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semiclassical properties of systems with mixed classical dynamics is a reach rapidly developing field of research . one of interesting results obtained in this direction is a chaos assisted tunneling . it was shown that the structure of the classical phase space of hamiltonian systems can influence such purely quantum processes as the tunneling  @xcite . it was demonstrated in numerical simulations that existence of chaotic motion region in the classical phase space of the system can increase or decrease tunneling rate by several orders of magnitude  @xcite . typically one considers tunneling between kam - tori embedded into the `` chaotic sea '' . the region of chaotic motion affects tunneling rate because compared to direct tunneling between tori it is easier for the system to penetrate primarily into the chaotic region , to travel then along some classically allowed path and to tunnel finally to another kam - torus  @xcite . chaos assisted tunneling phenomenon as well as the closely related coherent destruction of tunneling were experimentally observed in a number of real physical systems . the observation of the chaos assisted tunneling between whispering gallery - type modes of microwave cavity having the form of the annular billiard was reported in the ref . the study of the dynamical tunneling in the samples of cold cesium atoms placed in an amplitude - modulated standing wave of light provided evidences for chaos - assisted ( three - state ) tunneling as well  @xcite . recently , the coherent destruction of tunneling was visualized in the system of two coupled periodically curved optical waveguides  @xcite . the most popular methods which are used to investigate the chaos assisted tunneling are numerical methods based on floquet theory  @xcite . among other approaches to chaos - assisted tunneling we would like to mention the path integral approach for billiard systems  @xcite and quantum mechanical amplitudes in complex configuration space  @xcite . in this paper we will consider the original approach based on instanton technique , which was proposed in @xcite and numerically tested in  @xcite . instanton is a universal term to describe quantum transition between two topologically distinct vacuum states of quantum system . in classical theory the system can not penetrate potential or dynamical barrier , but in quantum theory such transitions may occur due to tunneling effect . it is known that tunneling processes can be described semiclassically using path integrals in euclidean ( imaginary ) time . in this case instantons are soliton - like solutions of the euclidean equations of motion with a finite action . for example , in euclidean yang - mills theory distinct vacuum states are the states of different chern - symons classes , and instanton solutions emerge due to topologically nontrivial boundary conditions at infinity . in this paper we will use a much simpler system to investigate the connection between the tunneling , instantons and chaos , namely the kicked system with double well potential . our attention will be focused on the behavior of the quasienergy spectrum when perturbation of the kicked type is added to the system . hamiltonian of the particle in the double - well potential can be written in the following form : @xmath0 where @xmath1 - mass of the particle , @xmath2 - parameters of the potential . we consider the perturbation @xmath3 where @xmath4 - value of the perturbation , @xmath5 - period of the perturbation , @xmath6 - time . full hamiltonian of the system is the following : @xmath7    euclidean equations of motion of the particle in the double - well potential have a solution - instanton . in phase space of nonperturbed system instanton solution lies on the separatrix . perturbation destroys the separatrix forming stochastic layer . in this layer a number of chaotic instantons appears . chaotic instanton can be written in the following form : @xmath8 it is a solution of the euclidean equations of motion . here @xmath9 and @xmath10 - chaotic and nonperturbed instanton solutions , respectively , @xmath11 - stochastic correction . it is convenient to work in the action - angle variables . using standart technique @xcite we obtain expressions for this variables in the following form : @xmath12 where @xmath13 and @xmath14 - action and angle variables , @xmath15 - energy of the particle , @xmath16 and @xmath17 - full elliptic integrals of the first and second kinds , respectively , @xmath18 - elliptic integral of the first kind , where @xmath19 is integral argument , @xmath20 - modulus which is in the following way expresses through energy @xmath15 @xmath21    we will use in our analytical estimations expression for the action of the chaotic instanton . we assume this action is equal to the nonperturbed instanton action corresponding