Theorem 3.1 (Fractional-order independent instability results)

1. If det(A) < 0, system (3.1) is unstable, regardless of the fractional ordersq₁andq₂.

2. If det(A) > 0, system (3.1) is unstable regardless of the fractional ordersq₁andq₂if and only if one of the following conditions holds:

   {a11+a22≥det(A)+1 ora11>0, a22>0, a11a22≥det(A).

Theorem 3.2 (Fractional-order-independent stability results)

System (3.1) is asymptotically stable, regardless of the fractional ordersq₁, q₂ ∈ (0, 1] if and only if the following inequalities are satisfied:

Remark 3.2

In the classical integer order case (i.e. q₁ = q₂ = 1), it is well-known that a two-dimensional linear autonomous system of the form x′ = Ax, with constant matrix A ∈ ℝ^2×2 is asymptotically stable if and only if Tr(A) < 0 and det(A) > 0. Based on Theorem 3.2, a supplementary inequality

is required to guarantee that system (3.1) is asymptotically stable, regardless of the choice of fractional orders q₁, q₂ ∈ (0, 1].

Remark 3.3

Due to Theorem 3.1, when det(A) = δ > 0 is arbitrarily fixed, system (3.1) is unstable for any choice of the fractional orders q₁, q₂ ∈ (0, 1] if and only if (a₁₁, a₂₂) ∈ R_u(δ). On the other hand, based on Theorem 3.2, system (3.1) is asymptotically stable for any q₁, q₂ ∈ (0, 1] if and only if (a₁₁, a₂₂) ∈ R_s(δ). Therefore, if (a₁₁, a₂₂) ∈ ℝ²\(R_s(δ) ∪ R_u(δ)) (e.g. white region in Fig. 1), the stability properties of system (3.1) depend on the considered fractional orders.

Lemma 3.1

Letδ > 0, q₁, q₂ ∈ (0, 1] and consider the smooth parametric curve in the (a₁₁, a₂₂)-plane defined by

where:

with the functionsρ₁(q₁, q₂) andρ₂(q₁, q₂) defined forq₁ ≠ q₂as

The following statements hold:

1. The curve Γ(δ, q₁, q₂) is the graph of a smooth, decreasing, concave bijective functionϕδ,q1,q2:ℝ→ℝin the (a₁₁, a₂₂)-plane.

2. The curve Γ(δ, q₁, q₂) lies outside the third quadrant of the (a₁₁, a₂₂)-plane.

Proof

Let δ > 0 and q₁, q₂ ∈ (0, 1] arbitrarily fixed.

Proof of statement (i). The real-valued function ω ↦ h(ω, q₁, q₂) is bijective and monotonous on ℝ: strictly decreasing if q₁ ≤ q₂ and strictly increasing otherwise. Therefore, the particular form of the parametric equations implies that the curve Γ(δ, q₁, q₂) is the graph of a smooth decreasing bijective function ϕδ,q1,q2:ℝ→ℝ in the (a₁₁, a₂₂)-plane.

If q₁ ≠ q₂, using the chain rule, we compute:

Assuming that q₁ < q₂, the expression above is strictly negative, as ρ₁ > 0, ρ₂ > 0 and q1q2≤ρ1ρ2≤1 (since the function x↦sinxx is decreasing on (0, π)). A similar argument holds in the case q₁ > q₂ as well. Hence, ϕδ,q1,q2 is a concave function.

Proof of statement (ii). Assume the contrary, i.e. that there exists (a₁₁, a₂₂) ∈ Γ(δ, q₁, q₂) such that a₁₁ < 0 and a₂₂ < 0, or equivalently, that there exists ω ∈ ℝ such that h(±ω, q₁, q₂) < 0. As the case q₁ = q₂ is trivial, we assume without loss of generality that q₁ < q₂. The inequalities h(±ω, q₁, q₂) < 0 are equivalent to

which leads to ρ₂(q₁, q₂) < ρ₁(q₁, q₂), or equivalently to q₂ < q₁, which is absurd. Hence, the curve Γ(δ, q₁, q₂) does not have any points in the third quadrant.

