Canards Existence in Memristor’s Circuits

Abstract

The aim of this work is to propose an alternative method for determining the condition of existence of “canard solutions” for three and four-dimensional singularly perturbed systems with only one fast variable in the folded saddle case. This method enables to state a unique generic condition for the existence of “canard solutions” for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is perfectly identical to that provided in previous works. Application of this method to the famous three and four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of “canard solutions” in such Memristor Based Chaotic Circuits.

Appendices

Appendices

Change of coordinates leading to the normal forms of three and four-dimensional singularly perturbed systems with one fast variable are given in the following section.

1.1 Appendix 1: Normal form of 3D Singularly Perturbed Systems with One Fast Variable

Let’s consider the three-dimensional singularly perturbed dynamical system (11) with $k=2$ slow variables and $m=1$ fast and let’s make the following change of variables:

$$\begin{aligned} x_1 = \alpha ^2 x, \quad x_2 = \alpha y, \quad y_1 = \alpha z \quad \hbox {where} \quad \alpha \ll 1. \end{aligned}$$

(86)

By taking into account Benoît’s generic hypothesis Eqs. (20), (21) and while using Taylor series expansion the system (11) becomes:

$$\begin{aligned} \dot{x}&= \dfrac{\partial f_1}{\partial y} y + \dfrac{\partial f_1}{\partial z} z, \nonumber \\ \dot{y}&= f_2 ( x, y, z ), \nonumber \\ \dfrac{\varepsilon }{\alpha ^2} \dot{z}&= \dfrac{\partial g_1}{\partial x} x + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial y^2} y^2 + \dfrac{\partial ^2 g_1}{\partial y \partial z} y z + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial z^2} z^2. \end{aligned}$$

(87)

Then, let’s make the standard polynomial change of variables:

$$\begin{aligned} X&= A x + B y^2, \nonumber \\ Y&= \dfrac{y}{f_2}, \nonumber \\ Z&= Cy + Dz. \end{aligned}$$

(88)

From (88) we deduce that:

$$\begin{aligned} x&= \frac{X - B f_2^2 Y^2}{A}, \nonumber \\ y&= f_2 Y,\nonumber \\ z&= \frac{Z - C f_2 Y}{D}. \end{aligned}$$

(89)

The time derivative of system (88) gives:

$$\begin{aligned} \dot{X}&= A \dot{x} + 2 B v \dot{y}, \nonumber \\ \dot{Y}&= \dfrac{\dot{y}}{f_2}, \nonumber \\ \dot{Z}&= C\dot{y} + D\dot{z}. \end{aligned}$$

(90)

Then, multiplying the third equation of (90) by $(\varepsilon / \alpha ^2)$ and while replacing in (90) $\dot{x}$, $\dot{y}$ and $\dot{z}$ by the right-hand-side of system (87) leads to:

$$\begin{aligned} \dot{X}&= A \left( \dfrac{\partial f_1}{\partial y} y + \dfrac{\partial f_1}{\partial z} z \right) + 2 B y f_2,\nonumber \\ \dot{Y}&= 1,\nonumber \\ \dfrac{\varepsilon }{\alpha ^2} \dot{Z}&= \dfrac{\varepsilon }{\alpha ^2} C f_2 + D\left( \dfrac{\partial g_1}{\partial x} x + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial y^2} y^2 + \dfrac{\partial ^2 g_1}{\partial y \partial z} y z + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial z^2} z^2 \right) , \end{aligned}$$

(91)

Since $\varepsilon / \alpha ^2 \ll 1$, the first term of the right-hand-side of the third equation of (91) can be neglected. Then, replacing in (91) x, y and z by the right-hand-side of (89) and identifying with the following system in which we have posed: $( \varepsilon / \alpha ^2) = \epsilon$:

$$\begin{aligned} \dot{X}&= a Y + b Z + O \left( X, \varepsilon , Y^2, Y Z, Z^2 \right) ,\nonumber \\ \dot{Y}&= 1 + O ( X, Y, Z, \varepsilon ),\nonumber \\ \epsilon \dot{Z}&= -( X + Z^2 ) + O ( \varepsilon X, \varepsilon Y, \varepsilon Z, \varepsilon ^2, X^2 Z, Z^3, X Y Z ), \end{aligned}$$

