On singularly perturbed problems and exchange of stabilities.

The nature of the boundary equation obtained as the small parameter tends to zero plays a major role in understanding the solution behavior of particular perturbed problems. In this thesis, we analyze the behavior of solving perturbed problems in a special way in the following cases.

Basic concepts of stability

Introduction

Pasiz is an arbitrary solution of the equation P(D)z = 0, we havez =c1z1+· · ·+cnzn for some constant cj. Let P(λ) be the characteristic polynomial of A. The roots of the characteristic equation P(λ) = 0 are the same as the eigenvalues of A.

Structural stability

If h preserves the time parameterization, then the dynamics generated by f and g is said to be Ck conjugate, [9]. Saddle coupling occurs when an orbit connects a saddle point to itself or to another saddle point, i.e. unstable and stable partitions are connected, [10].

Regular and Singular Perturbations

Introduction
A simple example
Regular perturbation
Singular Perturbations

Here the numerical simulation shows the convergence on (0, T] of the solution x(t) of the forced damped Duffing oscillator, given by (1.14) to the solution x(t)˜ of the degenerate equation given by (1.15) .We see that as zero approaches (1.14), its solution converges to the solution of the unperturbed equation (1.15).

Figure 1.1: The phase portrait of the forced damped oscillator given by (1.7).

Tikhonov theorem

Introduction

Therefore, the main assumption of Theorem 1.2.1, namely the continuity of the right-hand side, no longer holds. In other words, we can say that the right-hand side of equation (1.28) depends on in an irregular way or in a singular way.

Assumptions of the Tikhonov theorem

If assumptions A1−A5 in the Tikhonov theorem are satisfied, then there exists a unique solution (x(t), y(t)) of (1.29) on the closed interval [0, T] such that. Then one can include the initial layer term to achieve uniform convergence in the closed interval [0, T].

A simple example

By adding the initial layer term y(τˆ )−ϕ(0, x0) to the second expression of (1.43), we obtain uniform convergence in the closed interval [0, T] of the solution y(t) of the second equation of (1.36), as shown in the following numerical simulation. Here, the numerical simulation shows the uniform convergence in the closed interval[0, T] of the solution y(t) given by the second equation of (1.36) when the initial layer is added.

Figure 1.9: The uniform convergence in the closed interval [0, T ] of the solution x (t) given by the first equation of (1.36) to the solution x(t)¯ given by (1.41).

Centre manifold

Introduction
Simple example
Simple examples
Centre manifold theorems

Therefore, if we replace ω1 with its value in (1.54), we get an approximate equation of the central manifold. First, we put (1.68) into a suitable form for using the central manifold theorems.

Asymptotics expansions

From the third identity in (1.78) and the initial condition of the second equation of (1.75) we have. Therefore, when approaches zero, the first term in (1.87) converges to the solution of the boundary equation (1.41), and the second term in (1.87) converges to the quasi-steady state.

Theorem (Implicit Function Theorem)

Many mathematical models describing population growth assume that only an increase in population density has a negative effect on the life of a single individual within that population. In the following, we assume a one-to-one gender ratio in order not to explicitly model the male population, [13]. Thus, introducing a new dimensionless time s=µt in (2.2) creates an equivalent system in unit lifetime.

The case of short satiation time

Numerical simulation

Here, the numerical simulation illustrates the convergence of the solution z(t) given by (2.8) and the solution z(t)¯ given by the limit equation (2.10).

The case of short satiation time and high searching efficiency

Numerical simulation

This shows that the quasi-stationary state is approximately close to equation (3.23) of the central manifold. It shows that the quasi-steady state of Tikhonov's theorem is the 0th order approximation of the central manifold. To analyze the limit behavior of the solutions of (4.2), we consider two cases.

Figure 2.5: Comparison of the total female population searching for a mate y given by the second equation of (2.17) with the quasi-steady state (2.19).

