18) Integration of linear inhomogeneous system with quasi-polynomial right side.

(x)=A (x)+ (x), (L[ ]= (x) ), x€(a,b)

Theorem: Condition: The vector-function (x) has the form (x) +…+ (x) and vector-function (x) +…+ (x) is the solution of followin inhomogeneous systems

(x) =A(x) + (x), … (x) =A(x) + (x),

Statement: vector-function (x) +…+ (x) is the solution of inhomogeneous systems

(x) =A(x) + (x)=A(x) (x) …+ (x)

Proof: adding the equality

(x) =A(x) + (x), … (x) =A(x) + (x),

We get that (x) +…+ (x) is the solution of inhomogeneous systems (x) =A(x) + (x)

Def:Vector-function (x) called quasi-polinom if it has the form

(x)= (x)* , where (x)- quasi-polinom

Statement: If (x)= (x)* + (x)* (1) then there (x) and (x) polinoms of r=k degree such that Vector-function (x)= (x)* + (x)* is particular solution of inhomogeneous linear system.

Proof: right size of (1) we can submit (x)= (x)* + (x)*

Particular sol. of system (x)= (x) * and (x)= (x) *

(x), (x) quasi-polinoms

Solution of sourse function is (x)+ (x) =

(x) * + (x) * =)= (x)* + (x)*

19) Studying of different cases (resonance and in resonance cases) Theorem 1 (Non-resonance). Assume that 'ϕ(λ) has no roots on the imaginary axis, g 2 BC and p 2 C. Then equation has at least one bounded solution if and only if p€ C0 + BC: To state the result in the case of resonance we follow and define the notion of upper and lower average. Given p €C0 + BC,

(p) :=

(p) :=

It is easy to verify that

−1< AL(p) <= AU(p) < +1:

Moreover, if p is periodic the identity AL(p) = AU(p) = p holds and the concept of average is recovered.

Theorem 2 (Resonance). Assume that λ = 0 is a simple root of ϕ(λ) and there are no other roots on the imaginary axis. In addition, g € BC and p € C. Then a sufficient condition for the existence of a bounded solution

p € C0 + BC; (−∞) < (p) <= (p) < g(+∞):

As mentioned in the introduction, this theorem extends results.

+ c + g(y) = p(t) (c > 0);

and it was proved that if g €BC there exists a bounded solution if and only if p= p*+p** with p*€C0 g(−∞) < inf p**<= sup p**<g(+∞)

20.Boundary problem for system of the second order. Green function.

We consider a system of the second order boundary value problem of the

type

with the boundary conditions u(a) = and u(b) = (2)

and the continuity conditions of u and u’ at c and d. Here, f and g are continuous functions on [a; b] and [c; d], respectively. The parameters r; and

are real finite constants. Such type of systems arises in connection with the Green function. L[y] = 0, U1[y] = 0, U2[y] = 0. has only trivial solution.

Let (λ1, λ2) 6= (0, 0) be such that a1λ1 + a2λ2 = 0 and let φ1 be solution of L[y] = 0 satisfying φ1(a) = λ1 and φ′1(a) = λ2. Choose another solution φ2 of L[y] = 0 similarly. This way of choosing φ1 and φ2 make sure that both are non-trivial solutions.

Note that φ1 and φ2 form a fundamental pair of solutions of L[y] = 0, since we assumed that homogeneous BVP has only trivial solutions.

By Lagrange’s identity (5.20), we get d/dx[p(φ′1φ2 − φ1φ′2)= 0. This implies p(φ′1φ2 − φ1φ′2) ≡ c, a constant and non-zero, (5.44)

as a consequence of (φ′1φ2 − φ1φ′2) being the wronskian corresponding to a fundamental pair of solutions.

Green’s function is then given by

G(x, ξ) :=1/c

21) Reduction of equation to canonical form.

Canonical form in a differential form that is defined in a natural (canonical) way; Finding a canonical form is called canonization. Canonical differential forms include the canonical one-form and canonical symplectic form For instance, the expression f(x) dx from one-variable calculus is called a 1-form, and can be integrated over an interval [a,b] in the domain of  f and similarly the expression: f(x,y,z) dx∧dy + g(x,y,z) dx∧dz + h(x,y,z) dy∧dz is a 2-form that has a surface integral over an oriented surface S:

Likewise, a 3-form f(x, y, z) dxdydz represents something that can be integrated over a region of space

Canonical one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. The exterior derivative of this form defines a symplectic form giving T*Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. In canonical coordinates, the tautological one-form is given by _idqⁱ

Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential (exact form), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.

The canonical symplectic form is given by ⁱdp_i

The extension of this concept to extended to general fibre bundles is known as the solder form.

22) Autonomic system properties. Trajectory, phase space.

A system of ordinary differential equations which does not explicitly contain the independent variable (time). The general form of a first-order autonomous system in normal form is:

= ( ) j=1,…,n

or, in vector notation, =f(x). (1)

A non-autonomous system = f(t,x) can be reduced to an autonomous one by introducing a new unknown function =t . Historically, autonomous systems first appeared in descriptions of physical processes with a finite number of degrees of freedom. They are also called dynamical or conservative systems

A complex autonomous system of the form (1) is equivalent to a real autonomous system with 2n unknown functions

(Re x)= Re f(x), (Imx)=Im f(x).

The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic f(x)

Consider an analytic system with real coefficients and its real solutions. Let x Φ (t) be an (arbitrary) solution of the analytic system (1), let ∆ =( ) be the interval in which it is defined, and let x(t; , ) be the solution with initial data x = . Let G be a domain in and f (G) . The point G is said to be an equilibrium point, or a point of rest, of the autonomous system (1) if f( ) 0. The solution , Φ(t) , t corresponds to such an equilibrium point

Local properties of solutions.

1) If Φ(t) is a solution, then Φ(t+c) is a solution for any c .

2) Existence: For any , G ,a solution x(t; , ) exists in a certain interval .

3) Smoothness: If f⋲ , then Φ(t)⋲ .

4) Dependence on parameters: Let , f=f(x,α),α⋲ C R,where is a domain; if f⋲ (G* ),p≥1 , x(t; , )⋲ (∆* )

5) Let be a non-equilibrium point; then there exist neighbourhoods V,W of the points , f( )respectively, and a diffeomorphism y=h(x): VW such that the autonomous system has the form = const in W .

A substitution of variables x = Φ(y) in the autonomous system (1) yields the system (2)

f(Φ(y)), where (y) is the Jacobi matrix.

where is the Jacobi matrix.

23) Solution properties of autonomic systems.