The paper deals with the specific cases of the changing of stability of slow integral manifold of singularly perturbed systems of ODE via the dynamic model of nerve conduction. It is shown that the proper choice of the additional parameters of the system allows us to construct the slow integral manifold with multiple change of its stability.
The usual assumption of the singular perturbation theory [
The first case corresponds to the transition of one real eigenvalue of the linearized fast
subsystem through zero when the slow variables are changed. This scenario for
delaying of the loss of stability in the singularly perturbed systems is associated with the
canards or duck-trajectories [
All three scenarios of change of stability are discussed via the Hindmarsh-Rose model
of nerve conduction [
Consider the system of ordinary differential equations with a small positive
parameter:
x X (x, y, ), y Y (x, y, ),
where x and y are vectors in Euclidean spaces R m and Rn , respectively, ε is a small
positive parameter, the vector-functions X and Y are sufficiently smooth and their
values are O(1) as → 0. The slow and fast subsystems are represented by the first
and the second equation of (1), respectively. Such systems are called the singularly
perturbed systems [
A smooth surface S in Rm Rn R is called an integral manifold of the system (1) if any integral curve of the system that has at least one point in common with S lies entirely on S. Only the integral manifolds of system (1) with dimension m (the dimension of the slow variable x) that can be represented as graphs of functions are discussed here.
We will assume that h(x, ) is a sufficiently smooth function of . Such integral manifolds are called the manifolds of slow motions.
The surface described by the degenerate equation
(1)
(2)
(3)
determines a breakdown surface (or jump points in the scalar case) [
A slow integral manifold can change its stability in some specific cases. The mechanism of two cases of change of stability is described below. Y (x, y, 0) 0 J Y x, (x), 0
y det Y x, (x), 0 0
y is called a slow surface (or a slow curve when the dimension of this surface is equal to one). The slow surface can be considered as a zero-order ( 0) approximation of the slow integral manifold, i.e., h(x, 0) (x), where (x) is an isolated root of the degenerate equation (2).
The subset of the slow surface is stable (or attractive) if the spectrum of the Jacobian matrix is located in the left half-plane. If there is at least one eigenvalue of the Jacobian matrix (3) with a positive real part then the subset of the slow surface is unstable (or repulsive). The subset of the slow surface given by
Consider the autonomous singularly perturbed system x X (x, y, , ), y Y (x, y, , ), where is small positive parameter, is an additional parameter, for which an equilibrium of the fast equation becomes unstable with transition of one real eigenvalue of (3) through zero when the slow variables are changed.
For simplicity we consider the case when the variables x and y are scalar. The stable and unstable subsets of the slow curve are separated by the jump point(s). The presence of the additional scalar parameter provides the possibility of gluing the stable and unstable slow invariant manifolds at a jump point to form a single trajectory, the canard.
A canard is a trajectory of a singularly perturbed system of differential equations if it follows at first a stable invariant manifold, and then an unstable one. In both cases the length of the trajectory is more than infinitesimally small.
The term “canard” (or duck–trajectory) had been originally given by French
mathematicians to the intermediate periodic trajectories of the van der Pol equation between
the small and the large orbits due to their special shapes [
It is clear that the trajectory y 0 is a canard. The left part (x < 0) is stable and the right part (x > 0) is unstable. These two parts are divided by a jump point, which separates stable and unstable parts of the slow curve, at x 0 . (4) (5)
Consider the slow-fast system
x X (x, y, ), y Y (x, y, ),
for which a singular point of the equation of fast motions becomes unstable with
transition of a pairs of eigenvalues through the imaginary axis when the slow variables
are changed [
For analytical systems the positive semi-trajectories in a certain region of phase space tend to the curves of the degenerate system having a comparable length of motions near the stable and unstable parts of the slow surface. This describes the trajectories similar to canards, which are described above.
Slow surface of the system (5) is divided into stable and unstable regions. The first one consists of a stable equilibrium of the fast subsystem and the other one consists of the unstable steady states, their common border is called as the breakdown surface. There is an open set of points on the stable part of the slow surface starting from which the phase curves of the slow system intersect the breakdown surface and, under changing of x along the slow curve, the pair of eigenvalues of a singular point of the equation of fast motions passes through the imaginary axis with a non-zero rate. The phase point of the system (5) starting close to the stable part of the slow surface fast (during the time of O(|ln ε|) as ε → 0) tends to the ε-neighborhood of the slow surface and then moves along the slow trajectory. If the system (4) is analytic, then the further movement has an interesting and unusual phenomenon - the delay. It consists in the fact that the phase point continues to move along the unstable part of the slow surface in the ε-neighborhood of the slow path, more time on the order O(ε−1) after crossing by the slow path of the stability border. And this slow path of the trajectory along the unstable part of the slow surface has a distance of order O(1). Only then can happen jump, i.e., for a time of order O(|ln ε|) (slow variables are changed by a small amount of the order O(ε|ln ε|) as ε → 0), avoiding the slow surface at a distance of the order of O(1).