The case of short satiation time, high searching efficiency and mortality rate

Numerical simulation

Here, numerical simulations show the extinction of the entire population of females given by the first equation (2.30) and the increase and then the extinction of the entire population of mate-seeking females given by the second equation (2.30). Furthermore, in the following simulation we illustrate the extinction of the entire population of recently mated females obtained by subtracting the second equation (2.30) from the first.

Figure 2.10: Approximation of the population of females searching for a mate given by the solution y(s, ) of (2.30) and by the solution y(s/)ˆ of (2.33).

Comment on the numerical simulations

Here, we note that by the classical Tikhonov theorem we obtain that the boundary equation is a Lotka-Volterra system and we find that in any given time interval [0, T] the solutions of the pre-predator system converge to the solutions of the Lotka-Volterra system. The advantage of this method is that it can use the multiple center theory to infer even the long-term behavior of the approximation provided that the boundary equation is structurally stable. This improves the Tikhonov approximation of the predator-prey system so that it rests uniformly on [0,∞).

Application of the Tikhonov Theorem

Replacing the unknowns in the first equation of (3.2) with the known quasi-steady state¯ gives the boundary equation.

Numerical simulation

Initial layer

Application of the centre manifold theory

Then we replace w0, y0 and 0 in (3.12) by their respective values and we use conditions of the tangency of the center manifold with the center subspace at the origin, introduced in Remark 1.4.1, to obtain the following system which must be solved to calculate the equation of the center manifold. Moreover, (3.20) ensures the uniform convergence of the solution of (3.1) to the solution of the approximate limit equation on the center manifold, uniformly in time. Note that the uniform convergence is only local in y; this is due to the local nature of the middle multiple result.

Numerical simulation

The following numerical simulation shows the comparison between the solution of the limit equation (3.4) and the solution of the equation governing the stability of small solutions on the central manifold (3.18). The fact that we observe a worse approximation for small times in Figure 3.4 is, as mentioned earlier in this chapter, due to the fact that we have used only local expansion of the central manifold, which yields good approximations only for small values of y. we see that long-term behavior is not affected.

Figure 3.4: The comparison between the solutions given by Equation (3.18) and the solutions given by (3.4) for small and large values of y.

Models with not structurally stable limit equation

Numerical simulation
Equilibrium States
Application of the centre manifold theory
Equilibrium states

Thus, the limit system (3.29) cannot be used to describe the long-term dynamics of the original system (3.25) as it approaches zero. It follows that the roots of the characteristic equation, which are the eigenvalues of M, have For each small perturbation, the behavior of the equilibrium point of the reduced system (3.29) is affected.

Figure 3.5: The dynamics of the prey and predators given by the reduced system (3.29).

Discussion

Well-posedness of the model

Note that it is possible to introduce a partial order in Y by stating that x ≤ y if and only if y−x∈Y+. In particular, in Rn we can consider Rn+ as a positive cone that induces the next partial order from Rn.

Aggregated model

Inequalities (4.6) show that the infected population evolves more slowly than the healthy population, but faster than the healthy population with the disease-specific mortality, [19], and implies that neither i nor s can explode in finite time, becausein1, =s+i and i, s ≥0.

Analysis of quasi-steady states

It follows that the quasi-stationary state s¯1 = n1 is stable if n1 < ν and the quasi-stationary state. Note that if n1 = ν, then the two quasi-stationary states coincide; this violates assumption A2 of Tikhon's theorem. The second example consists of the analysis of the behavior of solutions (4.2) when passing near the intersection from a stable branch of one quasi-stationary state to a stable branch of another, [19].