Remark 3.4

If q₁ = q₂ := q, Γ(δ, q₁, q₂) represents the straight line:

Lemma 3.2

Letδ > 0, q₁, q₂ ∈ (0, 1] be arbitrarily fixed. The following statements hold:

1. The characteristic function Δ(s; a₁₁, a₂₂, δ, q₁, q₂) has at most a finite number of roots satisfying ℜ(s) ≥ 0.

2. The function (a₁₁, a₂₂) ↦ N(a₁₁, a₂₂, δ, q₁, q₂) is continuous at all points (a₁₁, a₂₂) that do not belong to the curve Γ(δ, q₁, q₂). Consequently, N(a₁₁, a₂₂, δ, q₁, q₂) is constant on each connected component of ℝ²\Γ(δ, q₁, q₂).

Proof

The first step of the proof (see Appendix A.1.) consists of showing that there exist a strictly decreasing function lδ,q1,q2:ℝ+→ℝ+ and a strictly increasing function Lδ,q1,q2:ℝ+→ℝ+ such that any unstable root of Δ(s; a₁₁, a₂₂, δ, q₁, q₂) is bounded by

where p=q1+q22min{q1,q2}≥1 and a = (a₁₁, a₂₂). Moreover, ‖·‖_p denotes the p-norm in ℝ².

Proof of statement (i). Assuming the contrary, that there exists an infinite number of unstable roots, the Bolzano-Weierstrass theorem implies that there exists a convergent sequence of unstable roots (s_j) with the limit s₀ ≠ 0 (since δ > 0), such that ℜ(s₀) ≥ 0. As the function Δ(s; a₁₁, a₂₂, δ, q₁, q₂) is analytic in ℂ∖ℝ−, by the principle of permanence it follows that it is identically zero, which is absurd. Therefore, we obtain that N(a₁₁, a₂₂, δ, q₁, q₂) is finite.

Proof of statement (ii). Let a0=(a110,a220)∈ℝ2∖Γ(δ,q1,q2) and consider r > 0 such that the open neighborhood B_r(a⁰) = {a = (a₁₁, a₂₂) ∈ ℝ²: ‖a − a⁰‖_p < r} of the point a⁰ is included in ℝ²\Γ(δ, q₁, q₂).

For any a = (a₁₁, a₂₂) ∈ B_r(a⁰), we have:

and hence, inequality (3.4) implies that any root s of Δ(s; a₁₁, a₂₂, δ, q₁, q₂) such that ℜ(s) ≥ 0 satisfies:

Denoting m=lδ,q1,q2(r+‖a0‖p) and M=Lδ,q1,q2(r+‖a0‖p), let us consider in the complex plane the simple closed curve (γ), oriented counterclockwise, bounding the open set

The above construction shows that for any a = (a₁₁, a₂₂) ∈ B_r(a⁰) all unstable roots of Δ(s; a₁₁, a₂₂, δ, q₁, q₂) are inside the open set D.

As Δ(s;a110,a220,δ,q1,q2)≠0 for any s ∈ (γ), it is easy to see that

Moreover, we consider q ≥ 1 such that 1p+1q=1 and denote:

Based on Hölder’s inequality, it follows that for any s ∈ (γ) and for any a ∈ B_r′(a⁰) ⊂ B_r(a⁰), we have:

The Rouché theorem implies Δ(s; a₁₁, a₂₂, δ, q₁, q₂) and Δ(s;a110,a220,δ,q1,q2) have the same number of roots in the domain D, and hence

Therefore, the function (a₁₁, a₂₂) ↦ N(a₁₁, a₂₂, δ, q₁, q₂) is continuous on ℝ²\Γ(δ, q₁, q₂), and as it is integer-valued, it follows that it is constant on each connected component of ℝ²\Γ(δ, q₁, q₂).

Theorem 3.3 (Fractional-order-dependent stability and instability results)

Let det(A) = δ > 0 andq₁, q₂ ∈ (0, 1] arbitrarily fixed. Consider the curve Γ(δ, q₁, q₂) and the functionϕδ,q1,q2:ℝ→ℝgiven by Lemma 3.1.

1. The characteristic equation (3.2) has a pair of pure imaginary roots if and only if (a₁₁, a₂₂) ∈ Γ(δ, q₁, q₂).

2. System (3.1) isO(t−q)-asymptotically stable (withq = min{q₁, q₂}) if and only if

   a22<ϕδ,q1,q2(a11).

3. Ifa22>ϕδ,q1,q2(a11), system (3.1) is unstable.

Proof

Assume that δ > 0 and q₁, q₂ ∈ (0, 1] are arbitrarily fixed.

