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We study electronic properties of the energy spectrum in a one-dimensional tight-binding model with site energies arranged in the Thue-Morse sequence. Using the trace map, we obtain the branching rules of the spectrum and perform a scaling analysis of bandwidths. It is shown that the spectrum consists of absolutely continuous parts and singular continuous parts. We consider a Kronig-Penney model to study the properties of the spectrum and eigenstates in a Thue-Morse superlattice. For this purpose, we calculate the density of states and the resistance using the symmetry of the Thue-Morse lattice, and wave functions via the Poincare map. The calculations reproduce the results obtained in the tight-binding model. We also discuss transport properties. Our results for the two models clearly show that the Thue-Morse lattice is intermediate between periodic and quasiperiodic lattices.

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We study nonequilibrium phenomena in a globally coupled oscillator system with a third harmonic pinning force in the presence of an additive noise and a fluctuating interaction. The system shows a subcritical saddle-node bifurcation from an asymmetric state to a symmetric state at a critical noise intensity leading to multistable states. The fluctuating interaction increases the critical noise intensity and thus enhances the multistability drastically. We show phase diagrams and discuss the nature of the phase transition.[S0031-9007 (97)02587-8] PACS numbers: 05.45. + b, 02.50.Ey, 05.40. + j, 05.70.FhNoise effect on a dynamical system has been studied extensively in the context of equilibrium and nonequilibrium phenomena. The study of phase transition, originally limited to equilibrium systems, was extended to nonequilibrium systems [1]. While an additive noise provides equilibrium phenomena such as a disordering effect and a symmetry-breaking transition, a multiplicative noise coupled to the state of the system induces nonequilibrium phenomena such as a change of the stability of the system. The multiplicative noise remains the focus of current research [1][2][3]. The question of the interplay between multiplicative and additive noises in the systems has also been raised continuously [3,4]. While second-order transition induced by the multiplicative noise has been studied [1,5], its effect on first-order transition remains to be investigated.In this paper we study the effect of the multiplicative noise on the multistability investigating the nonequilibrium phenomena of the globally coupled oscillator systems with a third harmonic pinning force subject to a fluctuating interaction and an additive noise. It is shown that the additive noise and the fluctuating interaction induces a subcritical saddle-node bifurcation from an asymmetric state to a symmetric state at a critical noise intensity leading to multistable states. The fluctuating coupling increases the critical noise intensity and thus enhances the multistability drastically. We show phase diagrams and discuss the nature of the phase transition.In the presence of additive and multiplicative noises, a model of N coupled oscillators with a third harmonic pinning force under study is expressed by the Langevin equationwhere f i , i 1, 2, . . . , N, is the phase of the ith oscillator. On the right-hand side of Eq. (1) the first term is a third harmonic pinning force, and the second term describes global coupling which depends on the phase difference of two oscillators with fluctuating interaction. j i ͑t͒ and h i ͑t͒ are independent Gaussian white noises characterized byand s A and s M measure the intensities of the additive noise and fluctuating interaction, respectively. Throughout this paper we set K 1 using a suitable time unit. Equation (1) is invariant under the global finite translationfor all f i 's and under the global inversion f i ! 2f i for all f i 's. In the absence of the noises the system has three stable fixed points synchronized perfectly at 0, ...