to the euclidean energy which is less than one of the nonperturbed instanton solution . we expand expression ( [ action ] ) in powers of the ( euclidean ) energy difference from the separatrix @xmath22 , where @xmath15 - energy , @xmath23 - energy on separatrix , and neglect terms higher than linear one . as a result we has the following linear expression @xmath24 = \pi j(e_{sep } - \xi ) =   s[x_{inst}(\tau , 0 ) ] - \alpha \ , \sqrt{\frac{m}{a_2 } } \ ; \xi,\ ] ] where @xmath25 = 2 \sqrt{m } \ , a^{3/2}_2 /(3 \ , a_0)$ ] - nonperturbed instanton action , @xmath26 - numerical coefficient . when stochastic layer in euclidean phase space is formed under the action of the perturbation a number of chaotic instanton solutions appear , see fig . [ fig : strob0p01 ]  ( b ) . these solutions implement the saddle points of euclidean action functional and thus give significant contribution to the tunneling amplitude . in the some sense their role is analogous to the role of the multiinstanton configurations in the case of the nonperturbed system . in the framework of the chaotic instanton approach the width of the stochastic layer determines the contribution of these chaotic solutions of euclidean equations of motion to the tunneling amplitude . the stochastic layer width estimated using resonances overlap criterion . for this purpose we should expand the expression for the perturbation  ( [ eq : v ] ) in the series of exponents . at first we evaluate the coordinate @xmath27 of the particle using expression  ( [ angle ] ) in the following way : @xmath28 and rewrite it as a series of sines @xmath29,\ ] ] where @xmath30 $ ] - parameter ( @xmath31 ) , @xmath32 - the frequency of the particle oscillations . leaving only the first term in expression ( [ eq : x ] ) and performing simplifications of this relation we obtain the following form for the coordinate @xmath33.\ ] ] using this relation and performing the fourier transformation of the series of delta functions in the perturbation ( [ eq : v ] ) one can obtain the following equations of motion in action - angle variables @xmath34,\\   \dot{\theta } = \omega(j),\label{dott}\end{aligned}\ ] ] where @xmath14 - angle variable for the particle in the system , @xmath35 is a perturbation frequency .    to obtain width of stochastic layer we calculated the parameter of the resonances overlap . it is equal to relation of the resonance width in the frequency scale ( @xmath36 ) and the distance between resonances ( @xmath37 ) . first one is estimated as a frequency of oscillations of the resonance angle variable @xmath38 . the result is obtained from the equations ( [ dotj ] ) and ( [ dott ] ) using standart technique ( see chapter 1 and 5 in  @xcite ) . the resonance width is the following @xmath39 where @xmath40 - distance from the separatrix . the distance between resonances is calculating using the expression for the resonance levels @xmath41 . thus one can obtain @xmath42    using two last expressions ( [ eq : dw ] ) and ( [ eq : dw ] ) the parameter of the resonances overlap can be written in the following form @xmath43^{1/2 } \gtrsim \left[\frac{\epsilon \ , \nu}{\delta h } \right]^{1/2}.\ ] ] overlap parameter is equal to unity on the boundary of the stochastic layer . using equation ( [ eq : k ] ) we can write the expression for the width of the stochastic layer in the following way @xmath44 where @xmath45 is the energy on the border between stochastic and regular regions , @xmath46 - some numerical parameter which can not be obtained in the framework of the criterion used . the tunneling amplitude for the perturbed system is a sum of the amplitude in the nonperturbed case and the amplitude of tunneling via chaotic instantons . the later can be evaluated by integration over action of the tunneling amplitude in nonperturbed system . using expression ( [ eq : s ] ) this integral can be transformed to the integral over the energy difference from zero up to the width of the stochastic layer  ( [ eq : dh ] ) : @xmath47 } \ , exp\ , ( - s[x_{chaos}(\tau , \xi)]),\ ] ] where @xmath48 is a normalize factor . to calculate contribution of chaotic instantons we use approximate expression for the chaotic instanton action ( [ eq : s ] ) and listed above assumptions . integration over @xmath49 gives the contribution of zero modes  @xcite . as the result we get the following expression for the amplitude : @xmath50 where @xmath51 is tunneling amplitude in the nonperturbed system , @xmath52 - a time of the tunneling which is put to infinity at the end . the last exponential factor in the expression  ( [ eq : a ] ) is responsible for the tunneling enhancement in the perturbed system . in the nonperturbed case the width of the stochastic layer is equal to zero and the expression ( [ eq : a ] ) coincides with the known expression describing the ordinary tunneling .    