restrict its tendency to run away from the channel , and overcome theshortcoming that the halo regeneration phenomena as scraping halos by some conventional methods, e.g., by the means of collimator or shelves system, namely, zero halo intensity can be obtain always .This perfect result is much better than the approximate amount of 0.1087 that provided in reference [73],our result is agree with the minimum limitation( I 04)of halo intensity of factual high intensity accelerator .The research result offers a valuable reference for the application in practicable intensity accelerator.In the second part ,we study the generalized synchronization of hyperchaotic circuit .Theoretical analysis demonstrates that driven system can be achieved if a matrix whose eigenvalues' real part is negative and its commutative matrix can be built .When transmitting the signal to the driven system ,generalized synchronization of driving system and driven system can be realized ,A 4-D hyperchaotic oscillator is taken as a typical example .The results of numerical simulation are presented ,the research results offer valuable reference for realizing the practical generalized synchronization of hyperchaotic circuit ,improving the degree of security in the field of chaotic secure communication.