Direct application of the Tikhonov theorem

The case of stable population

The first example consists of determining the invariant subsets In1×In2 that do not contain the intersection of quasi-stationary states and in which the quasi-stationary states are isolated. Then we use Tikhon's theorem within these sets. Therefore, the set V1 is invariant with respect to the current generated by (eAt)t≥0. If (0,0), which is the only stable equilibrium (4.10), is asymptotically stable (ie, if we have a strict inequality in (4.11) )), then any trajectory starting below the isocline2 = (a/µ2 )n1, increased and reached its maximum on the isocline, then converges to (0,0) and any trajectory starting above the isocline n2 = (a/µ2)n1 will decrease to (0,0). This shows that each initial condition s0 from (4.9) belongs to the region of attraction of the stable root of equation (4.7) when n1 < ν.

The case of unstable population

Comments on Stable-Unstable Case

It follows that by the linear transformation (4.23) the eigenvectors corresponding to the eigenvalues of the Jacobian matrix of (4.2) at (0,0,0) are given by. In the next subsection we present numerical simulations describing the behavior of the solutions of (4.2) when n1 < ν and when n1 > ν. In the next section we analyze the behavior of the solutions of (4.2) when they come close to the intersection of the quasi-stable states.

Figure 4.1: The solution n ¯ 1 (t) of the limit equation (4.10) uniformly attracting solutions n 1, (t) of (4.2) when n 1 < ν .

Immediate exchange of stabilities

Composed stable solution of the degenerate equation

4.35) If we replace the unknown u in the second equation of (4.32) by the known compound stable root ϕ(v, t), we get the limit equation. As for the initial value u0 for u(t), we consider, as in the standard Tikhonov theorem, the following assumption. The initial value u0 lies in the basin of attraction of the equilibrium point ϕ1(v0,0) of the auxiliary equation (4.34) for v =v0 and t= 0, i.e.

Initial Layer

Delayed exchange of stabilities

In I¯u ×I¯T the solution of the degenerate equation (4.50) consists of two roots u = 0 and u = ϕ(t), where ϕ is twice continuously differentiable on I¯T. For a more detailed explanation of the meaning of these assumptions, we refer the reader to the monograph of Butuzov, [2]. Then for u0 ≥ 0 and sufficiently small there exists a unique solution u(t) of the initial value problem (4.49) on [0, T], which is positive and satisfies the conditions.

SIS Model with Vital Dynamics

The case of increasing population
The case of decreasing population

For a growing population, Figure 4.9 shows that the root i = 0 of the degenerate equation is stable and that the solution of the original equation remains close to the quasi-stationary state ¯i(t) = 0 when 0< t < t∗ . For a decreasing population, the root i = n − ν of the degenerate equation is stable when 0 < t < tc and the solution of the original equation remains close to the quasi-steady state. The root i = 0 of the degenerate equation now becomes stable and the solution of the original equation converges to the quasi-stationary state ¯i(t) = 0.

Figure 4.9: Delayed exchange of stabilities for an increasing population given by the solution i (t) of (4.53) and the composed stable solution when r > 0 and n 0 < ν .

SIS Model with Age Structure

The case of increasing population

The population grows when δ <0. 4.80) The proof of this result follows from the one-dimensional case in Section 4.2, when the population grows. Then we have the following result Proposition The proof of this result follows from the one-dimensional case in Section 4.2, when the population grows. Then we have the following result Proposition 4.6.4. Then, for η, % sufficiently small and ˆt sufficiently close to ¯t∗, lim→0. 4.130) The proof of this result follows from the one-dimensional case in Section 4.2, when the population is increasing.

Figure 4.13: Upper solution of (4.3) given by the first equation of (4.73), when N 1 0 < ν and δ < 0.

Comment on the case of decreasing population

Discussion

Here we have shown that the zero-order approximation provided by Tikhonov's theorem allows for the extension of the concern result to the entire half-line. The exchange of stability occurs when the quasi-steady states of the problem intersect and the solutions of the perturbed problem pass near the intersection. Sakamoto.Invariant Manifolds in Singular Perturbation Problems for Ordinary Differential Equations, Proceedings of the Royal Society of Edinburgh.