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four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of “canard solutions” in such Memristor Based Chaotic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.

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Notes

  1. 1.

    For more details see Robinson [39] and Nelson [35].

  2. 2.

    Canards in French.

  3. 3.

    See Callot et al. [11], Benoît et al. [2], Benoît [4, 5] and Ginoux et al. [22].

  4. 4.

    Canard \(=\) false report, from the old-French “vendre un canard moitié” (sell the half of duck).

  5. 5.

    This concept has been originally introduced by José Argémi [1] See Sect. 3.7.

  6. 6.

    See Fenichel [1518], O’Malley [36], Jones [30] and Kaper [31]

  7. 7.

    Let’s notice that Eq. (73) corresponds exactly to what Itoh and Chua [28, p. 3188] have called in their previous paper on “Memristor Oscillators” the fourth-order memristor-based canonical Chua’s circuit equation (35).

  8. 8.

    In certain applications these functions will be supposed to be \(C^r\), \(r \geqslant 1\).

  9. 9.

    It represents the approximation of the slow invariant manifold, with an error of \(O(\varepsilon )\).

  10. 10.

    The set D is overflowing invariant with respect to (2) when \(\varepsilon = 0\). See Kaper [31] and Jones [30].

  11. 11.

    See Benoît [7, 10], Szmolyan and Wechselberger [42] and Wechselberger [47, 48].

  12. 12.

    In the three-dimensional case \(det(D_{\vec {y}} \vec {g}) = \partial g_1 / \partial y_1\).

  13. 13.

    This result will be proved below.

  14. 14.

    Keep in mind that \(c_2\) is generally negative so that the characteristic curve admits a negative slope.

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Acknowledgments

We would like to thank to Ernesto Pérez Chavela for previous discussions related with this work. The authors are partially supported by a MINECO/FEDER grant number MTM2008-03437. The second author is partially supported by a MICINN/FEDER grants numbers MTM2009-03437 and MTM2013-40998-P, by an AGAUR grant number 2014SGR-568, by an ICREA Academia, two FP7+PEOPLE+2012+IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378.

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Correspondence to Jean-Marc Ginoux.

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.

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.

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.

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.

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.

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Ginoux, JM., Llibre, J. Canards Existence in Memristor’s Circuits. Qual. Theory Dyn. Syst. 15, 383–431 (2016). https://doi.org/10.1007/s12346-015-0160-1

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Keywords

  • Geometric singular perturbation theory
  • Singularly perturbed dynamical systems
  • Canard solutions