Weak Deformation Research Articles

Compression and Stretching of a Self-Avoiding Chain in Cylindrical Nanopores

Force-induced deformations of a self-avoiding chain confined inside a cylindrical cavity, with diameter D, are probed using molecular dynamics simulations, scaling analysis, and analytical calculations. We obtain and confirm a simple scaling relation -fD approximately R(-9/4) in the strong-compression regime, while for weak deformations, we find fD = -A(R/R0) + B(R/R0)(-2), where A and B are constants, f the external force, and R the chain extension (with R0 its unperturbed value). For a strong stretch, we present a universal, analytical force-extension relation. Our results can be used to analyze the behavior of biomolecules in confinement.

Long term prediction of nonlinear viscoelastic creep behaviour of elastomers: extended Schapery model

Herein we develop a creep test method for characterizing nonlinear viscoelastic behaviour of an elastomer through weak deformation. This approach is based on an expanded Schapery model. Short term creep tests at various stress levels were performed upon specimens composed of a synthetic elastomer. A method to predict the long term creep behaviour using short term creep tests is presented. Experimental results and theoretical predictions are compared.

The instability of the spiral wave induced by the deformation of elastic excitable media

There are some similarities between the spiral wave in excitable media and in cardiac tissue. Much evidence shows that the appearance and instability of the spiral wave in cardiac tissue can be linked to one kind of heart disease. There are many models that can be used to investigate the formation and instability of the spiral wave. Cardiac tissue is excitable and elastic, and it is interesting to simulate the transition and instability of the spiral wave induced by media deformation. For simplicity, a class of the modified Fitzhugh–Nagumo (MFHN) model, which can generate a stable rotating spiral wave, meandering spiral wave and turbulence within appropriate parameter regions, will be used to simulate the instability of the spiral wave induced by the periodical deformation of media. In the two-dimensional case, the total acreage of elastic media is supposed to be invariable in the presence of deformation, and the problem is described with Lx × Ly = N × ΔxN × Δy = L′xL′y = N × Δx′N × Δy′. In our studies, elastic media are decentralized into N × N sites and the space of the adjacent sites is changed to simulate the deformation of elastic media. Based on the nonlinear dynamics theory, the deformation effect on media is simplified and simulated by perturbing the diffusion coefficients Dx and Dy with different periodical signals, but the perturbed diffusion coefficients are compensatory. The snapshots of our numerical results find that the spiral wave can coexist with the spiral turbulence, instability of the spiral wave and weak deformation of the spiral wave in different conditions. The ratio parameter ε and the frequency of deformation forcing play a deterministic role in inducing instability of the spiral wave. Extensive studies confirm that the instability of the spiral wave can be induced and developed only if an appropriate frequency for deformation is used. We analyze the power spectrum for the time series of the mean activator of four sampled sites which are selected symmetrically in different cases, such as the condition that the spiral wave coexists with the spiral turbulence, spiral wave without evident deformation, complete instability of the spiral wave (turbulence) and weak deformation of the spiral wave. It is found that more new peaks appear in the power spectrum and the distribution of frequency becomes sparser when the spiral wave encounters instability.

Motion of spiral waves induced by local pacing

Abstract The motion of spiral waves in excitable media driven by a weak pacing around the spiral tip is investigated numerically as well as theoretically. We presented a Bifurcations diagram containing four types of the spiral motion induced by different frequencies of pacing: rigidly rotating, inward-petal meandering, resonant drift, and outward-petal meandering spiral. Simulation shows that the spiral resonantly drifts when the frequency of pacing is close to that of the spiral rotation. We also find that the speed and direction of the drift can be efficiently controlled by means of the strength and phase of the local pacing, which is consistent with analytical results based on the framework of the weak deformation approximation.

Valorization of poly(butylene terephthalate) wastes by blending with virgin polypropylene: Effect of the composition and the compatibilization

AbstractBlends of recycled poly(butylene terephthalate) (PBT) parts obtained from scrapped cars, and virgin polypropylene (PP), were prepared in a twin‐screw extruder at different compositions. Selected compositions were also prepared with the presence of ethylene‐co‐glycidyl methacrylate copolymer (E‐GMA) and ethylene/methyl acrylate/glycidyl methacrylate terpolymer (E‐MA‐GMA) compatibilizers. The effect of the composition and the type of compatibilizer, as well as the mixing conditions, on the morphology phase, thermal, viscoelastic behavior, and mechanical properties of the blends has been investigated. Blends PP/PBT of various composition exhibit a coarse morphology and a poor adherence between both phases, resulting in the decrease of ductility, whereas at weak deformation, PBT reinforced the tensile properties of PP. Addition of E‐GMA and E‐MA‐GMA to the PP/PBT blend exhibited a significant change in morphology and improved ductility because of interfacial reactions between PBT end chains and epoxy groups of GMA that generate EG‐g‐PBT copolymer. Moreover, thermal and viscoelastic study indicated that the miscibility of PP and PBT has been improved further and the reactions were identified. The E‐MA‐GMA results in the best improvement of ductility. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers

