American Institute of Mathematical Sciences

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2015, 2015(special): 340-348. doi: 10.3934/proc.2015.0340

Bifurcation without parameters in circuits with memristors: A DAE approach

Ignacio García de la Vega 1, and  Ricardo Riaza 1,

1. 

Depto. Matemática Aplicada a las Tecnologías de la Información, ETSI Telecomunicación, Universidad Politécnica de Madrid, Ciudad Universitaria s/n - 28040 Madrid, Spain, Spain

Received  September 2014 Revised  August 2015 Published  November 2015

Bifurcations without parameters describe qualitative changes in the local dynamics of nonlinear ODEs when normal hyperbolicity of a manifold of equilibria fails. Non-isolated equilibrium points are systematically exhibited by nonlinear circuits with memristors; a memristor is a nonlinear device recently introduced in circuit theory and which is expected to play a key role in electronics in the near future. In this communication we provide a graph-theoretic analysis of the transcritical bifurcation without parameters in memristive circuits, owing to the presence of a locally active memristor. The results are crucially based on the use of differential-algebraic circuit models.
Keywords: di erential-algebraic equations., transcritical bifurcation, Bifurcation without parameters, nonlinear circuit, memristor.
Mathematics Subject Classification: Primary: 34C23, 94C05. Secondary: 34A0.
Citation: Ignacio García de la Vega, Ricardo Riaza. Bifurcation without parameters in circuits with memristors: A DAE approach. Conference Publications, 2015, 2015 (special) : 340-348. doi: 10.3934/proc.2015.0340
References:
[1]

B. Andrásfai, Introductory Graph Theory, Akadémiai Kiadó, Budapest, 1977.

[2]

B. Andrásfai, Graph Theory: Flows, Matrices, Adam Hilger, 1991.

[3]

A. Ascoli, T. Schmidt, R. Tetzlaff and F. Corinto, Application of the Volterra series paradigm to memristive systems, in R. Tetzlaff (ed.), Memristors and Memristive Systems, pp. 163-191, Springer, 2014.

[4]

B. Bao, Z. Ma, J. Xu, Z. Liu and Q. Xu, A simple memristor chaotic circuit with complex dynamics, Internat. J. Bifurcation and Chaos, 21 (2011), 2629-2645.

[5]

D. Biolek, Z. Biolek and V. Biolkova, SPICE modeling of memristive, memcapacitive and meminductive systems, Proc. Eur. Conf. Circuit Theor. Design 2009, (2009) 249-252.

[6]

B. Bollobás, Modern Graph Theory, Springer-Verlag, 1998.

[7]

L. O. Chua, Memristor - The missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.

[8]

L. O. Chua, C. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, 1987.

[9]

F. Corinto, A. Ascoli and M. Gilli, Analysis of current-voltage characteristics for memristive elements in pattern recognition systems, Internat. J. Circuit Theory Appl., 40 (2012), 1277-1320.

[10]

M. Di Ventra, Y. V. Pershin and L. O. Chua, Circuit elements with memory: memristors, memcapacitors and meminductors, Proc. IEEE, 97 (2009), 1717-1724.

[11]

B. Fiedler, S. Liebscher, and J. C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters: I. Theory, J. Differential Equations, 167 (2000), 16-35.

[12]

B. Fiedler and S. Liebscher, Generic Hopf bifurcation from lines of equilibria without parameters: II. Systems of viscous hyperbolic balance laws, SIAM J. Math. Anal., 31 (2000), 1396-1404.

[13]

B. Fiedler, S. Liebscher, and J. C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters: III. Binary oscillations, Internat. J. Bifur. Chaos, 10 (2000), 1613-1622.

[14]

M. Itoh and L. O. Chua, Memristor oscillators, Internat. J. Bifur. Chaos, 18 (2008), 3183-3206.

[15]

M. Itoh and L. O. Chua, Memristor Hamiltonian circuits, Internat. J. Bifur. Chaos, 21 (2011), 2395-2425.

[16]

L. Jansen, M. Matthes and C. Tischendorf, Global unique solvability for memristive circuit DAEs of index 1, Int. J. Circuit Theory Appl., in press, 2014.

[17]

D. Jeltsema and A. Doria-Cerezo, Port-Hamiltonian formulation of systems with memory, Proc. IEEE, 100 (2012), 1928-1937.

[18]

O. Kavehei, A. Iqbal, Y. S. Kim, K. Eshraghian, S. F. Al-Sarawi and D. Abbott, The fourth element: characteristics, modelling and electromagnetic theory of the memristor, Proc. R. Soc. A, 466 (2010), 2175-2202.

[19]

M. Messias, C. Nespoli and V. A. Botta, Hopf bifurcation from lines of equilibria without parameters in memristors oscillators, Internat. J. Bifur. Chaos, 20 (2010), 437-450.

[20]

B. Muthuswamy and L. O. Chua, Simplest chaotic circuit, Internat. J. Bifur. Chaos, 20 (2010), 1567-1580.

[21]

Y. V. Pershin and M. Di Ventra, Neuromorphic, digital and quantum computation with memory circuit elements, Proc. IEEE, 100 (2012), 2071-2080.

[22]

Y. V. Pershin and M. Di Ventra, Memory effects in complex materials and nanoscale systems, Advances in Physics, 60 (2011), 145-227.

