Periodic Differential Equations

Equations (1.1) are periodic in the time variable. As this is the first encounter with such equations, it is appropriate to review the basic strategy for dealing with them. For the purposes of this review, consider the general periodic system 

Just as the stability of rest points can be determined by linearization, the stability of a periodic solution x(t) can often be determined by linearizing (2.1) about x(t). The linearization, or variational equation, corresponding to x(0 is 

This system is periodic and therefore the Floquet theory described in Section 4, Chapter 3, applies. Let 4 (/) be the fundamental matrix solution of (2.2). The Floquet multipliers of (2.2) are the eigenvalues of 4 (w) if /i is a Floquet multiplier and /i = e then A is called a Floquet exponent. Only the real part of a Floquet exponent is uniquely defined. 

In terms of Floquet exponents, the condition for stability is 9J(A) 0 for all exponents and the condition for instability is that 9f(A) 0 for some exponent A. Here, 9i(A) denotes the real part of A. 

P simply advances a point one period along the trajectory through the point. The basic properties of P follow from standard theorems for ordinary differential equations P is continuous and one-to-one, and x t, Xq) is a periodic solution of (2.1) with period w if and only if Pxq =Xq. The orbit of a point Xq under Pis the set O (Xo) = (Xq, Pxq, P Xq,. .., where P denotes the -fold composition of P with itself. Note that P xq = x( co, Xq), so that the orbit of Xq under P is just a sampling of the solution of (2.1) through Xq at integral multiples of w. The map P captures all of the dynamical features of (2.1). For example, if lim coP xo = x then, by the continuity of P, Px = x. Therefore x t, x) is a periodic solution and lim, x(/, Xq)-x(/, x) = 0 that is, x(t, Xq) tends asymptotically to the periodic solution x(t, x). 

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F. M. Arscott, Periodic Differential Equations, Pergamon, Oxford, 1964. 

Thoulouze-Pratt, E., 1983, Numerical analysis of the behaviour of an almost periodic solution to a periodic differential equation, an example of successive bifurcations of invariant tori. In Rhythms in Biology and Other Fields of Application, Lect. Notes in Biomath, Vol. 49, pp. 265-271. 

Mathieusche Funktionen und Sphaeroidfunktionen (Springer, Berlin, 1954) F. M. Arscott, Periodic Differential Equations (Pergamon Press, Oxford, 1964). 

Fink, A. M. [1974] Almost Periodic Differential Equations Lecture Notes in Mathematics 377 (Springer-Verlag Berlin, New York). 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

These equations form a fourth-order system of differential equations which cannot be solved analytically in almost all interesting nonseparable cases. Further, according to these equations, the particle slides from the hump of the upside-down potential — V(Q) (see fig. 24), and, unless the initial conditions are specially chosen, it exercises an infinite aperiodic motion. In other words, the instanton trajectory with the required periodic boundary conditions,... 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

This development suggested to Poincar6 the possibility of enlarging the problem by attempting to establish conditions of periodicity directly from the differential equations.1... 

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9. Moreover, as Eqs. (6-96) contain the small factor (i on the right-hand side, the quantities, a and 9 are small, that is, both a and 9 are slowly varying functions of time and one can assume that during one period T = 2nfca, the trigonometric functions vary but slightly. 

If the process lasts long enough in comparison with the duration of the period 2v, one may consider (approximately) A , Ap, and A

difference equations (6-124) to the stroboscopic differential equations. 

Once the existence of subharmonics is ascertained, one can easily conclude that if one of them is near the period of the system, a corresponding subharmonic resonance must appear. Unfortunately while the physical nature of this phenomenon is simple, its mathematical expression is not. In fact, one is generally given, not the subharmonic, but the differential equation, and the establishment of the existence of a stable subharmonic is usually not a simple matter. 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

Hyclic elimination method. We now focus the reader s attention on periodic solutions to difference schemes or systems of difference schemes being used in approximating partial and ordinary differential equations in spherical or cylindrical coordinates. A system of equations such as... 

