Ghost ILDM-Manifolds
SOFIA BOROK, IGOR GOLDFARB, VLADIMIR GOL’DSHTEIN
Department of Mathematics
Ben-Gurion University of the Negev
P.O.Box 653, Beer-Sheva, 84105
ISRAEL
Abstract: Peculiarities of an application of the method of intrinsic low-dimensional manifolds (ILDM) are considered and analyzed in the present paper. It is shown that even for two-dimensional singularly perturbed system ILDM-technique may produce some artificial objects, which do not represent invariant manifolds of the original singularly perturbed systems. It is proved that a reason of their (that may be conditionally called “ghost” manifolds) appearance belongs to a structure of ILDM algorithm. A practical approach allowing to determine the “ghost” manifolds and to distinguish them from the “correct” ones is suggested in the paper. The approach is based on special properties of the “correct” invariant manifolds. A number of simple examples of “ghost” phenomena is considered and analyzed, reasons of “ghost” manifolds’ appearance are discussed, and applications of the suggested approach for “ghost” manifolds identification is demonstrated.
Key-Words: multi-scale systems, reduction methods, slow manifolds, intrinsic low-dimensional manifolds (ILDM), method of invariant manifolds (MIM).
1. Introduction
Mathematical models of chemical, biochemical, and mechanical systems are often formulated as a large sets of differential equations. For the purpose of numerical and analytical analysis, it is often desirable to reduce them to smaller systems. Generally, a large set of differential equations describing complex phenomenon has a number of essentially different time scales of sub-processes. This hierarchical structure considerably complicates numerical treatment of these systems and also allows applying a number of asymptotic approaches for analysis of their behavior. At present there is a number of asymptotical and numerical tools able to treat such multi-scale systems. Not complete list contains the integral (invariant) manifolds method (MIM) [3], [4], [10], the iteration method by to Fraser and Roussel [12], [13], and the computational singular perturbation method [17]. One of the numerical tools is the method of intrinsic low-dimensional manifolds (ILDM) [1], [2], [5], [7], [9], [11]. The reader is referred to the article [7] for more complete list of the reduction methods.
As any other algorithm, ILDM has its own restrictions, which are partly demonstrated in the present paper on a number of elementary examples. It is shown, that ILDM can not successfully treat the regions of the phase space, where the leading eigenvalues of the Jacobian are equal. In particular, it means, that the ILDM approach may face problems in the vicinity of the turning surfaces, where the leading eigenvalues are normally complex (their real values are equal and there is no splitting in rates of change of the processes involved). As a result of the ILDM application in these regions of the phase space, so called “ghost” manifolds may appear. The problem of the determination and elimination of the “ghost” manifolds is of high importance. A numerical criterion allowing to distinguish the “ghost” manifolds from the “true” ones is suggested. The criterion is based on the unique properties of the “true” invariant manifolds. The efficiency of the suggested criterion is demonstrated on the introduced examples.
2. Analytical background
In this section the method of invariant (integral) manifolds (MIM) and the method of intrinsic low dimensional manifolds (ILDM) are briefly described.
2.1. Method of Invariant Manifolds
Consider the singularly perturbed system (SPS) of ordinary differential equations
(1)
(2)
where is positive small parameter , x, y – vectors. The system (1) is called the fast subsystem and the system (2) is called the slow one. The dimension of the fast subsystem is and the dimension of the slow one is , (). By putting into the fast subsystem the degenerated system is obtained
(3)
(4)
The theory of invariant (integral) manifolds allows to reduce the original system to another system, on an invariant manifold, which has the dimension of the slow subsystem . Remind that a smooth surface S in is called an invariant manifold of the system (1)-(2) if any trajectory of the system that has al least one joint point with S lies entirely in S. The only invariant manifolds are of interest here that have the dimension of slow variables and can be represented as the graphs of vector-valued functions . One of the methods to find such function is an expansion into power series of small parameter
(5)
The functions are isolated solutions of the system (3) i.e. . The Implicit Function Theorem is not correct with respect to so-called turning surfaces (curves, points) representing solutions of the following system of functional equations:
(6)
On the turning surfaces and within their close neighborhoods the asymptotic expansion is inapplicable (the zero approximation have sense only). For these regions more delicate asymptotic expansions are correct (see, for example [10]).
The graph of the function represents the zero approximation of exact invariant manifold. Their graphs are called slow manifolds of the system (1)-(2). The slow manifolds represent the machinery that will allow us to check the correctness of the ILDM manifolds.
2.2 Method of Intrinsic Low Dimensional Manifolds
The method of intrinsic low-dimensional manifolds was constructed especially for fast numerical simulation of system with high number of ODE that are typical for practical kinetic and combustion problems. The essence of this approach is construction of a nonlinear coordinates system for which an original system of ODEs becomes a singularly perturbed one.
Consider the system of ordinary differential equations
(7)
where and are -dimensional vectors.
The method is based on the number of specific assumptions. The main of them is that the eigenvalues of the Jacobi matrix of can be sub-divided into two distinct groups: “fast” and “slow” in the following way
(8)
at any point z. Then this division can be used for a representation of the original system as a singularly perturbed system, where is dimension of a fast subsystem and is dimension of a slow subsystem. Correspondingly, -dimensional slow manifold can exist and our aim is to determine its location.
The authors of ILDM have suggested [9] that dynamics of the overall system from arbitrary initial condition should decay very quickly onto this - dimensional slow manifold.