TECTONO-METAMORPHIC HISTORY OF THE DUARTE TERRANE (JARABACOA AREA, HISPANIOLA ISLAND): INSIGHTS ON THE TECTONIC EVOLUTION OF THE NORTHERN RIM OF THE CARIBBEAN OCEANIC PLATEAU

The northern margin of the Caribbean Plate is characterized by a transpressional belt consisting of an assemblage of several terranes derived from Late Cretaceous to Tertiary oblique convergence between the Northern American and Caribbean Plates. Fragments of the oceanic plateau belonging to the Caribbean Plate are incorporated in these belts as strongly deformed and metamorphosed sequences, as recognized in the Jarabacoa area of the Hispaniola Island. In this area the Duarte Terrane consists of two different units showing different metamorphic grades. The lower unit consists of greenschist facies metabasites, whereas the upper one is represented by amphibolite facies metabasites topped by metasedimentary rocks, whose protoliths were clastic deposits supplied by a magmatic arc. Both units suffered four deformation phases, from D1 to D4, that can be strictly correlated. In both units the D1 phase is characterized by a syn-metamorphic foliation associated to a mineral lineation and very rare rootless isoclinal folds. The D2 phase is characterized by a well developed foliation and isoclinal folds acquired under retrograde metamorphic conditions. The D1 and D2 phases developed in the Cenomanian-Turonian time span. The following D3 and D4 phases produced weak deformations without metamorphic imprint. The D3 and D4 phase are probably connected with transpression tectonics of Early Oligocene to Early Miocene age. All these deformations developed in an arc setting connected with southward oblique subduction of the oceanic lithosphere of the North America Plate beneath the Caribbean oceanic plateau.

Blending PP with PA6 industrial wastes: Effect of the composition and the compatibilization

AbstractBlending polypropylene to recycled PA6 industrial wastes at different compositions, with and without compatibilizer PPgMA was produced in a corotating twin screw extruder where, polypropylene acts as the polymer matrix and polyamide as the dispersed phase. Several techniques were used to investigate the morphology, thermal, viscoelastic and tensile properties of these blend. Binary PP/PA6 blends showed the presence of PA6 particles dispersed in the PP continuous phase and exhibited a coarse morphology. Increasing PA6 contents in the blend increased their crystallinity and their size and improved the tensile properties at weak deformation. In addition to compatibilizer PPgMA, the morphology shows lower diameters and a decrease in size of the dispersed PA6 particles. The interfacial adhesion was also improved, as a result of the creation of an interphase that was formed by the interaction between the formed PPgPA6 copolymer in situ and both phases. This interphase induced an improvement in tensile properties. The PPgPA6 copolymer generated by the interphase was identified with DMA analysis thanks to an additional transition in loss modulus curves. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2008

Deformation-controlled cation diffusion in tourmaline: A microanalytical study on trace elements and boron isotopes

aBS tract Plastic deformation of variable intensity has modified primary textures, composition, and element distribution in tourmaline from a mylonitic pegmatite from the NW Argentinean Sierras Pampeanas. The primary concentric tourmaline zonation has been overprinted and partly destroyed by synkinematic cation diffusion that occurred under amphibolite-facies P-T conditions. Textural changes include bending of crystals, formation of subgrains, and subgrain rotation, leading locally to recrystallization of tourmaline. Some parts of deformed crystals were less affected or unaffected by plastic deformation and secondary cation diffusion. Two principal processes, intracrystal trace-element diffusion and matrix exchange of trace-elements, caused the compositional variations in deformed tourmaline domains. Intracrystal diffusion partly or completely destroyed the primary trace-element zonation by cation diffusion from zones of high to low concentration. Small cations occupying the Y-position are readily mobilized even by weak deformation, whereas large cations on the X-site require higher deformation intensity to be mobilized and show less pronounced or no homogenization of the primary zonation. Matrix exchange between tourmaline and co-existing solid or fluid phases generally leads to trace-element loss from deformed tourmaline domains. The extent of the matrix exchange is primarily controlled by the compatibility of mobilized cation species with matrix phases during mylonitization. In contrast, the B-isotopic compositions in deformed and undeformed crystal domains are uniform, indicating absence of significant B diffusion and isotope fractionation during crystal growth and plastic deformation; i.e., the activation energy to move B in tourmaline must be too high even for intense plastic deformation to overcome.