This situation observed is similar to the situation with canard. The canard also moves at first along the stable part of the slow surface, and then moves along the unstable part. But canards are found in two-dimensional system with an additional parameter and they are quite rare: they exist for an exponentially small range of parameter values. And canards are the result of the transition of one real eigenvalue from the left half-plane to the right one.
The Hindmarsh-Rose model [
The dimensionless form of the Hindmarsh-Rose model is
x y ax3 bx2 z I ,
y c dx2 y,
z s(x ) z,
where х is a transmembrane neuron potential, y and z are the characteristics of ionic
currents dynamic, I is ambient current, other parameters reflect the physical features
of the neurons. The typical values of the positive parameter are small ( 1) [
Slow curve of system (7) is described by: y ax3 bx2 z I 0, c dx2 y 0.
The plot of the slow curve is presented on Fig. 1. (6) (7) which allow us to investigate a projection of the slow curve on the XOZ-plane, see (8) 2 (1 3ax2 2bx) 3ax2 2(d b)x 0.
The necessary condition of stability (which is also the sufficient in this case) for the polynomial (8) is 1 3ax2 2bx 0, 3ax 2d 2b x 0. (9) From (9) we can determine the abscissas of the points of an expected change of stability of the slow curve:
These points divide the slow curve into several parts, see Fig. 3. We check the sign of real part of eigenvalues of the matrix J for all parts to find out whether the region is stable or unstable. One of the two real eigenvalues of the Jacobian matrix of the fast system changes its sign at the points A1 and A2 , and at these points the slow curve changes its stability. Thus, A1 and A2 are the jump points. The proper choice of an additional parameter of the system allows us to glue the stable and the unstable integral manifolds of the system that exist in the ε-neighborhood of the stable and unstable regions of the slow curve. As a result of this gluing we obtain a canard.
However, there are other points of the stability’s change (see B1 and B2 on Fig. 3), which are not jump points, because the trajectory does not immediately escape the slow integral manifold as soon as it reaches these points, compared to the previous case. One can find them by equating the real parts of the complex conjugate eigenvalues of J (10) (11) (1 3ax2 2bx) (1 3ax2 2bx)2 4(3ax2 2dx 2bx)
2 to zero. The phenomenon of the delay of loss of stability occurs as the trajectory goes through these points.
There are points C1 (between points A2 and B1 ) and C2 (to the right from the point B2 ) on the slow curve, at which the pair of real eigenvalues of the Jacobian matrix J becomes the pair of the complex conjugate eigenvalues, see Fig. 3.
From (10) it is possible to find the relations between parameters values d
3a 2 b b2 3a
, under which the points A2 , B1 and C1 coincide. In that case the real and the imaginary parts of the pair of complex conjugate eigenvalues become equal to zero simultaneously, and a multiple zero root arises with the following emergence of the pair of real eigenvalues with the opposite sign.
Under condition (11) and the proper choice of the values of the parameters it is
possible to construct the slow integral manifold with multiple change of stability: we need
one parameter, say s, to glue the stable and unstable slow integral manifolds at the
point A1 using the techniques of canards (see Figs. 4 and 5), and we need two more
parameters, say a and I, for gluing integral manifolds at the point A2 with help the
method described in [
In this paper we investigated the different scenarios of the changing of stability of slow integral manifold via the Hindmarsh-Rose model. We have shown that in this singularly perturbed system the equilibrium of the fast subsystem loses its stability with the passage of one or a pair of complex conjugate eigenvalues through the imaginary axis when the slow variables are changed. Moreover, it was shown that there is the scenario of change of stability when the real parts as well as the imaginary parts of a pair of complex conjugate eigenvalues become zero followed by the appearance of multiple zero root and then by the birth of a pair of real eigenvalues of opposite signs. The crucial result of present investigation is that it is possible to construct the slow integral manifold with multiple change of its stability for some values of the additional parameters of the singularly perturbed system.
This work is supported in part by the Russian Foundation for Basic Research (grant 14-01-97018-p) and the Ministry of Education and Science of the Russian Federation under the Competitiveness Enhancement Program of Samara University (2013–2020).