References
315
Since manifolds are parametrized by the n M -dimensional vector k, we conclude
that there are ∞ n M nonintersecting manifolds. To see that they fill the whole state
space, it is enough to note that, given any point w, we have w ∈ M(K(w)).
Appendix 3: Proof of Property 7.2
Due to (7.29), considering that ˙
x M (t) = ˙
x(t), we have ˙
k(t; t 0 , w 0 ) = S 22 ˙
x(t) +
J F (x M (t))˙ x(t) + M x ¨
x(t) − H 12 H
−1
22 M y ¨
y(t). Using such equation, the expression
in (7.33) is easily obtained by the following relationships derived from (7.23):
J F (x M (t))˙ x(t) + M x ¨
x(t) = −H 11 ˙
x(t) − H 12 ˙
y(t) − ˙
u x (t)
M y ¨
y(t) = −H 21 ˙
x(t) − H 22 ˙
y(t) − ˙
u y (t).
References
1. M. Itoh, L.O. Chua, Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)
2. R. Riaza, C. Tischendorf, Semistate models of electrical circuits including memristors. Int. J.
Circuit Theory Appl. 39(6), 607–627 (2011)
3. A. Ascoli, F. Corinto, R. Tetzlaff, Generalized boundary condition memristor model. Int. J.
Circuit Theory Appl. 44(1), 60–84 (2016)
4. J. Ma, F. Wu, G. Ren, J. Tang, A class of initials-dependent dynamical systems. Appl. Math.
Comput. 298, 65–76 (2017)
5. B. Bao, T. Jiang, Q. Xu, M. Chen, H. Wu, Y. Hu, Coexisting infinitely many attractors in active
band-pass filter-based memristive circuit. Nonlinear Dyn. 86(3), 1711–1723 (2016)
6. A. Buscarino, C. Corradino, L. Fortuna, M. Frasca, L.O. Chua, Turing patterns in memristive
cellular nonlinear networks. IEEE Trans. Circuits Syst. I Regul. Pap. 63(8), 1222–1230 (2016)
7. V.-T. Pham, S. Vaidyanathan, C.K. Volos, S. Jafari, N.V. Kuznetsov, T.M. Hoang, A novel
memristive time-delay chaotic system without equilibrium points. Eur. Phys. J. Spec. Top.
225(1), 127–136 (2016)
8. R. Riaza, Manifolds of equilibria and bifurcations without parameters in memristive circuits.
SIAM J. Appl. Math. 72(3), 877–896 (2012)
9. B. Bao, Z. Ma, J. Xu, Z. Liu, Q. Xu, A simple memristor chaotic circuit with complex
dynamics. Int. J. Bifurc. Chaos 21(9), 2629–2645 (2011)
10. Q. Li, S. Hu, S. Tang, G. Zeng, Hyperchaos and horseshoe in a 4D memristive system with
a line of equilibria and its implementation. Int. J. Circuit Theory Appl. 42(11), 1172–1188
(2014)
11. F. Corinto, A. Ascoli, M. Gilli, Analysis of current–voltage characteristics for memristive
elements in pattern recognition systems. Int. J. Circuit Theory Appl. 40(12), 1277–1320 (2012)
12. L.O. Chua, Dynamic nonlinear networks: state-of-the-art. IEEE Trans. Circuits Syst. 27(11),
1059–1087 (1980)
13. H.C. So, On the hybrid description of a linear n–port resulting from the extraction of arbitrarily
specified elements. IEEE Trans. Circuit Theory CT–12(3), 381–387 (1965)
14. F. Zhang, The Schur Complement and Its Applications, vol. 4 (Springer, Berlin, 2006)
15. F. Corinto, P.P. Civalleri, L.O. Chua, A theoretical approach to memristor devices. IEEE J.
Emerg. Sel. Topics Circuits Syst. 5(2), 123–132 (2015)
315
Since manifolds are parametrized by the n M -dimensional vector k, we conclude
that there are ∞ n M nonintersecting manifolds. To see that they fill the whole state
space, it is enough to note that, given any point w, we have w ∈ M(K(w)).
Appendix 3: Proof of Property 7.2
Due to (7.29), considering that ˙
x M (t) = ˙
x(t), we have ˙
k(t; t 0 , w 0 ) = S 22 ˙
x(t) +
J F (x M (t))˙ x(t) + M x ¨
x(t) − H 12 H
−1
22 M y ¨
y(t). Using such equation, the expression
in (7.33) is easily obtained by the following relationships derived from (7.23):
J F (x M (t))˙ x(t) + M x ¨
x(t) = −H 11 ˙
x(t) − H 12 ˙
y(t) − ˙
u x (t)
M y ¨
y(t) = −H 21 ˙
x(t) − H 22 ˙
y(t) − ˙
u y (t).
References
1. M. Itoh, L.O. Chua, Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)
2. R. Riaza, C. Tischendorf, Semistate models of electrical circuits including memristors. Int. J.
Circuit Theory Appl. 39(6), 607–627 (2011)
3. A. Ascoli, F. Corinto, R. Tetzlaff, Generalized boundary condition memristor model. Int. J.
Circuit Theory Appl. 44(1), 60–84 (2016)
4. J. Ma, F. Wu, G. Ren, J. Tang, A class of initials-dependent dynamical systems. Appl. Math.
Comput. 298, 65–76 (2017)
5. B. Bao, T. Jiang, Q. Xu, M. Chen, H. Wu, Y. Hu, Coexisting infinitely many attractors in active
band-pass filter-based memristive circuit. Nonlinear Dyn. 86(3), 1711–1723 (2016)
6. A. Buscarino, C. Corradino, L. Fortuna, M. Frasca, L.O. Chua, Turing patterns in memristive
cellular nonlinear networks. IEEE Trans. Circuits Syst. I Regul. Pap. 63(8), 1222–1230 (2016)
7. V.-T. Pham, S. Vaidyanathan, C.K. Volos, S. Jafari, N.V. Kuznetsov, T.M. Hoang, A novel
memristive time-delay chaotic system without equilibrium points. Eur. Phys. J. Spec. Top.
225(1), 127–136 (2016)
8. R. Riaza, Manifolds of equilibria and bifurcations without parameters in memristive circuits.
SIAM J. Appl. Math. 72(3), 877–896 (2012)
9. B. Bao, Z. Ma, J. Xu, Z. Liu, Q. Xu, A simple memristor chaotic circuit with complex
dynamics. Int. J. Bifurc. Chaos 21(9), 2629–2645 (2011)
10. Q. Li, S. Hu, S. Tang, G. Zeng, Hyperchaos and horseshoe in a 4D memristive system with
a line of equilibria and its implementation. Int. J. Circuit Theory Appl. 42(11), 1172–1188
(2014)
11. F. Corinto, A. Ascoli, M. Gilli, Analysis of current–voltage characteristics for memristive
elements in pattern recognition systems. Int. J. Circuit Theory Appl. 40(12), 1277–1320 (2012)
12. L.O. Chua, Dynamic nonlinear networks: state-of-the-art. IEEE Trans. Circuits Syst. 27(11),
1059–1087 (1980)
13. H.C. So, On the hybrid description of a linear n–port resulting from the extraction of arbitrarily
specified elements. IEEE Trans. Circuit Theory CT–12(3), 381–387 (1965)
14. F. Zhang, The Schur Complement and Its Applications, vol. 4 (Springer, Berlin, 2006)
15. F. Corinto, P.P. Civalleri, L.O. Chua, A theoretical approach to memristor devices. IEEE J.
Emerg. Sel. Topics Circuits Syst. 5(2), 123–132 (2015)