Generally, the closure problem reflects the idea of a spatially periodic porous media, whereby the entire structure can be described by small portions (averaging volumes) with well-defined geometry. Two limitations of the method are therefore related to how well the overall media can be represented by spatially periodic subunits and the degree of difficulty in solving the closure problem. Not all media can be described as spatially periodic [6,341 ]. In addition, the solution of the closure problem in a complex domain may not be any easier than solving the original set of partial differential equations for the entire system. 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

The calculation becomes more difficult when the polarization resistance RP is relatively small so that diffusion of the oxidized and reduced forms to and from the electrode becomes important. Solution of the partial differential equation for linear diffusion (2.5.3) with the boundary condition D(dcReJdx) = —D(d0x/dx) = A/sin cot for a steady-state periodic process and a small deviation of the potential from equilibrium is... 

Export processes are often more complicated than the expression given in Equation 7, for many chemicals can escape across the air/water interface (volatilize) or, in rapidly depositing environments, be buried for indeterminate periods in deep sediment beds. Still, the majority of environmental models are simply variations on the mass-balance theme expressed by Equation 7. Some codes solve Equation 7 directly for relatively large control volumes, that is, they operate on "compartment" or "box" models of the environment. Models of aquatic systems can also be phrased in terms of continuous space, as opposed to the "compartment" approach of discrete spatial zones. In this case, the partial differential equations (which arise, for example, by taking the limit of Equation 7 as the control volume goes to zero) can be solved by finite difference or finite element numerical integration techniques. 

When oxidation occurs for a long period of time, the hydrocarbon is consumed and this influences on the concentration of hydroperoxide and the rate of hydrocarbon oxidation. The exact solution for the description of hydrocarbon consumption during oxidation can be found by the common integration of two differential equations one for hydrocarbon consumption and another for hydroperoxide accumulation. The approximation for the time of oxidation t > tmax, where tmax is the moment when [ROOH] = [ROOH]max gives the following equation [3,56] ... 

This problem was first approached in the work of Denisov [59] dealing with the autoxidation of hydrocarbon in the presence of an inhibitor, which was able to break chains in reactions with peroxyl radicals, while the radicals produced failed to contribute to chain propagation (see Chapter 5). The kinetics of inhibitor consumption and hydroperoxide accumulation were elucidated by a computer-aided numerical solution of a set of differential equations. In full agreement with the experiment, the induction period increased with the efficiency of the inhibitor characterized by the ratio of rate constants [59], An initiated inhibited reaction (vi = vi0 = const.) transforms into the autoinitiated chain reaction (vi = vio + k3[ROOH] > vi0) if the following condition is satisfied. 

Equations 1 and 2 can be solved numerically using an algorithm which handles stiff differential equations (28). Two sets of boundary conditions are required. For 0reduced catalyst is exposed to NO, the inlet gas composition is given by... 

The E-Z Solve software may also be used to solve Example 12-7 (see file exl2-7.msp). In this case, user-defined functions account for the addition of fiesh glucose, so that a single differential equation may be solved to desenbe the concentration-time profiles over the entire 30-dry period. This example file, with die user-defined functions, may be used as the basis for solution of a problem involving the nonlinear kinetics in equation (A), in place of the linear kinetics in (B) (see problem 12-17). 

Since there is no axial mixing, clearly any input is delayed at the outlet by the period of the residence time. This result also can demonstrated formally by solving the pertinent differential equations. 

Assuming that a number of NMR data sets (e.g., 2-D or 3-D maps of displacement vectors resulting from an external periodic excitation) from an object are acquired, the remaining difficulty is their reconstruction into viscoelastic parameters. As written in Section 2 the basic physical equation is a partial differential equation (PDE, Eq. (3)) relating the displacement vector to the density, the attenuation, Young s modulus and Poisson s ratio of the medium. The reconstruction problem is indeed two-fold ... 

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