Nonexistence of invariant manifolds in fractional-order dynamical systems

Abstract

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional-order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional-order systems.

Appendix

1.1 Literature review

Consider a planar polynomial vector field (13).

The second part of Hilbert’s sixteenth problem [68] is related to the number of limit cycles in the polynomial system (13). The literature review of the planar quadratic system is taken by Coppel [69]. In [70], authors studied the classification of phase portraits of a quadratic system in a region surrounded by separatrix cycle.

In [71], Ye proposed the following conjecture:

Conjecture 1

When n is odd, the system (13) has at most $M_n=2n+2$ invariant lines; when n is even, the system (13) has at most $M_n=2n+1$ invariant straight lines.

For $n=2, 3$ and 4, this conjecture is proved by Sokulski [72]. However, the conjecture is false [73] if $n>4$. It should be noted that the system (13) can have infinitely many invariant straight lines (see Example (iv)).

Artes [73] proposed the following important result:

Theorem 6.1

Assume that the polynomial differential system (13) of degree n has finitely many invariant straight lines. Then, the following statements hold for the system (13).

• 1. Either all the points on an invariant line are equilibrium or the line contains no more than n equilibrium points.

• 2. No more than n invariant straight lines can be parallel.

• 3. The set of all invariant straight lines through a single point cannot have more than $n+1$ different slopes.

• 4. Either it has infinitely many finite equilibrium points, or it has at most $n^2$ finite equilibrium points.

Theorem 6.2

There exists invariant parabola ${\varvec{x}} ={\varvec{my}}^\mathbf{2 }$ for the system (21) if and only if

• 1. $a_2=0,\,a_3=2b_5,\,b_3=0$   and

• 2. One of the conditions $(a),\, (b)$,  (c) and (d) hold.

  • (a) $a_4=0$,  $a_1=2b_2$,  $a_5\ne 2b_4$  and  $b_1\ne 0$.  (In this case $m=\frac{a_5-2b_4}{2b_1}$.)

  • (b) $a_5=2b_4$,  $b_1=0$,  $a_4\ne 0$  and  $a_1\ne 2b_2$.  (In this case $m=\frac{-a_4}{a_1-2b_2}$.)

  • (c) $a_4\ne 0$,  $a_1\ne 2b_2$,  $a_5\ne 2b_4$,  $b_1\ne 0$  and 

    $a_1a_5-2a_1b_4-2b_2a_5+4b_2b_4+2a_4b_1=0$.  (In this case $m=\frac{a_5-2b_4}{2b_1}=\frac{-a_4}{a_1-2b_2}$.)

  • (d) $a_4=0$,  $a_1=2b_2$,  $a_5=2b_4$  and $b_1=0$. (In this case, $x=my^2$,  $\forall \, m\in {\mathbb {R}}$.)

Example 6.1

Consider

$$\begin{aligned} \begin{aligned} {\dot{x}}&=2x-2x^2+y^2+6xy\\ {\dot{y}}&=-y+3y^2-xy. \end{aligned} \end{aligned}$$

(56)

This system satisfies the conditions 1 and 2(b) of Theorem 6.2, and the invariant parabola is $x=-\frac{1}{4}y^2$.

The corresponding vector field is sketched in Fig. 7a.

In the following example, we can see that the general parabola is invariant under the flow.

Example 6.2

Consider the following system

$$\begin{aligned} \begin{aligned} {\dot{x}}&=\frac{1}{2}x-\frac{5}{2}y+2\sqrt{2}x^2+2\sqrt{2}y^2\\ {\dot{y}}&=-\frac{5}{2}x+\frac{1}{2}y+\frac{3}{\sqrt{2}}x^2+\frac{7}{\sqrt{2}}y^2-\sqrt{2}xy. \end{aligned} \end{aligned}$$

(57)

Here,  $5x^2-10xy+5y^2-8\sqrt{2}x-8\sqrt{2}y=0$ is invariant under the flow of this system (see Fig. 7b).

Fig. 7

Invariant parabolas $x=my^2$

1.2 Some other invariant curves

The Hamiltonian H(x, y) of a Hamiltonian system, passing through a saddle equilibrium point, works as separatrix [4].

Example 6.3

Consider a planar quadratic system

$$\begin{aligned} \begin{aligned} {\dot{x}}&=y-\sqrt{2}xy\\ {\dot{y}}&=\frac{1}{2}(2x+\sqrt{2}x^2+\sqrt{2}y^2) . \end{aligned} \end{aligned}$$

(58)

This system is a Hamiltonian system, and the Hamiltonian is given by

$$\begin{aligned} H(x,\,y)=-\frac{1}{2}x^2+\frac{1}{2}y^2-\frac{1}{3\sqrt{2}}x^3-\frac{1}{\sqrt{2}}xy^2. \end{aligned}$$

Here,  $-\frac{1}{2}x^2+\frac{1}{2}y^2-\frac{1}{3\sqrt{2}}x^3-\frac{1}{\sqrt{2}}xy^2=0$  gives separatrix for this system and it is shown in Fig. 8.

Fig. 8

Vector field of system (58)

Theorem 6.3

The cubic curve $y=x^3+mx^2+ux$  is invariant under planar quadratic system (21) if and only if

$$\begin{aligned} a_4=a_2=b_4=a_5=0,\,\, b_5=3a_3, \end{aligned}$$

$$\begin{aligned}&6a_1^3+2a_3^2b_1-11a_1^2b_2-b_2^3-a_3b_2b_3\\&\quad +\,a_1(6b_2^2+a_3b_3)=0, \end{aligned}$$

$a_3\ne 0$ and $a_1\ne b_2$.

In this case,

$$\begin{aligned} m=\frac{3a_1-b_2}{a_3}\quad \mathrm {and}\quad u=\frac{b_1}{a_1-b_2}. \end{aligned}$$

Example 6.4

The planar quadratic system

$$\begin{aligned} \begin{aligned} {\dot{x}}&=x+x^2\\ {\dot{y}}&=x+2y+2x^2+3xy, \end{aligned} \end{aligned}$$

(59)

has invariant curve, viz. $y=x^3+x^2-x$ (see Fig. 9).

Fig. 9

Vector field of system (59)

Theorem 6.4

The curves $y=mx^k$, (for any $m\in {\mathbb {R}}$ and $k>0$) are invariant under the flow of planar quadratic system (21) if and only if

$$\begin{aligned} b_1=b_3=a_4=a_2=0,\,\, b_4=ka_5, \end{aligned}$$

$$\begin{aligned} b_5=ka_3\quad \mathrm {and}\quad b_2=ka_1. \end{aligned}$$

The system (21) can have invariant curves other than polynomial curves also. The following theorem provides conditions for the existence of an exponential curve as an invariant.

Theorem 6.5

The system (21) has exponential curve $y=m e^x$ for all $m\in {\mathbb {R}}$, as an invariant curve if

$$\begin{aligned} a_3= & {} a_4=a_5=b_1=b_2=b_3=0, \\b_4= & {} a_2\quad \mathrm {and}\quad b_5=a_1. \end{aligned}$$

Example 6.5

Consider the planar quadratic system

$$\begin{aligned} \begin{aligned} {\dot{x}}&=-2x+3y\\ {\dot{y}}&=3y^2-2xy. \end{aligned} \end{aligned}$$

(60)

It can be verified that the curve $y=m e^x$ (for all $m\in {\mathbb {R}}$) is invariant under the flow of system (60).