A theory of algebraic n-ports

Chua, Leon O.; Lam, Ying-Fai

doi:10.1109/tct.1973.1083715

Cited by 42 publications

(13 citation statements)

References 26 publications

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“…The no-complexity theorem is valid also for the case m < n, where some diffusion coefficients are set to zero. The proof of this generalization requires some in-depth background from nonlinear circuit theory [Chua & Green, 1976a, 1976bChua & Lam, 1972, 1973 beyond the scope of this paper. 2.…”

Section: Remarksmentioning

confidence: 99%

Local Activity Is the Origin of Complexity

Chua

2005

Int. J. Bifurcation Chaos

250

179

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Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction-diffusion system possess to make the Turing interacting system oscillate.The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability.Among other things, a mathematical proof is given which shows none of the complexityrelated jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge.

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Section: Remarksmentioning

confidence: 99%

Local Activity Is the Origin of Complexity

Chua

2005

Int. J. Bifurcation Chaos

250

179

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“…In this section, the capacitor and inductor are allowed to be nonlinear, as shown in Figure 6. The functions f (·) and h(·) are assumed to satisfy the following standard assumptions [8,9,10]:…”

Section: Dual Circuitmentioning

confidence: 99%

Thermal Noise Behavior of the Bridge Circuit

Coram

Anderson

Wyatt

2000

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This paper considers a connection between the deterministic and noisy behavior of nonlinear networks. Specifically, a particular bridge circuit is examined which has two possibly nonlinear energy storage elements. By proper choice of the constitutive relations for the network elements, the deterministic terminal behavior reduces to that of a single linear resistor. This reduction of the deterministic terminal behavior, in which a natural frequency of a linear circuit does not appear in the driving-point impedance, has been shown in classical circuit theory books (e.g. [1,2]). The paper shows that, in addition to the reduction of the deterministic behavior, the thermal noise at the terminals of the network, arising from the usual Nyquist-Johnson noise model associated with each resistor in the network, is also exactly that of a single linear resistor. While this result for the linear time-invariant (LTI) case is a direct consequence of a well-known result for RLC circuits, the nonlinear result is novel. We show that the terminal noise current is precisely that predicted by the Nyquist-Johnson model for R if the driving voltage is zero or constant, but not if the driving voltage is time-dependent or the inductor and capacitor are time-varying.

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“…A regular algebraic n -port N is said to admit a C'hybrid representation (I s k ) if it can be represented by Y = h(x) ( 5 ) where h( -) is a C'-function on R", and where A and B are diagonal n x n matrices satisfying the property that either Aii = 1, Bi, = 0 or Ajj = 0, For a hybrid representation, we will refer to the vector x as the controlling port vector and to the vector y as the controlled port vector. It should be clear that transmission representations are not hybrid because x, and y j are dynamically independent variables associated with the jth port.…”

Section: Definitionmentioning

confidence: 99%