The manifold can be found in the following manner. Suppose that a local basis is formed by the eigenvectors of the Jacobian at a point . The corresponding transformation matrix Q is built from the eigenvectors of J: the eigenvectors are sorted in order of decreasing of absolute value of real parts of the eigenvalues. Q and its inverse can be written as
,
where is matrix (corresponding to the “fast” motion), is matrix (corresponding to the “slow” motion), is and is matrix. The matrix is coordinates transformation matrix from the original basis to the basis of eigenvectors of . According to the definition of ILDM [7], [9], the desired slow manifold can be found as a solution of the system of functional equations with variables
(9)
Since the eigenvectors used by ILDM-algorithm are not to be orthonormal the matrix may be ill-conditioned and the inversion can have singularities. To avoid partially this problem the real Schur decomposition [6] is used to find orthogonal matrix such that
where is upper triangular matrix. The matrix may be chosen such that the eigenvalues appear in any necessary order along the diagonal of .
2.3 The ILDM-method for Singularly Perturbed System
Now it will be of interest to compare the ILDM and invariant manifolds for the SPS system.
Consider the system (1)-(2) and rewrite the ILDM equation (6). In this case the fast part of the matrix and the vector field F can be rewritten as
, ,
where is a and is matrixes.
The ILDM-equation (9) gets the form
In the zero approximation () the equation can be simplified
(10)
If then the last equation has additional solutions comparatively with slow manifolds .
3. ILDM - backsides
As any other algorithm, the ILDM method has a number of its own backsides. Their appearance depends on a specific type of non-linearity of the considered system. In particular, an application of the ILDM algorithm can produce additional objects having no connection to the invariant manifolds of the original systems of ODEs. We will call these artificial objects “ghost” manifolds. Some of the backsides will be listed and illustrated by examples in this section.
The present study focuses on the analysis of the reasons of the “ghost” manifolds appearance and the ways of their detection.
An analysis shows that there are a number of possible reasons of the appearance of the “ghost” manifolds. Their incomplete list looks as follows.
1). Incorrect division of the considered system into “fast” and “slow” subsystems.
2). Incorrect description of “turning zones”. Typically in these zones complex eigenvalues exist and ILDM-algorithm cannot treat them.
3). Degeneration of the ILDM-equation (eigenvectors, transition matrix). This situation is typical for regions, where the leading eigenvalues are equal one to another.
Let us illustrate mentioned above problematic points of the ILDM approach by simple examples.
3.1. Examples
Three distinct examples are considered and analyzed in this Subsection. Two of them are theoretical and they illustrate problems 1 and 3 listed above. The third example is practical one and it will illustrate the problems occurring within vicinity of the turning surfaces.
1) Theoretical example 1
This example will illustrate the problems 1, 3. It will be shown that the algorithm of ILDM produces incorrect fast direction of the system and this is the reason for existence a large number of “ghosts” manifolds.
Consider the following SPS of ODEs
(11)
and apply the ILDM method. The Jacobian of the system is easily calculated
Eigenvalues of the Jacobian are: and . At first glance can be accepted as fast eigenvalue (it is comparable with ). But it is easy to see that there is infinite number of domains where . In these domains the ILDM algorithm changes the real fast direction of the system to the slow one. Therefore the two ILDM equations should be treated together, namely
(12)
(13)
The equation (13) appears on the phase plane as segments along the -axis located in the domains and represents “ghost” solution. Figure 1 shows the ILDM run (solid line) and arbitrary trajectory of the system (dashed line). It can be seen the part of the fast motion (almost parallel to -axis) and after it the slow motion along the attractive manifold. The value of ε it was taken 0.01.
Now our aim is to verify this result. Apply MIM conception for this system. According to the MIM the slow curve of the system is given by the equation
(14)
Fig. 1
Figure 2 represents the slow curve
Fig. 2.
One can see that the central curves on the Figures 1 and 2 are identical. Consider the ILDM manifold (12). The central curve represents the slow curve. Additional small curves are result of non-correct division to “fast” and “slow” eigenvectors as was mentioned in the problem 1.
ILDM provides us with slow manifold, but also with large number of “ghost” objects far from it.
2) Theoretical example 2
This example will demonstrate essential perturbations produced by the ILDM-method on the unique stable invariant manifold.
Consider another SPS of ODEs
(15)
and apply the ILDM-technique. The Jacobian of the system is easily calculated:
Eigenvalues are and . Applying the original ILDM-algorithm the two ILDM-manifolds can be found for different domains with different hierarchy of the eigenvalues. Namely,
(16)
(17)
Figure 3 illustrates the ILDM-running of the system (solid line) and the arbitrary trajectory of the system (dashed line). The small parameter is taken 0.001.
Fig. 3
To realize which of these lines represents the slow manifold let us apply the MIM technique for this system. By substituting into the fast equation of the system the slow manifold (curve) is obtained
(18)
Figure 4 depicts the slow curve. The slow manifold in this case has special “almost flat” parts. Theory of invariant manifold provides us with the conclusion that it is attractive (stable) manifold.
Comparison of Figures 3 and 4 shows that ILDM-algorithm essentially changes the shape of the unique slow attractive manifold in regions where it has the flat form. Figure 3 demonstrates also phenomena similar to the previous example. The fast motion (almost parallel to the -axes) reflects correct dynamics of the system. The trajectory (dashed line) goes through an artificial object - “ghost” manifold, which appearance has the same source, as in the previous example.
Fig. 4
ILDM provides us with an incorrect slow manifold, and also with a large number of “ghost” objects. The ILDM slow manifold has a different shape than the standard slow manifold.
3) Semenov’s model of thermal explosion
Consider the dimensionless model of the thermal explosion in a gas (see, for example [4])
,
,
.
Here is a dimensionless temperature, is a dimensionless concentration, and are positive small parameters, is the small parameter of the system. This system was investigated in [4] by the invariant manifold method; an application of ILDM was analyzed in [2,5].
According to the ILDM-method the Jacobian of the system reads
.
This provides us with two eigenvalues , , that can be calculated as
,