Computing the stability regions of Hill's equation
Introduction
A linear second-order ordinary differential equation with a periodic coefficient of the formwhere ψ(t+T,α1,α2,…,αn)=ψ(t,α1,α2,…,αn) is periodic and α1,α2,…,αn are parameters characterizing the state of the system, is called Hill's equation. The study of Hill's equation has a rich history. In the early 1880s, Floquet [1], [2] developed his well-known theory of the behavior of solution of these equations. Hill [3] studied what is now called Hill's equation in his analysis of the orbit of the moon. Since then Hill's equation has appeared in many applications including Schrodinger's equation, studies of parametric resonance, to name but a few, and has been the subject of numerous analytic studies [4], [5], [6], [7], [8].
To motivate our study of Hill's equation, we present the model of an inverted pendulum [9]. We consider a pendulum of mass m and length L subject to gravity g. The pivot point is allowed to move along the y-axis and is subject to the vertical oscillation p(t), p″(t)=r(t), as shown in Fig. 1.
The equation governing the angular displacement θ(t) isHere we assume that r(t) is a periodic, i.e. r(t+T)=r(t). The rest points of (2) are (θ,θ′)=(0,0) and (π,0). The rest point (0,0) corresponds to the down position of the pendulum and (π,0) to the up position. A local stability analysis near (0,0) and (π,0) leads to the linearized equations (Hill's equations)where the `+' is for (0,0) and the − for (π,0). Depending on the parameters in the equations, all the solutions may be bounded (rest point is stable) or all/some unbounded (rest point is unstable). This leads to the result that the upward or downward rest points may be either stable or unstable. Hence, it is of interest to characterize the stability/instability by a stability diagram as a function of the parameters in the equations.
The original analysis assumed that pivot point moves vertically according to p(t)=Acosωt, where A is an amplitude and ω is the frequency of the oscillations of the pivot point. Then for an inverted pendulum, the small angular displacements of the pendulum about the vertical upward position are described by (3) (using the − sign), which we give as
We introduce the following new variables:into the above equation to obtain the Mathieu equationThis equation has been studied extensively.
We now return to the general form of Hill's equation (1). There are many stability results for this type of equation, which are based on Floquet theory [4] (see Section 2 for a review). The theory characterizes the problem of stability of solutions in terms of transition surfaces in parameter space that separate regions of stability and instability. Crossing one of these surfaces leads to a change of stability.
There are numerous approaches for locating the stability regions of (1). Floquet theory can be applied directly, but this requires a fundamental solutions set of (1). Constructing a fundamental solution set is only feasible for a restricted form of the equation. For the Mathieu equation (6), a fundamental solution set can be constructed using Fourier series [3], [10]. Approximations to the boundaries of the stability regions can be obtained by truncating the infinite determinant in the equation |K|=1, see (14). Another approach is to view the Mathieu equation as an eigenvalue or spectral problem and then use perturbation methods for small ε to obtain the boundaries δ(ε) as described in [10], [11]. More recently in [6], the stability regions were studied using perturbation methods for large δ and ε but with a more general periodic function. Finally, the stability boundaries can be located numerically. The use of numerical schemes allows a broader range of problems to be analyzed. A numerical approach based on spectral methods is presented in [13]. Here the second derivative term is replaced by an appropriate differentiation matrix and the eigenvalues of the linear algebra problems are constructed.
We present and expand on a method for locating the boundaries between the stable and unstable regions developed in [14]. The idea is to convert the differential equation spectral problem into an integral equation spectral problem. We then present a numerical scheme based on the Schwarz iteration procedure to locate the eigenvalues and hence the boundaries of the stability regions in parameter space. We feel that this method has advantages over other schemes since it has a strong analytic foundation and it can be easily adapted to cases in which the periodic function is discontinuous and applied to more general types of equations.
The paper is organized as follows. In Section 2, we briefly review Floquet theory, which forms the basis of analytic results for (1). We formulate the problem of computing the stability regions of (1) in terms of an integral operator in Section 3 based of the approach in [14]. Our main result is in Section 4, where we introduce a numerical scheme based on a spectral problem for the integral operator. We then illustrate a numerical implementation of our method on several examples in Section 5.
Section snippets
Floquet theory
We now present a brief summary of the important aspects of Floquet theory following the presentation in [10]. Let {y1(t),y2(t)} be a fundamental solution set of (1) which satisfy the following initial conditions:It is easy to verify that their Wronskian is equal to one for all t values.
One of the consequences of periodicity of ψ(t,α1,α2,…,αn) is that {y1(t+T),y2(t+T)} is also a fundamental solution set, since the functions satisfy the equation and their
Integral operator formulation
We now present and improve on a different approach to analyze the stability of Hill's equation using an integral operator formulation described in [14]. We first formulate the problem in integral operator form and then give some of its important properties.
We consider the particular type of the function ψ(t,α1,α2,…,αn) in (1):where φ(t,β1,β2,…,βn−1) is a T-periodic function, φ(t,β1,β2,…,βn−1)∈L2(0,T), and ω is a parameter. Since the set of T
Numerical scheme based on Schwarz iteration procedure
We consider a nonsingular operator , compact and self-adjoint on the Hilbert space H. The spectrum of the operator can be found from the spectral problem (nontrivial solutions)where is an identity operator, λ is an eigenvalue and z is a corresponding eigenfunction of the operator . Consider the equationWe define , μ=λ2 then is obviously a positive, real, compact, self-adjoint on H operator. Clearly, we see that satisfies the
Implementation and applications
In this section we explain how to implement the procedure and then apply it to several examples. We formulate a concrete stability problem in the integral operator form. Then we construct a numerical algorithm based on the improved Schwarz iteration procedure. We then display the stability diagrams.
We start with the stability problem for the Mathieu equation:Our goal is to get the stability diagram in the (δ,ε)-plane so we formulate the problem in the operator form.
We
Conclusion
We introduced and implemented an integral operator approach for constructing the stability regions of Hill's equation. We can consider the integral formulation as a generalization of the classical differential set up. The periodic coefficient in Hill's equation can be taken from wider functional space. It is also possible to use higher order numerical schemes to compute the integrals and improve accuracy. Since the integral operator is compact and self-adjoint, the rich theory of such operators
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The work was completed while the author was at and supported by the University of Illinois at Chicago.