substituting @xmath53 in formula ( [ eq : a ] ) by the expression ( [ eq : dh ] ) and expanding it in powers of the perturbation strength we can write phenomenological formula for the quasienergy splitting @xmath54 where @xmath55 we fix this phenomenological parameter value using the results of numerical simulations . for this purpose we perform the linear fitting of the numerical data for the dependencies on the perturbation strength and take average value of the parameter over these dependencies . as the result we have the _ single _ numerical parameter  @xmath56 for our phenomenological formula explaining all these dependencies . for the computational purposes it is convenient to choose as basis vectors the eigenvectors of harmonic oscillator . in this representation matrix elements of hamiltonian ( [ eq : h ] ) and the perturbation ( [ eq : v ] ) are real and symmetric . they have the following forms ( @xmath57 ) : @xmath58,\\   v_{m\ , n } & = \epsilon \ ; \delta_{m + 1 \ ; n } \;\sqrt{\frac g2 } \ ; \sqrt{m + 1},\end{aligned}\ ] ] where @xmath59 and @xmath60 , @xmath61 - planck constant which we put equal to @xmath62 , @xmath32 - frequency of harmonic oscillator which is arbitrary , and so may be adjusted to optimize the computation . we used the value @xmath63 with parameters @xmath64 @xmath65 @xmath66 . in numerical simulations size of matrices was chosen to be equal to @xmath67 . simulations with larger matrices give the same results . we calculate eigenvalues of the one period evolution operator @xmath68 and obtain quasienergy levels which are related with the evolution operator eigenvalues through the expression @xmath69 . then we get the ten levels with the lowest one period average energy which is calculated using the formula @xmath70 ( @xmath71 are the eigenvectors of the one period evolution operator ) . the dependence of quasienergies of this ten levels on the strength of the perturbation is shown in the figure  [ fig : qes ] . quasienergies of two levels with the minimal average energy ( thick lines in the figure  [ fig : qes ] ) has a linear dependence on the strength of the perturbation in the considered region . they are strongly influenced by the perturbation while some of the quasienergy states are not . performed numerical calculations give the dependence of the quasienergy splitting on the strength ( fig.[fig : de](a ) ) and the frequency ( fig.[fig : de](b ) ) of the perturbation . these dependencies are linear as it predicted by the obtained analytical formula ( [ eq : de ] ) . using least square technique we calculate the _ single _ numerical parameter @xmath56 which describes all numerical dependencies demonstrated in the figures [ fig : de](a ) and [ fig : de](b ) . relative error in determining of the parameter @xmath56 from numerical results is less than @xmath72 . analytical results are plotted in the figures  [ fig : de ]  ( a )  and  ( b ) by lines . numerical points lie close to this lines . the agreement between numerical simulations and analytical expression is good in the parametric region considered . double - well system is investigated in presence of external kick perturbation . analytic chaotic instanton approach is applied for this system in order to obtain the phenomenological formula for the ground quasienergy splitting . the formula has the single numerical parameter which is determined from the numerical results . this formula describes ground quasienergy splitting as a function of strength and frequency of perturbation . it predicts linear dependence of the ground quasienergy splitting on these parameters . numerical results for the quasienergy splitting as a function of the perturbation frequency and strength demonstrate linear dependence as well . they are in a good agreement with the analytical formula . analytical determination of the parameter @xmath56 at the moment seems to be impossible due to estimative character of the analytical method used . however , it is not a large problem since this parameter can be evaluated with the high accuracy using the single dependence of the ground quasienergy splitting for any particular system parameters values . parameter @xmath56 determined in such a way allows to describe correctly all other dependencies of the ground quasienergy splitting on the perturbation strength and frequency for any other values of the system parameters . the only restriction is that the perturbation strength has to be sufficiently small ( in the case considered @xmath73 ) in order the chaotic instanton approach used to be valid .
kicked double - well system is investigated both analytically and numerically . 
 phenomenological formula for ground quasienergy splitting is obtained using resonances overlap criterion in the framework of chaotic instanton approach . 
 results of numerical calculations of quasienergy spectrum are in good agreement with the phenomenological formula .