thank you for your answers Patrick. may be i was not clear but i've already change Ns to their difference. but unfortunately it makes no effect:

> restart;

unprotect(gamma);

gamma := 1.322;

m0 := 7.59*10^(-26);

d[1] := 0.1e-2;

d[2] := 0.2e-1;

A[1] := (1/4)*Pi*d[1]^2;

A[2] := (1/4)*Pi*d[2]^2;

k[B] := 1.38*10^(-23);

T[res] := 300;

kp := (1/760)*10^5;

P[res] := (20*760)*kp;

V[1] := .125-3;

V[2] := 0.1e-2; kappa := 0.1e23;

V[3] := 0.251e-3; U[p] := 0.6664e-2;

a := U[p]/V[3];

alpha[1] := k[B]*T[res]*x(t)*kappa/V[1];

P[1] := 10*kp;

T[1] := T[res]*(P[1]/P[res])^((gamma-1)/gamma); alpha[2] := k[B]*T[1]*y(t)*kappa/V[2];
> sys := {x(0) = 0.1e-7, y(0) = 0.1e-7, diff(x(t), t) = (A[1]*sqrt(gamma*(2/(gamma+1))^((gamma+1)/(gamma-1))*P[res]^2*(1-(alpha[1]/P[res])^gamma)/(m0*k[B]*T[res]))-A[2]*sqrt(gamma*(2/(gamma+1))^((gamma+1)/(gamma-1))*P[1]^2*(1-(alpha[2]/P[1])^gamma)/(k[B]*T[1]*m0)))/kappa, diff(y(t), t) = (A[2]*sqrt(gamma*(2/(gamma+1))^((gamma+1)/(gamma-1))*P[1]^2*(1-(alpha[2]/P[1])^gamma)/(k[B]*T[1]*m0))-a*y(t)*kappa)/kappa};
> dsn1 := dsolve(sys, numeric);
> dsn1(10)[2]; dsn1(10)[3];
Error, (in dsn1) cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up
Error, (in dsn1) cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up

Moreover rescaling of variables also does not help:

> restart;

unprotect(gamma);

gamma := 1.322;

m0 := 7.59*10^(-26);

d[1] := 0.1e-2;

d[2] := 0.2e-1;

A[1] := (1/4)*Pi*d[1]^2;

k[B] := 1.38*10^(-23);

T[res] := 300;

kp := (1/760)*10^5;

P[res] := (20*760)*kp;

V[1] := .125-3;

V[3] := 0.251e-3; U[p] := 0.6664e-2;

a := U[p]/V[3]; kappa := 0.1e26;

alpha[1] := k[B]*T[res]*(`#msub(mi("N"),mo("in",fontweight = "bold"))`(t)-N[pump](t))*kappa/V[3];

P[1] := P[res]*(alpha[1]/P[res])^gamma;

T[1] := T[res]*(P[1]/P[res])^((gamma-1)/gamma);
> sys := diff(`#msub(mi("N"),mo("in",fontweight = "bold"))`(t), t) = A[1]*sqrt(gamma*(2/(gamma+1))^((gamma+1)/(gamma-1))*P[res]^2*(1-(alpha[1]/P[res])^gamma)/(m0*k[B]*T[res]))/kappa, diff(N[pump](t), t) = a*(`#msub(mi("N"),mo("in",fontweight = "bold"))`(t)-N[pump](t));
> ics := `#msub(mi("N"),mo("in",fontweight = "bold"))`(0) = 0.1e-7, N[pump](0) = 0.1e-7;
> dsn1 := dsolve({ics, sys}, numeric);
> dsn1(750)[2]; dsn1(750)[3];
Error, (in dsn1) cannot evaluate the solution further right of 590.57010, maxfun limit exceeded (see ?dsolve,maxfun for details)
Error, (in dsn1) cannot evaluate the solution further right of 590.57010, maxfun limit exceeded (see ?dsolve,maxfun for details)

Dear Patrick,

N has a meaning a number of particles in the interconnected system of 3 chambers and vacuum pump. Therefore its range is quite big. it is of order 10^21-10^24. may be i have to rescale system somehow to manage with big numbers? t is time. i've noticed that if P_1 is assumed constant then the system can be solved but solution is avaliable only until some limit time ~270 sec. which is actually is not long enough to study whole dynamics which i am interested in.

i tried to solve the problem in close vicinity of singular point as you advised, also i tried to introduce new variables. but every time i receive the same Error, (in dsn1) cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up

I already second week stuck in this problem while i am planning to generalize it for larger number of chambers and take into account spatial distribution of the flow which goes from the orifice/duct between chambers. apparently this will lead to the partial derivatives system. but this is the final step. first of all i have to solve this one.

Dear Patrick,

thank you for your answer.

i've changed the intial conditions:ics := `#msub(mi("N"),mo("in",fontweight = "bold"))`(0) = 1, N[pump](0) = .1, N[out](0) = 5. In this case problem disappeared but new problem occured: 

> dsn1(.1)[1]; dsn1(.1)[2];
Error, (in dsn1) cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up
Error, (in dsn1) cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up