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four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of “canard solutions” in such Memristor Based Chaotic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
Canards in French.
Canard \(=\) false report, from the old-French “vendre un canard moitié” (sell the half of duck).
In certain applications these functions will be supposed to be \(C^r\), \(r \geqslant 1\).
It represents the approximation of the slow invariant manifold, with an error of \(O(\varepsilon )\).
In the three-dimensional case \(det(D_{\vec {y}} \vec {g}) = \partial g_1 / \partial y_1\).
This result will be proved below.
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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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:
By taking into account Benoît’s generic hypothesis Eqs. (20), (21) and while using Taylor series expansion the system (11) becomes:
Then, let’s make the standard polynomial change of variables:
From (88) we deduce that:
The time derivative of system (88) gives:
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:
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: \(( \varepsilon / \alpha ^2) = \epsilon \):
we find:
where
Finally, we deduce:
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:
By taking into account extension of Benoît’s generic hypothesis Eqs. (40), (41) and while using Taylor series expansion the system (30) becomes:
Then, let’s make the standard polynomial change of variables:
From (98) we deduce that:
The time derivative of system (98) gives:
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:
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: \(( \varepsilon / \alpha ^2) = \epsilon \):
we find:
where
Finally, we deduce:
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:
where
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:
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:
The determinant \(D_2\) reads:
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:
where
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:
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:
The determinant \(D_3\) reads:
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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DOI: https://doi.org/10.1007/s12346-015-0160-1