(92)

we find:

$$\begin{aligned} a&= A \left( \dfrac{\partial f_1}{\partial x_2} - \dfrac{C}{D} \dfrac{\partial f_1}{\partial y_1} \right) f_2 + 2 Bf_2^2 ,\nonumber \\ b&= \dfrac{A}{D} \dfrac{\partial f_1}{\partial y_1}, \end{aligned}$$

(93)

where

$$\begin{aligned} A&= \frac{1}{2} \dfrac{\partial g_1}{\partial x} \dfrac{\partial ^2 g_1}{\partial z^2},\nonumber \\ B&= \dfrac{1}{4} \left[ \dfrac{\partial ^2 g_1}{\partial y^2}\dfrac{\partial ^2 g_1}{\partial z^2} - \left( \dfrac{\partial ^2 g_1}{\partial y \partial z}\right) ^2 \right] ,\nonumber \\ C&= - \frac{1}{2} \dfrac{\partial ^2 g_1}{\partial y \partial z}, \nonumber \\ D&= - \frac{1}{2} \dfrac{\partial ^2 g_1}{\partial z^2}. \end{aligned}$$

(94)

Finally, we deduce:

$$\begin{aligned} a =&\frac{1}{2} f_2^2 \left( \dfrac{\partial ^2 g_1}{\partial x_2^2 } \dfrac{\partial ^2 g_1}{\partial y_1^2} - \left( \dfrac{\partial ^2 g_1}{\partial x_2 \partial y_1}\right) ^2 \right) + \frac{1}{2} f_2 \dfrac{\partial g_1}{\partial x_1} \left( \dfrac{\partial ^2 g_1}{\partial y_1^2 }\dfrac{\partial f_1}{\partial x_2} - \dfrac{\partial ^2 g_1}{\partial x_2 \partial y_1 } \dfrac{\partial f_1}{\partial y_1} \right) ,\nonumber \\ b =&- \dfrac{\partial g_1}{\partial x_1}\dfrac{\partial f_1}{\partial y_1}, \end{aligned}$$

(95)

This is the result established by Benoît [9] and presented in Sect. 4.7.

1.2 Appendix 2: Normal Form of 4D Singularly Perturbed Systems with One Fast Variable

Let’s consider the four-dimensional singularly perturbed dynamical system (30) with $k=3$ slow variables and $m=1$ fast and let’s make the following change of variables:

$$\begin{aligned} x_1 = \alpha ^2 x ,\quad x_2 = \alpha y , \quad x_3 = \alpha z ,\quad y_1 = \alpha u\quad \hbox {where}\quad \alpha \ll 1. \end{aligned}$$

(96)

By taking into account extension of Benoît’s generic hypothesis Eqs. (40), (41) and while using Taylor series expansion the system (30) becomes:

$$\begin{aligned} \dot{x}&= \dfrac{\partial f_1}{\partial y} y + \dfrac{\partial f_1}{\partial z} z + \dfrac{\partial f_1}{\partial u} u, \nonumber \\ \dot{y}&= f_2 ( x, y, z, u ),\nonumber \\ \dot{z}&= f_3 ( x, y, z, u ),\nonumber \\ \dfrac{\varepsilon }{\alpha ^2} \dot{u}&= \dfrac{\partial g_1}{\partial x} x + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial y^2} y^2 + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial z^2} z^2 + \dfrac{1}{2} \dfrac{\partial ^2 g_1}{\partial u^2} u^2\nonumber \\&\quad + \dfrac{\partial ^2 g_1}{\partial y \partial z} y z + \dfrac{\partial ^2 g_1}{\partial y \partial u} y u + \dfrac{\partial ^2 g_1}{\partial z \partial u} z u . \end{aligned}$$

(97)

Then, let’s make the standard polynomial change of variables:

$$\begin{aligned} X&= A x + B y^2 + C z^2,\nonumber \\ Y&= \dfrac{y}{f_2},\nonumber \\ Z&= \dfrac{z}{f_3} + D y,\nonumber \\ U&= E y + F z + G u. \end{aligned}$$

(98)

From (98) we deduce that:

$$\begin{aligned} x&= \frac{X - B f_2^2 Y^2 - C f_3^2 ( Z - D f_2 Y )^2}{A},\nonumber \\ y&= f_2 y,\nonumber \\ z&= f_3 ( Z - D f_2 Y ),\nonumber \\ u&= \frac{U - E f_2 Y - F f_3 ( Z - D f_2 Y ) }{G}. \end{aligned}$$

(99)

