Characteristic exponent of 3. From 6 one obtains ,.
The characteristic exponent can be defined as the complex number for which 3 has a solution that is representable in the form. In applications, the coefficients of 1 often depend on parameters; in the parameter space one must distinguish the domains at whose points the solutions of 1 have desired properties usually these are the first four properties mentioned in the Table, or the fact that with given.
These problems thus reduce to the calculation or estimation of the characteristic exponents multipliers of 1. If the corresponding homogeneous equation. It can be determined by the formula. Suppose that 8 has linearly independent -periodic solutions. Then the adjoint equation. The inhomogeneous equation 7 has a -periodic solution if and only if that the orthogonality relations. If so, an arbitrary -periodic solution of 7 has the form.
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Under the additional conditions. Suppose that for the series.
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Then the transition matrix of 10 for fixed is an analytic function of for. Let be a constant matrix with eigen values ,. Let be the multipliers of equation 10 ,. If is a multiplier of multiplicity , then. If simple elementary divisors of the monodromy matrix correspond to this multiplier, or, in other words, if to each , , correspond simple elementary divisors of the matrix for example, if all the numbers are distinct , then is called an -fold characteristic exponent of equation 10 with of simple type.
It turns out that the corresponding characteristic exponents of 10 with small can be very easily computed to a first approximation. Namely, let and be the corresponding normalized eigen vectors of the matrices and ;.
Then for the corresponding characteristic exponents , , of 10 , which become for , one has series expansions in fractional powers of , starting with terms of the first order:. Here the are the roots written as many times as their multiplicity of the equation. If the root is simple, then and the corresponding function is analytic for. From 13 it follows that cases are possible in which the "unperturbed" that is, with system is stable all the are purely imaginary and simple elementary divisors correspond to them , but the "perturbed" system small is unstable for at least one.
This phenomenon of stability loss for an arbitrary small periodic change of parameters with time is called parametric resonance. Similar but more complicated formulas hold for characteristic exponents of non-simple type. Let be the distinct multipliers of equation 3 and let be their multiplicities, where. Suppose that the points on the complex -plane are surrounded by non-intersecting discs and that a cut, not intersecting these discs, is drawn from the point to the point. Suppose that with each multiplier is associated an arbitrary integer and that is the transition matrix of The branches of the logarithm are determined by means of the cut.
The matrix "matrix logarithmmatrix logarithm" can be defined by the formula. The set of numbers determines a branch of the matrix logarithm.
Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients
Also, for small. Generally speaking, formula 14 for all possible does not cover all the values of the matrix logarithm, that is, all solutions of the equation. However, the solution given by 14 has the important property of holomorphy: The entries of the matrix in 14 are holomorphic functions of the entries of. For equation 10 , formula 5 takes the form. If is determined in accordance with 14 , then. The main information about the behaviour of the solutions as which is usually of interest in applications is contained in the indicator matrix. Below a method for the asymptotic integration of 10 is given, that is, a method for successively determining the coefficients and in Suppose that in Although , generally speaking there is no branch of the matrix logarithm such that the matrix is analytic for and.
This branch of the logarithm will exist in the so-called non-resonance case, when among the eigen values of there are no numbers for which. In the resonance case when such eigen values exist equation 10 reduces by a suitable change of variable , where , to an analogous equation for which the non-resonance case holds.
ALMOST STABILITY OF HAMILTON'S EQUATIONS WITH QUASIPERIODIC OPERATOR COEFFICIENTS
The coefficients of the parametric excitation terms are not necessarily small in all cases. The proposed approximate approach would allow one to construct Lyapunov-Perron L-P transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.
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Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients
Click on an option below to access. Log out of ReadCube. The periodic solution of polynomial nonlinear equations is derived based on Galerkin's procedure.
An algorithm for evaluating deterministic equations and elements of Jacobian matrices is presented. This algorithm enables one to determine a periodic solution and to reduce the computation time and procedure. The result is applied successfully to periodic solutions of Duffing's equation, Van der Pols' equation, the pendulum equation, Minorsky's equation, the population growth model with time lag and the difference equation.
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