There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.
In the beginning of bifurcation problem, R. Thom developed catastrophe theory, which stands on a statical model, that is, it consists of multi-variable real functions with parameters ( [
Although there are many books written on “bifurcation problem”, “Catastrophe Theory and its Applications” by T. Poston and I. Stewart [
In the slow-fast system with parameters, we took up the pseudo-singular point having structural stability. It means the catastrophe is caused on a dynamic model directly. In fact, when analyzing a concrete example, the constrained surface ( ε = 0 ) in the slow-fast system reduces the Thom’s function classified. Throughout this paper, we shall describe the structure precisely. In our previous paper [
When the parameter value changes from negative sign to positive one, even if the pseudo singular point satisfies the conditions of canard, it is confirmed that the unstable pseudo-singular point is vanished. In other words, the canards are vanished. Then, the canards on the stable pseudo-singular point just appear. We call it “canards flying”. Since “canard” is a singular-limit solution ( ε tends to zero) in the slow-fast system, the behavior of the orbit is very complicated when ε takes nearly equal to zero. It gives us a new structure as “dynamical catastrophe”.
Consider the following system:
{ ε d x d t = h ( x , y , ε ) d y d t = g ( x , y , b ) (1)
where ε is infinitesimal, b is any constant and
x = ( x 1 , x 2 ) ∈ R 2 , y = ( y 1 , y 2 ) ∈ R 2 ,
h = ( h 1 , h 2 ) : R 4 → R 2 , g = ( g 1 , g 2 ) : R 4 → R 2 .
Assume that g ( x , y , b ) = g ( x , b y ) , for the simplicity, and the origin is a singular point.
Furthermore we assume that the System (1) sastisfies the following conditions (A1)-(A6):
(A1) h is of class C 1 and g is of class C 2 .
(A2) The slow manifold S = { ( x , y ) ∈ R 4 | h ( x , y ,0 ) = 0 } is a two-dimensional differential manifold and intersects the set
T = { ( x , y ) ∈ R 4 | det [ ∂ h ∂ x ( x , y , 0 ) ] = 0 } (2)
transversely, where
∂ h ∂ x = [ ∂ h 1 ∂ x 1 ∂ h 1 ∂ x 2 ∂ h 2 ∂ x 1 ∂ h 2 ∂ x 2 ] (3)
Then, the pli set
P L = { ( x , y ) ∈ S ∩ T } (4)
is a one-dimensional differentiable manifold.
(A3) Either the value of g 1 or that of g 2 is nonzero at any point of PL.
Note that the pli set PL devides the slow manifolds S\PL into three parts
depending on the signs of the two eigenvalues of ∂ h ∂ x ( x , y ,0 ) .
First consider the following reduced system which is obtained from (1) with ε = 0 :
{ 0 = h ( x , y , 0 ) d y d t = g ( x , y , b ) (5)
By differentiating h ( x , y ,0 ) with respect to t, we have
∂ h ∂ x ( x , y , 0 ) d x d t + ∂ h ∂ y ( x , y , 0 ) g ( x , y , b ) = 0 (6)
Then (4) becomes the following:
{ d x d t = − [ ∂ h ∂ x ( x , y , 0 ) ] − 1 ∂ h ∂ y ( x , y , 0 ) g ( x , y , b ) d y d t = g ( x , y , b ) (7)
where ( x , y ) ∈ S \ P L . To avoid degeneracy in (6), we consider the time-scaled-reduced system:
{ d x d t = { − det [ ∂ h ∂ x ( x , y , 0 ) ] − 1 } [ ∂ h ∂ x ( x , y , 0 ) ] − 1 ∂ h ∂ y ( x , y , 0 ) g ( x , y , b ) d y d t = { det [ ∂ h ∂ x ( x , y , 0 ) ] − 1 } g ( x , y , b ) (8)
The phase portrait of the System (8) is the same as that of (7) except the region
where det [ ∂ h ∂ x ( x , y , 0 ) ] = 0 , but only the orientation of the orbit is different.
Definition 1. A singular point of (8), which is on PL, is called a pseudo singular point of (1). The set of pseudo singular points is denoted by PS.