An improvement of the too cold tongue in the tropical Pacific with the development of an ocean-wave-atmosphere coupled numerical model

The anisotropy of magnetic susceptibility (AMS) is a quick, effective and sensitive technique used to measure the weakly deformed sedimentary rocks, and also a reliable method to reveal the deforming mechanisms of fault-related folds. In Longmenshan front belt, a typical cross-section of fault-related folds is chosen to study the AMS. A total of 224 oriented specimens have been drilled at 23 different sampling sites which were distributed at the key structural positions of this structural section developed in the Xujiahe formation of the upper Triassic. Six elementary types of magnetic fabrics are recognized and established through this AMS study: 1) a sedimentary fabric; 2) an initial deformation fabric; 3) a pencil structure fabric; 4) a weak cleavage fabric; 5) a strong cleavage fabric; 6) a stretching lineation fabric. It has been found that most of magnetic fabrics are characterized by fabrics of weak deformation which belong to the pure-shear results of a pre-folding layer parallel shortening (LPS). In the fault-bend fold, almost all magnetic fabrics are the initial deformation fabrics of weak deformation, and denote that the deformation in the forelimb is stronger than that in the backlimb and no finite strain is shown in the footwall. While in the fault-propagation fold, finite strains are concentrated in the trishear zone where magnetic fabric results are approximately consistent with the estimated consequences of the kinematic model. The tectonic stress field indicated by the magnetic fabrics is basically the same along the whole structural section and shows a NW to SE compression and shortening which is accordant with the regional compressive stress field of the Longmenshan fold-thrust belt.

Superposed deformation in turbidites and syn‐sedimentary slides of the tectonically active Miocene Waitemata Basin, northern New Zealand

ABSTRACTThe Miocene Waitemata Basin was deposited on a moving base provided by the Northland Allochthon, which was emplaced in the Late Oligocene, as a new convergent plate boundary was established in northern New Zealand. The basin experienced complex interaction between tectonic and gravity‐driven shallow deformation. Spectacular examples of the resulting structures exposed on eastern Whangaparaoa Peninsula 50 km north of Auckland provide a world‐class example of weak rock deformation, the neglected domain between soft‐sediment and hard rock deformation. Quartz‐poor turbidite sequences display a protracted sequence of deformations: D1, synsedimentary slumping; D2, large scale deeper‐seated sliding and extensional low‐angle shearing, associated with generation of boudinage and broken formation; D3, thrusting and folding, indicating transport mostly to the SE; D4, thrusting and folding in the opposite direction; D5, further folding, including sinistral shear; D6, steep faults. The deformation sequence suggests continuous or intermittent southeastward transport of units with increasing sedimentary and structural burial. By phase D3, the rocks had passed from the soft‐sediment state to low levels of consolidation. However, with a compressive strength of ∼5 MPa they are weak rocks even today. Such weak‐rock deformation must be important in other sedimentary basins, especially those associated with active convergent plate boundaries and with immature source areas for their sediments.

Paleo-deviatoric stress magnitudes from calcite twins and related structural permeability evolution in minor faults: Example from the toarcian shale of the French Causses Basin, Aveyron, France

The relationship between deformation and so-called fluid paleotransfers in minor faults has been analysed in an argillaceous formation located in the Causses Basin in France. The fluid paleotransfers are related to the fault activity to a large extent. We attempt to estimate the intensity of paleo-deviatoric stress magnitudes under which the fault activity may have occurred and consequently, the change in the structural fault permeability. The paleo-deviatoric stress magnitudes were calculated with the inverse method of Etchecopar applied to calcite twinning. The measured crystals are contained within the core zone of minor faults and this study is based on a previous complete microtectonic and microstructural analysis of the faults. In this paper, analysis of calcite twinning has been applied for the first time to vein fillings associated small faults in a context of relatively weak deformation, a condition ensured by the tectonic and structural analysis. Calculation and discussion of the paleo-deviatoric stress tensors in relation to the evolution of the structural fault permeability and to the hydraulic behaviour of the faults are the aim of this paper. The analysed faults, created and active during the same tectonic event, were permeable under a ( σ 1– σ 3) mean value of 40–50 MPa. On the other hand, the reactivation of faults during a second tectonic event implies mean ( σ 1– σ 3) value higher than 40–50 MPa, especially for the faults that are poorly oriented with respect to the principal tectonic stress directions. The core zone of these faults remained sealed and impermeable or became permeable by development of microcracks inside the pre-existing fillings.