The time derivative of system (98) gives:

$$\begin{aligned} \dot{X}&= A \dot{x} + 2 B y \dot{y} + 2 C z \dot{z},\nonumber \\ \dot{Y}&= \dfrac{\dot{y}}{f_2},\nonumber \\ \dot{Z}&= \dfrac{\dot{z}}{f_3} + D \dot{y},\nonumber \\ \dot{U}&= E\dot{y} + F\dot{z} + G \dot{u}. \end{aligned}$$

(100)

Then, multiplying the fourth equation of (100) by $( \varepsilon / \alpha ^2)$ and while replacing in (100) $\dot{x}$, $\dot{y}$, $\dot{z}$ and $\dot{u}$ by the right-hand-side of system (97) leads to:

$$\begin{aligned} \dot{X}&= A \left( \dfrac{\partial f_1}{\partial y} y + \dfrac{\partial f_1}{\partial z} z + \dfrac{\partial f_1}{\partial u} u\right) + 2 B y f_2 + 2 C z f_3,\nonumber \\ \dot{Y}&= 1,\nonumber \\ \dot{Z}&= 1 + D f_2,\nonumber \\ \dfrac{\varepsilon }{\alpha ^2} \dot{U}&= \dfrac{\varepsilon }{\alpha ^2} E f_2 + \dfrac{\varepsilon }{\alpha ^2} F f_3 + G\left( \dfrac{\partial g_1}{\partial x} x + \ldots + \dfrac{\partial ^2 g_1}{\partial z \partial u} z u \right) , \end{aligned}$$

(101)

Since $\varepsilon / \alpha ^2 \ll 1$, the two first terms of the right-hand-side of the fourth equation of (101) can be neglected. Then, by replacing in (101) x, y, z and u by the right-hand-side of (99) and by identifying with the following system in which we have posed: $( \varepsilon / \alpha ^2) = \epsilon$:

$$\begin{aligned} \dot{X}&= {\tilde{a}} Y + {\tilde{b}} U + O ( X, \epsilon , Y^2, Y U, U^2 ),\nonumber \\ \dot{Y}&= 1 + O \left( X, Y, U, \epsilon \right) ,\nonumber \\ \dot{Z}&= 1 + O \left( X, Y, U, \epsilon \right) ,\nonumber \\ \epsilon \dot{Z}&= -\left( X + U^2 \right) + O \left( \epsilon X, \epsilon Y, \epsilon U, \epsilon ^2, X^2 U, U^3, X Y U \right) , \end{aligned}$$

(102)

we find:

$$\begin{aligned} {\tilde{a}}&= A \left( \dfrac{\partial f_1}{\partial x_2} - \dfrac{E}{G} \dfrac{\partial f_1}{\partial y_1} \right) f_2 + A \left( \dfrac{\partial f_1}{\partial x_3} - \dfrac{F}{G} \dfrac{\partial f_1}{\partial y_1} \right) + 2 B f_2^2 + 2 C f_3^2, \nonumber \\ {\tilde{b}}&= \dfrac{A}{G} \dfrac{\partial f_1}{\partial y_1}, \end{aligned}$$

(103)

where

$$\begin{aligned} A&= \frac{1}{2} \dfrac{\partial g_1}{\partial x} \dfrac{\partial ^2 g_1}{\partial u^2},\nonumber \\ B&= \dfrac{f_3}{2f_2} \left[ \dfrac{\partial ^2 g_1}{\partial u^2}\dfrac{\partial ^2 g_1}{\partial y \partial z} + \dfrac{\partial ^2 g_1}{\partial y \partial u}\dfrac{\partial ^2 g_1}{\partial z \partial u} \right] + \dfrac{1}{4} \left[ \dfrac{\partial ^2 g_1}{\partial u^2}\dfrac{\partial ^2 g_1}{\partial y^2} - \left( \dfrac{\partial ^2 g_1}{\partial y \partial u} \right) ^2 \right] ,\nonumber \\ C&= \dfrac{1}{4} \left[ \dfrac{\partial ^2 g_1}{\partial z^2}\dfrac{\partial ^2 g_1}{\partial u^2} - \left( \dfrac{\partial ^2 g_1}{\partial z \partial u}\right) ^2 \right] ,\nonumber \\ D&= - \frac{1}{f_2},\nonumber \\ E&= - \frac{1}{2} \dfrac{\partial ^2 g_1}{\partial y \partial u},\nonumber \\ F&= - \frac{1}{2} \dfrac{\partial ^2 g_1}{\partial z \partial u},\nonumber \\ G&= - \frac{1}{2} \dfrac{\partial ^2 g_1}{\partial u^2}. \end{aligned}$$