Proof of statement (i). It is easy to see that the characteristic equation (3.2) has a pair of pure imaginary roots if and only if there exists ω ∈ ℝ such that Δ(iδ1q1+q2eω;a11,a22,δ,q1,q2)=0. As iq=cosqπ2+isinqπ2, taking the real and the imaginary parts of the previous equation, one obtains:

If q₁ ≠ q₂, solving this system for a₁₁ and a₂₂ shows that the characteristic equation (3.2) has a pair of pure imaginary roots if and only if (a₁₁, a₂₂) belongs to the curve Γ(δ, q₁, q₂) given by Lemma 3.1.

In the particular case q₁ = q₂ := q, system (3.5) is compatible if and only if ω = 0. Moreover, the set of solutions of (3.5) is the straight line

which represents Γ(δ, q, q) given by Lemma 3.1 (see Remark 3.4).

Proof of statement (ii). Choosing a₁₁ = a₂₂ = −1, we argue that Δ(s; −1, −1, δ, q₁, q₂) does not have any roots in the right half plane. Indeed, assuming that there exists s∈ℂ such that ℜ(s) ≥ 0 and

it follows by division by sq1 that

As q₂ ∈ (0, 1], q₂ − q₁ ∈ [−1, 1] and −q₁ ∈ [−1, 0), it follows that the real part of each term from the left hand side of the above equality is positive, which leads to a contradiction. Hence, N (−1, −1, δ, q₁, q₂) = 0. From Lemma 3.1 (ii) and Lemma 3.2 it follows that N(a₁₁, a₂₂, δ, q₁, q₂) = 0, for any (a₁₁, a₂₂) from the region below the curve Γ(δ, q₁, q₂), which leads to the desired conclusion.

Proof of statement (iii). Let s(a₁₁, a₂₂, δ, q₁, q₂) denote the root of Δ(s; a₁₁, a₂₂, δ, q₁, q₂) satisfying s(a11⋆,a22⋆,δ,q1,q2)=iβ, with β=δ1q1+q2eω as in the proof of statement (i), where (a11⋆,a22⋆)∈Γ(δ,q1,q2). Taking the derivative with respect to a₁₁ in the equation

we obtain

We deduce:

and therefore

We have

where P(s)=(q1+q2)sq1+q2−1−a11⋆q2sq2−1−a22⋆q1sq1−1. A simple computation leads to

In a similar way, we compute ∂ℜ(s)∂a22|(a11⋆,a22⋆) and we finally obtain the gradient vector

From the parametric equations of the curve Γ(δ, q₁, q₂) and the properties of the function h it is easy to deduce that the gradient vector ∇ℜ(s)(a11⋆,a22⋆) is in fact a normal vector to the curve Γ(δ, q₁, q₂) that points outward from the region below the curve. We deduce that the following transversality condition is fulfilled for the directional derivative:

for any vector u¯ which points outward from the region below the curve Γ(δ, q₁, q₂). Therefore, as the parameters (a₁₁, a₂₂) cross the curve Γ(δ, q₁, q₂) into the region above the curve, ℜ(s) becomes positive and the pair of conjugated roots (s,s¯) crosses the imaginary axis from the open left half-plane to the open right half-plane. Hence, N(a₁₁, a₂₂, δ, q₁, q₂) = 2 for any (a₁₁, a₂₂) from the region above the curve Γ(δ, q₁, q₂), and the system (2.1) is unstable.

Remark 3.5

In Fig. 2, several curves Γ(δ, q₁, q₂) have been plotted for δ = 4, q₁ = 0.6 and q₂ ∈ (0, 1], together with the fractional-order-independent stability regions R_s(δ), R_u(δ). The regions below and above each curve represent the asymptotic stability region and instability region, respectively, provided by Theorem 3.3. Lighter shades of red and blue have been used to plot the parts of these regions for which system (3.1) is asymptotically stable / unstable for the particular values of the fractional orders q₁, q₂ which have been chosen, but not for all (q₁, q₂) ∈ (0, 1].

Example 3.1

We consider the system

where q₁, q₂ ∈ (0, 1].

We first verify if the fractional-order-independent stability or instability conditions given in Theorem 3.1 and Theorem 3.2 hold. First, as det(A) = 0.002201 > 0, a simple computation shows that a₁₁ + a₂₂ < det(A) + 1 and a₁₁a₂₂ < det(A). Hence, based on Theorem 3.1, we deduce that system (3.6) is not unstable regardless of the considered fractional orders q₁ and q₂. On the other hand, as a₁₁ + a₂₂ > 0, it is clear from Theorem 3.2 that system (3.6) is not asymptotically stable regardless of the considered fractional orders q₁ and q₂. In other words, the stability and instability of system (3.6) depend on the choice of the fractional orders q₁ and q₂, as shown in the following situations.

