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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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. Google Scholar

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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. Google Scholar

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