(104)

Finally, we deduce:

$$\begin{aligned} {\tilde{a}}&= \frac{1}{2} f_2^2 \left( \dfrac{\partial ^2 g_1}{\partial x_2^2 } \dfrac{\partial ^2 g_1}{\partial y_1^2} - \left( \dfrac{\partial ^2 g_1}{\partial x_2 \partial y_1}\right) ^2 \right) + \frac{1}{2} f_2 \dfrac{\partial g_1}{\partial x_1} \left( \dfrac{\partial ^2 g_1}{\partial y_1^2 }\dfrac{\partial f_1}{\partial x_2} - \dfrac{\partial ^2 g_1}{\partial x_2 \partial y_1 } \dfrac{\partial f_1}{\partial y_1} \right) \nonumber \\&\quad + \frac{1}{2} f_3^2 \left( \dfrac{\partial ^2 g_1}{\partial x_3^2 } \dfrac{\partial ^2 g_1}{\partial y_1^2} - \left( \dfrac{\partial ^2 g_1}{\partial x_3 \partial y_1}\right) ^2 \right) + \frac{1}{2} f_3 \dfrac{\partial g_1}{\partial x_1} \left( \dfrac{\partial ^2 g_1}{\partial y_1^2 }\dfrac{\partial f_1}{\partial x_3} - \dfrac{\partial ^2 g_1}{\partial x_3 \partial y_1 } \dfrac{\partial f_1}{\partial y_1} \right) \nonumber \\&\quad + f_2 f_3 \left( \dfrac{\partial ^2 g_1}{\partial x_2 \partial x_3 } \dfrac{\partial ^2 g_1}{\partial y_1^2} - \dfrac{\partial ^2 g_1}{\partial x_2 \partial y_1}\dfrac{\partial ^2 g_1}{\partial x_3 \partial y_1} \right) ,\nonumber \\ {\tilde{b}}&= - \dfrac{\partial g_1}{\partial x_1}\dfrac{\partial f_1}{\partial y_1}, \end{aligned}$$

(105)

This is the result we established in Sect. 5.7. Moreover, let’s notice that by posing $f_3=0$ in ${\tilde{a}}$ we find again a given in Sect. 4.7.

Routh–Hurwitz’ theorem and their application to the determination of the Hopf bifurcation parameter-value in the case of three and four-dimensional singularly perturbed system are presented in this appendix.

1.3 Appendix 3: Routh–Hurwitz’s Theorem for 3D Systems

According to (23) the Cayley–Hamilton eigenpolynomial associated with the Jacobian of a three-dimensional singularly perturbed system (11) reads:

$$\begin{aligned} \lambda ^3 - \sigma _1 \lambda ^2 + \sigma _2 \lambda - \sigma _3 = 0 \end{aligned}$$

(106)

where

$$\begin{aligned} \sigma _1&= \lambda _1 + \lambda _2 + \lambda _3,\nonumber \\ \sigma _2&= \lambda _1\lambda _2 + \lambda _2\lambda _3 + \lambda _1\lambda _3,\nonumber \\ \sigma _3&= \lambda _1\lambda _2\lambda _3. \end{aligned}$$

(107)

Let’s rewrite the eigenpolynomial (106) as: $a_3 \lambda ^3 + a_2 \lambda ^2 + a_1 \lambda + a_0 = 0$ ($a_0 > 0$). Routh–Hurwitz’ theorem [26, 40] states that the real parts of the eigenvalues of this eigenpolynomial are negative if and only if all the following determinants:

$$\begin{aligned} D_1 = a_1; \quad D_2 = \begin{vmatrix} a_1&\quad a_0 \\ a_3&\quad a_2 \end{vmatrix} = a_1 a_2 - a_0 a_3 \end{aligned}$$

(108)

are positive.