[23]

R. Riaza, Differential-Algebraic Systems, World Scientific, 2008.

[24]

R. Riaza, Nondegeneracy conditions for active memristive circuits, IEEE Trans. Circuits and Systems - II, 57 (2010), 223-227.

[25]

R. Riaza, Manifolds of equilibria and bifurcations without parameters in memristive circuits, SIAM J. Appl. Math., 72 (2012), 877-896.

[26]

D. B. Strukov, G. S. Snider, D. R. Stewart and R. S. Williams, The missing memristor found, Nature, 453 (2008), 80-83.

[27]

C. Tischendorf, Coupled systems of differential algebraic and partial differential equations in circuit and device simulation. Modeling and numerical analysis, Habilitationsschrift, Inst. Math., Humboldt University, Berlin, 2003.

show all references

References:
[1]

B. Andrásfai, Introductory Graph Theory, Akadémiai Kiadó, Budapest, 1977.

[2]

B. Andrásfai, Graph Theory: Flows, Matrices, Adam Hilger, 1991.

[3]

A. Ascoli, T. Schmidt, R. Tetzlaff and F. Corinto, Application of the Volterra series paradigm to memristive systems, in R. Tetzlaff (ed.), Memristors and Memristive Systems, pp. 163-191, Springer, 2014.

[4]

B. Bao, Z. Ma, J. Xu, Z. Liu and Q. Xu, A simple memristor chaotic circuit with complex dynamics, Internat. J. Bifurcation and Chaos, 21 (2011), 2629-2645.

[5]

D. Biolek, Z. Biolek and V. Biolkova, SPICE modeling of memristive, memcapacitive and meminductive systems, Proc. Eur. Conf. Circuit Theor. Design 2009, (2009) 249-252.

[6]

B. Bollobás, Modern Graph Theory, Springer-Verlag, 1998.

[7]

L. O. Chua, Memristor - The missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.

[8]

L. O. Chua, C. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, 1987.

[9]

F. Corinto, A. Ascoli and M. Gilli, Analysis of current-voltage characteristics for memristive elements in pattern recognition systems, Internat. J. Circuit Theory Appl., 40 (2012), 1277-1320.

[10]

M. Di Ventra, Y. V. Pershin and L. O. Chua, Circuit elements with memory: memristors, memcapacitors and meminductors, Proc. IEEE, 97 (2009), 1717-1724.

[11]

B. Fiedler, S. Liebscher, and J. C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters: I. Theory, J. Differential Equations, 167 (2000), 16-35.

[12]

B. Fiedler and S. Liebscher, Generic Hopf bifurcation from lines of equilibria without parameters: II. Systems of viscous hyperbolic balance laws, SIAM J. Math. Anal., 31 (2000), 1396-1404.

[13]

B. Fiedler, S. Liebscher, and J. C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters: III. Binary oscillations, Internat. J. Bifur. Chaos, 10 (2000), 1613-1622.

[14]

M. Itoh and L. O. Chua, Memristor oscillators, Internat. J. Bifur. Chaos, 18 (2008), 3183-3206.

[15]

M. Itoh and L. O. Chua, Memristor Hamiltonian circuits, Internat. J. Bifur. Chaos, 21 (2011), 2395-2425.

[16]

L. Jansen, M. Matthes and C. Tischendorf, Global unique solvability for memristive circuit DAEs of index 1, Int. J. Circuit Theory Appl., in press, 2014.

[17]

D. Jeltsema and A. Doria-Cerezo, Port-Hamiltonian formulation of systems with memory, Proc. IEEE, 100 (2012), 1928-1937.

[18]

O. Kavehei, A. Iqbal, Y. S. Kim, K. Eshraghian, S. F. Al-Sarawi and D. Abbott, The fourth element: characteristics, modelling and electromagnetic theory of the memristor, Proc. R. Soc. A, 466 (2010), 2175-2202.

[19]

M. Messias, C. Nespoli and V. A. Botta, Hopf bifurcation from lines of equilibria without parameters in memristors oscillators, Internat. J. Bifur. Chaos, 20 (2010), 437-450.

[20]

B. Muthuswamy and L. O. Chua, Simplest chaotic circuit, Internat. J. Bifur. Chaos, 20 (2010), 1567-1580.

[21]

Y. V. Pershin and M. Di Ventra, Neuromorphic, digital and quantum computation with memory circuit elements, Proc. IEEE, 100 (2012), 2071-2080.

[22]

Y. V. Pershin and M. Di Ventra, Memory effects in complex materials and nanoscale systems, Advances in Physics, 60 (2011), 145-227.

[23]

R. Riaza, Differential-Algebraic Systems, World Scientific, 2008.

[24]

R. Riaza, Nondegeneracy conditions for active memristive circuits, IEEE Trans. Circuits and Systems - II, 57 (2010), 223-227.

[25]

R. Riaza, Manifolds of equilibria and bifurcations without parameters in memristive circuits, SIAM J. Appl. Math., 72 (2012), 877-896.

[26]

D. B. Strukov, G. S. Snider, D. R. Stewart and R. S. Williams, The missing memristor found, Nature, 453 (2008), 80-83.

[27]

C. Tischendorf, Coupled systems of differential algebraic and partial differential equations in circuit and device simulation. Modeling and numerical analysis, Habilitationsschrift, Inst. Math., Humboldt University, Berlin, 2003.

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