(A4) rank [ ∂ h ∂ x ( x , y , 0 ) ] = 2 , rank [ ∂ h ∂ y ( x , y , 0 ) ] = 2 for any ( x , y ) ∈ S \ P L .
From (A4), the implicit function theorem guarantees the existence of a unique function y = φ ( x ) such that h ( x , φ ( x ) ,0 ) = 0 . By using y = φ ( x ) , we obtain the following system:
d x d t = { − det [ ∂ h ∂ x ( x , φ ( x ) , 0 ) ] − 1 } [ ∂ h ∂ x ( x , φ ( x ) , 0 ) ] − 1 ∂ h ∂ y ( x , φ ( x ) , 0 ) g ( x , φ ( x ) , b ) . (9)
(A5) All singular points of (8) are non-degenerate, that is, the linearization of (8) at a singular point has two nonzero eigenvalues.
Now, let us introduce a definition of “symmetry”. It is a key word through this paper.
Definition 2. If h 1 ( x 1 , x 2 , y 1 , y 2 , ε ) = h 2 ( x 2 , x 1 , y 2 , y 1 , ε ) , and g 1 ( x 1 , x 2 , y 1 , y 2 , b ) = g 2 ( x 2 , x 1 , y 2 , y 1 , b ) , then the system is “symmetric” for the subspace I = { ( x 1 , x 2 , y 1 , y 2 ) | x 1 = x 2 , y 1 = y 2 } .
(A6) I intersects PL transversely.
Definition 3. Let λ 1 , λ 2 be two eigenvalues of the linearization of (8) at a pseudo singular point. The pseudo singular point with real eigenvalues is called a pseudo singular saddle point if λ 1 < 0 < λ 2 and a pseudo singular node point if λ 1 < λ 2 < 0 or λ 1 > λ 2 > 0 .
The following Theorems 1 and 2 are established in [
Theorem 1. Let ( x 0 , y 0 ) be a pseudo singular saddle or node point. If
trace [ ∂ h ∂ x ( x 0 , y 0 , 0 ) ] < 0 , then there exists a solution which first follows the attractive part and the repulsive part after crossing PL near the pseudo singular
point.
Remark 1. The solution in Theorem 1 is called “canard”.
Theorem 2. If det [ ∂ f ∂ x ( 0 , 0 ) ] = 0 , and trace [ ∂ f ∂ x ( 0 , 0 ) ] < 0 , then canards
near the subspace I has a centre manifold.
Remark 2. The condition trace [ ∂ h ∂ x ( x 0 , y 0 , 0 ) ] < 0 implies that one of eigenvalues of [ ∂ h ∂ x ( x 0 , y 0 ,0 ) ] is equal to zero and the other one is negative. Notice
that the system has two kinds of vector fields: one is 2-dimensional slow and the other is 2-dimensional fast one. The condition provides the state of the fast vector field.
Remark 3. The singular solution in Theorem 1 is called a canard in R 4 with 2-dimensional slow manifold. As a result, it causes a delayed jumping. The study of canards requires still more precise topological analysis on the slow vector field.
Remark 4. On the subspace I, the following system is established for some b. I is an invariant manifold.
{ ε d x 1 d t = h 1 ( x 1 , y 1 , ε ) d y 1 d t = g 1 ( x 1 , y 1 , b ) g 1 ( x 1 , x 2 , y 1 , y 2 , b ) = (10)
Remark 5. On the set PL, det [ ∂ h ∂ x ] = 0 is satisfied and at ( x 0 , y 0 ) ∈ P S the
following equation is established:
{ ∂ h 1 ∂ x 1 ( x 0 , φ ( x 0 ) , 0 ) ∂ h 2 ∂ x 2 ( x 0 , φ ( x 0 ) , 0 ) − ∂ h 1 ∂ x 2 ( x 0 , φ ( x 0 ) , 0 ) ∂ h 2 ∂ x 1 ( x 0 , φ ( x 0 ) , 0 ) } g 1 ( x 0 , φ ( x 0 ) , b ) = 0. (11)
Note that there exists y = ϕ ( x ) because of assuming rank [ ∂ h ∂ y ] = 2 .
When and why does the pseudo singular point have structural stability? A geometrical point of view to make it clear is shown in this section.