Now, let suppose that the eigenpolynomial (106) has one real eigenvalue $\lambda _1 \ne 0$ and two complex conjugated $\lambda _{2,3} = a + \imath b$ (with $a\ne 0$ an $b \ne 0$). So, we have:

$$\begin{aligned} \sigma _1&= \lambda _1 + 2a,\nonumber \\ \sigma _2&= 2a\lambda _1 + a^2 + b^2,\nonumber \\ \sigma _3&= \lambda _1( a^2 + b^2 ). \end{aligned}$$

(109)

The determinant $D_2$ reads:

$$\begin{aligned} D_2 = -2 a (a^2 + b^2 + 2a \lambda _1 + \lambda _1^2 ) \end{aligned}$$

(110)

Moreover, if we consider that the real part of the complex conjugated eigenvalues $\lambda _{2,3}$ depends on a parameter, say $\mu$, we have $a = a( \mu )$. Then, determinant $D_2$ vanishes at the location of the points where the real part $a = a( \mu )$. So, it can be used to determine the Hopf-parameter value.

1.4 Appendix 4: Routh–Hurwitz’s Theorem for 4D Systems

According to (43) the Cayley–Hamilton eigenpolynomial associated with the Jacobian of a four-dimensional singularly perturbed system (30) reads:

$$\begin{aligned} \lambda ^4 - \sigma _1 \lambda ^3 + \sigma _2 \lambda ^2 - \sigma _3 \lambda + \sigma _4 = 0 \end{aligned}$$

(111)

where

$$\begin{aligned} \sigma _1&= \lambda _1 + \lambda _2 + \lambda _3 + \lambda _4,\nonumber \\ \sigma _2&= \lambda _1 \lambda _2 + \lambda _1 \lambda _3 + \lambda _2 \lambda _3 + \lambda _1 \lambda _4 + \lambda _2 \lambda _4 + \lambda _3 \lambda _4,\nonumber \\ \sigma _3&= \lambda _1 \lambda _2 \lambda _3 + \lambda _1 \lambda _2 \lambda _4 + \lambda _1 \lambda _3 \lambda _4 + \lambda _2 \lambda _3 \lambda _4,\nonumber \\ \sigma _4&= \lambda _1\lambda _2\lambda _3\lambda _4. \end{aligned}$$

(112)

Let’s rewrite the eigenpolynomial (111) as: $a_4 \lambda ^4 + a_3 \lambda ^3 + a_2 \lambda ^2 + a_1 \lambda + a_0 = 0$ ($a_0 > 0$). Routh–Hurwitz’ theorem [26, 40] states that the real parts of the eigenvalues of this eigenpolynomial are negative if and only if all the following determinants:

$$\begin{aligned} D_1 = a_1 ;\quad D_2 = \begin{vmatrix} a_1&\quad a_0 \\ a_3&\quad a_2 \end{vmatrix} = a_1 a_2 - a_0 a_3 ; \quad D_3 = \begin{vmatrix} a_1&\quad a_0&\quad 0 \\ a_3&\quad a_2&\quad a_1 \\ 0&\quad a_4&\quad a_3 \end{vmatrix} \end{aligned}$$

(113)

are positive.

Now, let suppose that the eigenpolynomial (111) has two real eigenvalues $\lambda _1$, $\lambda _2$ with $\lambda _1 \ne - \lambda _2 \ne 0$ and two complex conjugated $\lambda _{3,4} = a + \imath b$ (with $a\ne 0$ an $b \ne 0$). So, we have:

$$\begin{aligned} \sigma _1&= 2 a + \lambda _1 + \lambda _2,\nonumber \\ \sigma _2&= a^2 + b^2 + 2a \left( \lambda _1 + \lambda _2 \right) + \lambda _1 \lambda _2,\nonumber \\ \sigma _3&= 2 a \lambda _1 \lambda _2 + \left( a^2 + b^2\right) \left( \lambda _1 + \lambda _2 \right) ,\nonumber \\ \sigma _4&= \left( a^2 + b^2 \right) \lambda _1 \lambda _2. \end{aligned}$$

(114)

The determinant $D_3$ reads:

$$\begin{aligned} D_3 = 2 a \left( a^2 + b^2 + 2a \lambda _1 + \lambda _1^2 \right) \left( \lambda _1 + \lambda _2 \right) \left( a^2 + b^2 + 2a\lambda _2 + \lambda _2^2 \right) \end{aligned}$$

(115)

Moreover, if we consider that the real part of the complex conjugated eigenvalues $\lambda _{2,3}$ depends on a parameter, say $\mu$, we have $a = a( \mu )$. Then, determinant $D_3$ vanishes at the location of the points where the real part $a = a( \mu )$. So, it can be used to determine the Hopf-parameter value.