Case 1. The special case (q1,q2)=(12,14) has been considered in [11], and it has been shown, by transforming the corresponding system to a system of three fractional differential equations with the same order 14, that in this particular case, system (3.6) is globally asymptotically stable.

Indeed, applying Theorem 3.3, we deduce that system (3.6) with the fractional orders (q1,q2)=(12,14) is asymptotically stable if and only if

where a₁₁ = 0.00001, a₂₂ = 0.1, δ = det(A) = 0.002201 and, relying on the notations introduced in Lemma 3.1:

where ω^* is the unique root of the equation

From this algebraic equation, we numerically compute ω^* = 0.818108 and therefore, we also deduce ϕδ,q1,q2(a11)=0.208493. As a₂₂ = 0.1, it can be easily seen that the asymptotic stability condition a22<ϕδ,q1,q2(a11) is satisfied (as depicted in Fig. 3).

Case 2. We now consider a different special case: (q1,q2)=(14,12). Following the same steps as in the previous case, using Theorem 3.3, we now compute: ϕδ,q1,q2(a11)=0.0271274 and therefore, as a₂₂ = 0.1, it follows that a22>ϕδ,q1,q2(a11), and hence, system (3.6) is unstable (as shown in Fig. 3).

This can also be verified by the method proposed in [11]. Indeed, for (q1,q2)=(14,12), system (3.6) is equivalent to the following system of three fractional differential equations of the same order q=14:

The spectrum of eigenvalues of the matrix B is

and hence, based on Matignon’s theorem [25], as arg(λ₂) = arg(λ₃) = 0, it follows that system (3.7) is unstable.

In conclusion, it is easy to see from the previously considered cases that system (3.6) will be asymptotically stable for some pairs of fractional orders (such as (q1,q2)=(12,14) considered in Case 1.), while it will be unstable for other pairs of fractional orders (such as (q1,q2)=(14,12) considered in Case 2.). In fact, using the inequality provided by Theorem 3.3, the region of all pairs of fractional orders (q₁, q₂) ∈ (0, 1]² for which system (3.6) is asymptotically stable can be numerically computed (see Fig. 4).

Lemma 3.3

If (a₁₁, a₂₂) ∈ R_u(δ), then system (3.1) is unstable, regardless of the fractional ordersq₁andq₂.

Proof

Let (a₁₁, a₂₂) ∈ R_u(δ) and (q₁, q₂) ∈ (0, 1]² arbitrarily fixed. We will show that the characteristic function Δ(s; a₁₁, a₂₂, δ, q₁, q₂) has at least one positive real root.

First, it is easy to see that Δ(s; a₁₁, a₂₂, δ, q₁, q₂) → ∞ as s → ∞.

On one hand, let us notice that if a₁₁+a₂₂ ≥ δ + 1, it follows that

Hence, the function s ↦ Δ(s; a₁₁, a₂₂, δ, q₁, q₂) has at least one positive real root in the interval [1, ∞). Therefore, the system (3.1) is unstable.

On the other hand, if a₁₁ > 0, a₂₂ > 0 and a₁₁a₂₂ ≥ δ, as

we see that for s0=(a11)1/q1>0, we have

Hence, the function s ↦ Δ(s; a₁₁, a₂₂, δ, q₁, q₂) has at least one strictly positive real root. It follows that system (3.1) is unstable.

Lemma 3.4

If (a₁₁, a₂₂) ∈ R_s(δ) then system (3.1) is asymptotically stable, regardless of the fractional ordersq₁andq₂.

Proof

Let (a₁₁, a₂₂) ∈ R_s(δ) and (q₁, q₂) ∈ (0, 1]² arbitrarily fixed. As a₁₁+a₂₂ < 0, we may assume, without loss of generality, that a₁₁ < 0.

Assume by contradiction that Δ(s; a₁₁, a₂₂, δ, q₁, q₂) has a root s₀ = re^iθ in the right half-plane, where r > 0 and θ∈[0,π2].

