The exponential divergence or convergence of nearby trajectories (Lyapunov exponents) is conceptually the most basic indicator deterministic chaos. We propose a new method to determine spectrum several Lyapunov exponents (including positive, zero, and even negative ones) from observed time series single variable. have applied various known model systems also Rayleigh-B\'enard experiment, elucidated dependence on Rayleigh number.

Zinc metal leaves are grown two-dimensionally by electrodeposition. The structures clearly remind us of the random patterns simulated computer according to Witten-Sander diffusion-limited-aggregation model. scale invariance is tested computing density-density correlation function for digitized photographs. Hausdorff dimension averaged over many examples $D=1.66\ifmmode\pm\else\textpm\fi{}0.03$, which in excellent agreement with that two-dimensional model ($D\ensuremath{\cong}\frac{5}{3}$).

Pattern formation in the electrodeposition of Zn from a thin layer ZnS${\mathrm{O}}_{4}$ solution was studied as function electrolyte concentration and applied voltage. We found several qualitatively different growth forms this system. Most strikingly, transition dendritic crystals (where crystalline anisotropy dominates) to disordered ramified patterns is when reduced. The may be described fractal below concentration-dependent cutoff length.

Practical methods to extract the generalized dimension Dq and largest Lyapunov exponent from experimental data are proposed tested on examples. The measured values agree well with known values. In applications chaotic signals, convergence of is investigated for varying delay time embedding dimension.

We report the observation of self-limited layer-by-layer etching Si by alternated chlorine adsorption and low energy Ar+ ion irradiation using an ultraclean electron-cyclotron-resonance plasma apparatus. The etch rate per cycle increased with supplying time saturated to a constant value about 1/2 atomic layer for Si(100) 1/3 Si(111), which was independent partial pressure in range 1.3–6.7 mPa. These results indicate that determined chlorine. Moreover, found be described Langmuir-type...

We studied effects of chaos added to the dynamics a neural network model. By numerical simulations, we found with forcing by chaotic noise operated very efficiently solve an optimization problem. also showed that short time correlation was relevant and it could work effectively for global minima search.

Tip-stable, tip-oscillating, and tip-splitting types of dendrites were observed experimentally in a two-dimensional supersaturated N${\mathrm{H}}_{4}$Cl solution. The relation between the tip velocity $V$ curvature $K$ measured for tip-stable type. It was found that is proportional to instead ${K}^{2}$ which expected from marginal-stability hypothesis melt-growth system.

Zinc-metal ``trees'' are grown two dimensionally from a line electrode by electrodeposition. The deposits, which consist of trees all sizes, bear close resemblance to the patterns two-dimensional (d=2) diffusion-limited deposition on one-dimensional (${d}_{b}$=1) substrate computer simulated first Meakin. fractal nature deposits is confirmed with dimension ${d}_{f}$(2,1)=0.70\ifmmode\pm\else\textpm\fi{}0.06. number (trees) size s also found scale as...

Fractal properties of a random pattern formation produced by computer simulation have been analyzed. The controlling parameter was tip priority factor $R$ which growing grows further compared to any site on branch produce new side branch. found show an approximate self-similarity and associated with two fractal dimensions: One is the inner dimension measures fine structure other outer framework pattern. With increasing factor, decreases shows phase-transition-like behavior at $R=35$. It...

The authors have studied the information processing ability of stochastic logic neural networks, which constitute one pulse-coded artificial network families. These networks realize pseudoanalog performance with local learning rules using digital circuits, and therefore suit silicon technology. synaptic weights outputs neurons in are represented by pulse sequences. limited range reduces coding noise suppresses degradation memory storage capacity. To study effect on an optimization problem,...

From a far-sighted viewpoint for nonlinear non-equilibrium system including the reservoirs principle of maximum increasing rate total entropy is proposed to describe steady as well non-steady phenomena. The conditions validity this postulate are discussed and was compared with Prigogine's general evolution criterion minimum excess production principle.

Computer calculation for the numerical solution of Duffing's and generalized equations shows global scaling properties bifurcation in parameter space. These are discussed terms a one-dimensional map. The analysis based on piecewise linear approximation gave results good agreement with experimentally observed behavior.

Received 2 December 1966DOI:https://doi.org/10.1103/PhysRevLett.18.159©1967 American Physical Society

A discussion is presented of one the simplest models Josephson computer systems (phase mode system), in which quantized vortices magnetic flux (fluxoids) are employed as information bits and all logic functions achieved by interactions between fluxoids. In a phase system, device operation depends on existence many stable states differing from each other integer multiples 2 pi plane. As an example, authors propose simple model complete computing system discuss basic elements instruction this...

We propose a new global characterization of chaotic dynamics. It gives unified description some aspects chaos (e.g., the relationships among dimensions invariant measures, Lyapunov exponents, and entropies; strange attractor vs repeller; variational principle). apply this formalism to few examples employing practical algorithm.

We show that the equation which describes two- and three-dimensional arrays of small-area Josephson junctions is reduced to following form, −∇×(∇×φ)−∂2φ/∂t2−Γ∂φ/∂t = sin φ, where φ or phase-difference vector, vector whose components are sine functions respectively. Based on above we analyze numerically a two-dimensional square network junctions, discuss vortex motion network.

Highly pure single-crystal Nb films (thickness, ∼2000 Å) with high superconducting transition temperatures Tc of ∼9.3 K and resistance ratios R300/R10 up to ∼200 are successfully grown epitaxially on sapphire (α-Al2O3) MgO substrates at ∼500–∼700 °C by using an electron-beam evaporation technique. The most high-quality film (with the maximum 9.45 199) is obtained a (11̄02) substrate, which has thermal expansion coefficient very close that as well small lattice misfit Nb. quality found be...

The interactions of two coupled oscillating systems the Belousov–Zhabotinsky chemical reaction were investigated experimentally. A mapping diagram coupling was experimentally obtained in plane ω2/ω1 and S1–2 where ω1 ω2 natural frequencies reactor 1 2, area window that reactors. irregular synchronization region this may correspond to Tomita Kai’s χ a chaotic response has been predicted.

By computer simulation, interaction of two solitons with the same screw sense sine-Gordon equation, which retain their shapes and velocities upon collision other even in presence bias loss terms, are examined. It is confirmed that they can couple when greater than a critical value. The conditions mechanism coupling examined detail. They explained terms energy between ripple structures trailed by dissipative moving solitons. distance D coupled be expressed as D= (n−1/2−δ) λ (n an integer),...