Butterfly loop

Figure 1.26 displays the results of the dss coefficient as function of the bias field, the large signal 33 response ( butterfly loop ) as well as constructed x33trev curve of the same pzt... 

The electric field induced intrinsic strain for different crystallographic directions could be calculated from the shift in peak positions (see Figure 7.9). Figure 7.11 shows the result of the measurements for a rhombohedral pzt in [111] and [100] direction. Only one half of each cycle is shown for the sake of a clarity of the plot. The curve for the [100] direction reveals the typical shape of a butterfly loop for the electric field induced strain in ferroelectrics. However in [111] direction, which is parallel to the spontaneous polarization, strain is significantly smaller. From both curves, the so-called unipolar strain can be evaluated as the strain induced at the maximum electric field Emax with a reference to the remanent state (E = 0). The calculation gives strain values in [111] direction of 0.02% and for the [100] direction 0.15%. The observations are in good agreement with theoretical calculations made by Du et al. [22],... 

Figure 6.12 Strain versus applied electric field loops (a) symmetrical butterfly loop for a ferroelectric with a symmetrical P-E hysteresis loop (b) experimentally determined loop for a fine-grained ceramic 0.95Bi Nag iO -0.05KNbO ...

Soh, A.K., Song, Y.C., and Ni, Y. 2006. Phase field simulations of hysteresis and butterfly loops in ferroelectrics subjected to electro-mechanical coupled loading. Journal of the American Ceramic Society 89[2] 652-661. 

Figure 8.11 Butterfly loop of a piezostrictive material such as PZT, showing the strain S associated with a reversal of the polarization direction.

Figure 1.11 Butterfly loop observed in converse piezoelectric measurements. Reprinted with permission from Ok et al., Chem. Soc. Rev., 35, 710 (2006). Copyright (2006) Royal Society of Chemistry...

The Zoran 38000 has an internal data path of 20 bits as well as a 20 bit address bus. The two accumulators have 48 bits. It can perform a Dolby AC-3 [Vernon, 1995] five channel decoder in real time, although the memory space is also limited to one Megaword. It has a small (16 instruction) loop buffer as well as a single instruction repeat. The instruction set has support for block floating point as well as providing simultaneous add and subtract for FFT butterfly computation. 

Molecules with small spin have also been discussed. For example, time-resolved magnetization measurements were performed on a spin 1/2 molecular complex, so-called V15 [81]. Despite the absence of a barrier, magnetic hysteresis is observed over a time scale of several seconds. A detailed analysis in terms of a dissipative two-level model has been given, in which fluctuations and splittings are of the same energy. Spin-phonon coupling leads to long relaxation times and to a particular butterfly hysteresis loop [58, 82],... 

Several sophisticated physical studies of the Vis cluster have been published, for example I. Chiorescu, W Werns-dorfer, A. Muller et al., Butterfly Hysteresis Loop and Dissipative Spin Reversal in the S = 1/2, Vis Molecular Complex, Phys. Rev. Lett. 2000, 84, 3454-3457 B. Barbara, I. Chiorescu, W Wernsdorfer et al. The Vis Molecule, a Multi-Spin Two-Level System Adiabatic LZS Transi-tion with or without Dissipation and Kramers Theorem, Progr. Theor. Phys. Suppl. 2002, 145, 357-369. 

Arnaud (7) has developed a butterfly model that provides a diagrammatic view of the complex interrelationships among the three hormones (parathyroid, calcitonin, and vitamin D) that control calcium homeostasis (serum concentrations of ionic calcium) and their target organs (bone, kidney, and intestine) (Fig. 35.1). The right side (B loops) of the butterfly model describes the processes that increase the serum calcium concentration in response to hypocalcemia the left side (A loops) depicts the events that occur in response to hypercalcemia. 

A state of hypercalcemia (Table 35.6) will promote calcitonin biosynthesis and release. As a result, PTH biosynthesis and its secretion are inhibited, as is the production of vitamin D. The right wings of Arnaud s butterfly model (Fig. 35.1, B loops) would be activated to decrease serum calcium concentrations. In the presence of calcitonin. 

Dipole reorientation under the influence of an electric field was investigated in oriented nylon 11 films with polarised IR spectroscopy for the amide A (N-H stretching) and amide I (carbonyl stretching) bands. Butterfly-shaped hysteresis loops were obtained from peak intensity versus applied electric field strength. Least-squares Gaussian... 

Level control is critical because an unsteady level will result in varying discharge flow, which directly affects the brine neutralization control loop. The level transmitter should be a dual remote d/p cell with wetted parts of tantalum. The control valve should be a fully PTFE-lined butterfly valve in the pump discharge line. If powered by a UPS, this pump can be a source of flushing brine during a total power outage. TTie dechlorinated brine receiver may be elevated in order to provide more suction head to the discharge pump. 

The buffer-fly curve of FMR fi-equencies vs. applied electric field was observed and shown in Fig. 2 (b), which resembled the widely observed piezoelectric strain vs. electric field butterfly curves for piezoelectric materials and matched the ferroelectric P-E hysteresis loop of PMN-PT single crystal as well. This once again confirmed that the change of FMR fi equency of the FeGaB film results from the ME coupling induced strain in the FeGaB film. 

Figure 2. Permeability spectra of the FeGaB/Si/PMN-PT heterostnicture under different electric fields (left) The butterfly sh d hysteresis of the FMR fiequency vs. electric field of the FeGaB/Si/PMN-PT heterostructure, and the ferroelectric P-E hysteresis loop of the PMN-PT single...

The FMR field of the Fe304/PZT and Fe304/PMN-PT muitiferroic composites exhibited the characteristic butterfly shape in their FMR field vs. electric field curves as shown in Fig. 7, which coincided with the ferroelectric hysteresis loops of the PZT and PMN-PT respectively and were similar... 

Figure 7. Butterfly curves of resonance fields vs. electric fields and ferroelectric hysteresis loops of muitiferroic composite Fe304/PZT (a) and FesOVPMN-PT (b).

Figure 24.4 Possible structures for single-chain micelles of an ABABA pentablock copolymer in block-selective solvents for the B blocks. From left to right are a triple-beaded chain, a double-beaded loop, a tethered balloon, and a butterfly. The beads correspond to the A blocks, while the loops correspond to the B blocks.

In general, the bifurcation of a homoclinic butterfly is of codimension two. However, the Lorenz equation is symmetric with respect to the transformation (x y z) <-)> (—X, —y z). In such systems the existence of one homoclinic loop automatically implies the existence of another loop which is a symmetrical image of the other one. Therefore, the homoclinic butterfly is a codimension-one phenomenon for the systems with symmetry. 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

In the case of a saddle (the leading characteristic exponent Ai is real), the bifurcation diagram depends on the signs of the separatrix values Ai and A2, as well as on the way the homoclinic loops F1 and F2 enter the saddle at t = -f 00. Let us consider first the case where F1 and F2 enter the saddle tangentially to each other, i.e. bifurcations of the stable homoclinic butterfly. 

It is not hard to conclude from numerical experiments, which reveal the manner in which the separatrices converge to the homoclinic butterfly that A must be within the range (0,1). In this case, when a < 0, everything is simple the homoclinic butterfly splits into either two stable periodic orbits (Fig. C.7.8(g)), or just one stable symmetric periodic orbit (Fig. C.7.8(i)). It follows from Sec. 13.6 that when <7 > 0, two bifurcation curves originate from this codimensiomtwo point. They correspond to the saddle-node bifurcation (Fig. C.7.8(d)) and to the double homoclinic loop (Fig. C.7.8(f)). The... 

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