Lemma 1. The matrix [ ∂ h ∂ x ] is symmetric.
Proof. Because the system is symmetric for the set I, it is obvious from elementary calculus. □
From (A6), the subspace I intersects PL transversely. Lemma 1 ensures that I c also intersects PL transversely, where I c is the orthogonal complement of
I. Since the matrix [ ∂ h ∂ y ] is also symmetric, for the sake of simplicity, suppose that [ ∂ h ∂ y ] is identity without loss of generality.
Using Remark 5, the following lemma is established in [
Lemma 2. Let ( x 0 , y 0 ) ∈ P S be on I ∩ P L , then it depends on the parameter b. On the other hand, on I c ∩ P L , it is independent of b.
From Lemma 2, we establish the following thorems.
Theorem 3. There exists a pseudo singular point ( x 0 , y 0 ) ∈ P S , which is one of a coupled points near the subspace I, when satisfying b 0 > b > 0 . When b 0 = 0 , it is just on I.
Theorem 4. Let ( x 0 , y 0 ) ∈ P S be a saddle or node point. Then, if ( x 0 , y 0 ) ∈ I c ∩ P L , the pseudo singular point is structurally stable. If ( x 0 , y 0 ) ∈ I ∩ P L it is structurally unstable.
The next theorem is the main result.
Theorem 5. Let a critical value b 0 = inf 1 ≤ k ≤ N b k be positive, where each
b k ( k = 1 , 2 , ⋯ , N ) is also a critical value and N is the hyperfinte number. If the structurally stable pseudo-singular point for b < 0 and the unstable pseudo-singular point satisfy the canard conditions in Theorem 1, then there exists the canard flying.
Proof. If b 0 > b > 0 , then the pseudo-singular point on I is unstable by Lemma 2. If b < 0 , the pseudo-singular point on I c is stable and saddle or node because of the conditions in Theorem 1. Suppose the canard conditions in Theorem 1, then the other unstable point is vanishing, that is, the canard is flying. □
Theorem 5 plays a central role to establish canards flying under Theorems 3 and 4 which are already shown in K-T [
Consider the following typical example of modefied coupled FitzHugh-Nagumo equations. See [
{ ε d x 1 d t = x 2 + y 1 − x 1 3 3 ε d x 2 d t = x 1 + y 2 − x 2 3 3 d y 1 d t = − 1 c ( x 1 + b y 1 ) d y 2 d t = − 1 c ( x 2 + b y 2 ) (12)
The next equation is the time-scaled-reduced system corresponding to (19).
d x d t = − ( − x 2 2 − 1 − 1 − x 1 2 ) ( − 1 c ( x 1 + b y 1 ) − 1 c ( x 2 + b y 2 ) ) ≜ f = ( f 1 f 2 ) . (13)
There exsist pseudo singular points ( x 0 , y 0 ) ∈ P S of the System (9) which are obtained by the following. If ( x 0 , y 0 ) = ( x 01 , x 02 , y 01 , y 02 ) exists on neighborhood of I ∩ P L , then x 1 x 2 = 1 holds.
Remark 6. In the previous paper [
Remark 7. Notice that in Lemma 2, if ( x 0 , y 0 ) ∈ I ∩ P L , then the critical value
b = b 0 = 3 2 holds. When 0 < b < 3 2 = b 0 , the corresponding pseudo singular
points are on neighborhood of I but not on I.
If ( x 0 , y 0 ) ∈ I c ∩ P L , then x 1 x 2 = − 1 holds. Therefore,
{ x 01 = ± 1 x 02 = ∓ 1 y 01 = x 01 3 3 − x 02 y 02 = x 02 3 3 − x 01 (14)
Remark 8. The solution of (14) is structurally stable. Then
[ ∂ h ∂ x ] ( x 1 , x 2 ) = ( 1 , − 1 ) = ( − 1 1 1 − 1 ) (15)
and
[ ∂ f ∂ x ] ( x 1 , x 2 ) = ( 1 , − 1 ) = − 1 c ( 1 − 1 − 8 3 b − 1 − 8 3 b 1 ) . (16)
Then,
trace [ ∂ h ∂ x ] ( x 1 , x 2 ) = ( 1 , − 1 ) = trace ( − 1 1 1 − 1 ) = − 2. (17)
Therefore, the assumptions of Theorem 1 are satisfied and a canard exsits. On the other hand, the characteristic equation is
( 1 + λ ) 2 − ( 1 + 8 3 b ) 2 = 0. (18)
The solution of (18) is
λ = − 1 ± ( 1 + 8 3 b ) . (19)
Therefore, if b < − 3 4 , then the psudo singular point is saddle. On the other hand, if − 3 4 < b < 0 , then it is node.