Multiplying the characteristic equation by s0−q1, we get:

Taking the real part in the above equation and noticing that s0q2,s0q2−q1 and s0−q1 are in the right half-plane, we obtain:

It is important to remark that for any r > 0, q₁, q₂ ∈ (0, 1] and θ∈[0,π2], the following inequality holds:

Indeed, denoting q1θ=x∈[0,π2],q2θ=y∈[0,π2] and r1θ=α>0 this inequality is equivalent to

which is proved in Appendix A.2. It follows that:

On the other hand, as (a₁₁, a₂₂) ∈ R_s(δ), we have a₂₂< −a₁₁ and a₂₂ < min{1, δ}. Hence, a₂₂ < min{1, −a₁₁, δ}, which leads to a contradiction. Therefore, we deduce that all the roots of the characteristic function Δ(s; a₁₁, a₂₂, δ, q₁, q₂) are in the open left half-plane, and hence, system (3.1) is asymptotically stable.

Lemma 3.5

The following holds:

Proof

A proof by double inclusion is presented below.

Step 1. Proof of the inclusionQ(δ) ⊆ ℝ²\(R_s(δ) ∪ R_u(δ)).

In the case q₁ = q₂ = q, elementary inequalities and Remark 3.4 provide that Γ(δ, q, q) are straight lines which are included in ℝ²\(R_s(δ) ∪ R_u(δ)).

Let us now consider q₁ < q₂ (the opposite case is treated similarly) and show that Γ(δ, q₁, q₂) ⊂ ℝ²\(R_s(δ) ∪ R_u(δ)). Considering an arbitrary point (a₁₁, a₂₂) ∈ Γ(δ, q₁, q₂), it follows that there exists ω ∈ ℝ such that a11=δq1q1+q2 h(ω,q1,q2) and a22=δq2q1+q2 h(−ω,q1,q2), where the function h is given in Lemma 3.1.

Let us first show that (a₁₁, a₂₂) ∉ R_u(δ). On one hand, one can write:

where t=δ1q1+q2eω>0 and u(t)=ρ2tq1−ρ1tq2, where the arguments of the functions ρ₁, ρ₂ have been dropped for simplicity. The function u(t) reaches its maximal value at the point tmax=(q1ρ2q2ρ1)1q2−q1 and a straightforward calculation leads to:

We will next show that u_max < 1. Indeed, as the function v(x)=xln(xsinx) is positive and convex on [0, π] with lim_x→0v(x) = 0, it follows that v(x) is superadditive, and hence:

Considering x=q1π2 and y=q2π2 in the previous inequality, we obtain ln(u_max) < 0, and hence, u_max < 1 for any 0 < q₁ < q₂ ≤ 1. Therefore:

On the other hand,

and hence, combined with inequality (3.9) it follows that (a₁₁, a₂₂) ∉ R_u(δ).

Moreover, assuming by contradiction that (a₁₁, a₂₂) ∈ R_s(δ), Lemma 3.4 implies that all roots of the characteristic function Δ(s; a₁₁, a₂₂, δ, q₁, q₂) are in the open left half-plane, and hence, by Theorem 3.3 we obtain that (a₁₁, a₂₂) ∉ Γ(δ, q₁, q₂), which is absurd. Therefore, (a₁₁, a₂₂) ∉ R_s(δ).

Hence, the proof of the inclusion Q(δ) ⊆ ℝ²\(R_s(δ) ∪ R_u(δ)) is now complete.

Step 2. Proof of the inclusion ℝ²\(R_s(δ) ∪ R_u(δ)) ⊆ Q(δ).

Considering the function F_δ : ℝ × (0, 1] × (0, 1] → ℝ² defined by

it is easy to see that Q(δ) represents the image of the function F_δ, i.e.: Q(δ) = F_δ(ℝ × (0, 1] × (0, 1]).

From Remark 3.4, it easily follows that

Moreover, as h(−ω, q₁, q₂) = h(ω, q₂, q₁) for any q₁ ≠ q₂, it follows that Q(δ) is symmetric with respect to the first bisector a₁₁ = a₂₂ of the (a₁₁, a₂₂)-plane. Therefore, in order to determine Q(δ) it suffices to find its intersection with an arbitrary straight line l_m : a₁₁ − a₂₂ = m, m ∈ ℝ, which is parallel to the first bisector of the (a₁₁, a₂₂)-plane. First, Lemma 3.1 implies that each curve Γ(δ, q₁, q₂) is the graph of a smooth, decreasing, concave, bijective function in the (a₁₁, a₂₂)-plane, and hence, it intersects the line l_m exactly in one point. In other words, for arbitrarily fixed q₁, q₂ ∈ (0, 1] and m ∈ ℝ, the equation