Let us consider a stochastic differential equation for a slow-fast system with a Brownian motion B ( t ) as the random noises modifying the slow fast System (1): For t ∈ [ 0 , T ] , T > 0
{ ε d x = h ( x , y , ε ) d t d y = g ( x , y , ε ) d t + σ d B (20)
where B = ( B 1 B 2 ) ∈ R 2 is a 2-dimensional standard Brownian motion and σ > 0
is a positive constant which gives a standard deviation for the Brownian motion B ( t ) .
When the system (12) is disturbed by B ( t ) , which gives small noises, the canard solution changes to another orbit.
Anderson [
Definition 4. Let N t = t Δ t , 0 ≤ t ≤ T and N = N T . Assume that a sequence
of i.i.d. random variables { Δ B k , k = 1 , ⋯ , N } has the distribution
P { Δ B k = Δ t } = P { Δ B k = − Δ t } = 1 2 (21)
for each k = 1 , ⋯ , N . An extended Wiener process { B ( t ) , t ≥ 0 } is defined by
B ( t ) = ∑ k = 1 N t Δ B k , 0 ≤ t ≤ T . (22)
Rewriting the System (20) via step functions on the hyper finite time line, the following System (23) is obtained.
{ ε { x 1 ( t k ) − x 1 ( t k − 1 ) } = ( x 2 + y 1 − x 1 3 3 ) Δ t ε { x 2 ( t k ) − x 2 ( t k − 1 ) } = ( x 1 + y 2 − x 2 3 3 ) Δ t y 1 ( t k ) − y 1 ( t k − 1 ) = − 1 c ( x 1 + b y 1 ) Δ t + σ 1 Δ B 1 k y 2 ( t k ) − y 2 ( t k − 1 ) = − 1 c ( x 2 + b y 2 ) Δ t + σ 2 Δ B 2 k (23)
where
B 1 ( t ) = ∑ k = 1 N t Δ B 1 k , B 1 ( t ) = ∑ k = 1 N t Δ B 1 k .
When σ 1 = σ 2 = 0 , the System (23) is the nonstandard form of (12). For more details of the stochastic slow fast system, see [
In Figures 1-8, ε = 0.01 , c = 1 and Δ t = 0.0001 in (23). The curves, which satisfy x 1 x 2 = 1 and x 1 x 2 = − 1 , respectively, are Pli set.
singular point (1, −1). At the pseudo singular point (1, −1), the eigenvalue λ in (18) is negative and it is node.
The slow-fast system with a bifurcation parameter gives us structural stability under the key notion “symmetry”. In our previous paper published in Advances in Pure Mathematics, vol.12 (2022), it has been confirmed, then we emphasize that the stability for the pseudo singular point plays an important role.
In economic models [
When being ε = 0 , the constrained surface describes something like a potential function because of holding (A4). On the invariant manifold, it is “fold” in Thom’s function: only domestic case has such a function, and the two-region model has a multi-variable one much complicated. Especially, take a note that the pseudo singular point has no structural stability due to no bifurcation parameter in the original system [
The orbit ( ε tends to “zero”) after passing through near the pseudo singular point jumps out with delay, that is, the state itself is jumping and the behavior is complicated as “dynamical catastrophe”.
The first author is supported in part by Grant-in-Aid Scientific Research (C), No.18K03431, Ministry of Education, Science and Culture, Japan.
The authors declare no conflicts of interest regarding the publication of this paper.
Kanagawa, S. and Tchizawa, K. (2023) Canards Flying on Bifurcation. Advances in Pure Mathematics, 13, 412-424. https://doi.org/10.4236/apm.2023.136026