Magnetostratigraphic study of the Kuche Depression, Tarim Basin, and Cenozoic uplift of the Tian Shan Range, Western China

Magnetostratigraphic and rock magnetic results from 1006 horizons in Paleogene and Neogene sediments between the upper Kumugeliemu Formation and the base of the Kuche Formation within the Kuche Depression of the Tarim Basin, NW China, are used to evaluate postcollisional uplift and deformation in the Tian Shan Range. Characteristic remanences with both normal and reversed polarity were isolated by thermal demagnetization, generally between 300 and 690 °C, and a composite magnetostratigraphy is constructed from data at 969 accepted levels correlating with polarity chrons C12n to C3r dated between ∼ 31.0 and ∼ 5.5 Ma on the CK95 geomagnetic polarity time scale. Observed height-dependent changes of rock magnetic parameters ( k m and AMS ellipsoid parameters) indicate that these sediments were influenced by weak deformation with the succession accumulated before ∼ 15 Ma recording the effects of compressive deformation. This finding, together with two substantial increases in accumulation rate at ∼ 16–17 and ∼ 7 Ma, define a framework for Cenozoic uplift and deformation of the Tian Shan Range. We argue that thrusting in the south of the range was probably initiated at ∼ 20 Ma and followed by rapid uplift at ∼ 16–17 Ma. On the assumption that sediment accumulation in western China was still predominately tectonically controlled at ∼ 7 Ma, another later pulse of rapid uplift with a much larger regional influence is also recognized in the Tian Shan. The Kuche Depression was subjected to two postcollisional phases of rotational deformation with clockwise rotation prior to ∼ 12 Ma (∼ 1.36°/Ma) succeeded by counterclockwise rotation (at ∼ 1.64°/Ma) after ∼ 12 Ma and possibly continuing to the present day. This rotational deformation is not temporally related to the history of basin formation resolved from the magnetostratigraphy and may be linked to block interactions with adjoining terranes.

Potential Seismic Risk in Taiwan Strait Region

GPS observation indicates that the Fujian coastal region of China mainland, the region of Taiwan Strait and northern Taiwan island all show a generally homogenous horizontal motion with weak deformation. Historical earthquake record over more than 800 years and modern instrumental data reveal that there is potential seismic risk in and around Taiwan Strait region. After the National Seismic Zoning Map of China (2001) the expected seismic risk in northern part of the Taiwan Strait is lower than that in middle and southern part. The suggested northern route of the Taiwan Strait passage project seems to be relatively save seismically.

Late-Hercynian shearing during crystallization of granitoid magmas (Sila massif, southern Italy): regional implications

Shearing of regional extent, involving granitoids and underlying mid-crustal rocks of the Sila massif (Calabria, Italy), is analysed in this paper. The deformed granitoids are affected by a wide NNW-SSE oriented deformation zone, stretching for about 60 km, from the neighbourhood of Cecita Lake to Cropani village. Meso- and micro-structures in granitoids, close to the boundary with underlying migmatitic paragneiss, indicate that deformation developed from melt-present to solid-state conditions. Simultaneous tectonics and magmatism activated a plutonic accretionary process at mid-crustal levels. This took place at about 300 Ma and involved hybrid magmas with a dominat contribution from a mantle source. The deformation regime remained steady for a long time during magma crystallization and cooling in subsolidus conditions. The regional top-to-the-W sense of shear in the present geographic coordinates, recorded in the deformed granitoids, seems geometrically consistent with the coeval direction of maximum extension found in another sector of the southern Hercynian belt, suggesting the original position of the Sila basement in this context. Magmatic ativity ended with the intrusion of mafic and felsic magams affected by a very weak deformation, ongoing during the final strain increments of the late-Hercynian stage.

Cylindrical implosion from explosive: Experiment of validation of the constitutive equations– Experimental approach

The validation of constitutive equations currently based on sphere expansion or Taylor test is difficult to obtain for fragile materials, which are fractured in traction for weak deformations. It is thus preferable, for these materials, to choose a test such as the cylindrical implosion for which the material is under compression. The study presented relates to the experimental validation of a cylindrical device of implosion, which was developed on titanium (TA6V4). The major innovations related to this experimental configuration are the use of high explosive and the instrumentation implemented. The velocity of a conical titanium sample is measured using IDL and VISAR techniques. The results indicate that deformations close to 20% were thus obtained. They enable us to validate the technology and the instrumentation implemented. A first comparison simulation-experiment is carried out.

Weak long-lived ground deformation related to Iwate volcanism revealed by Bayesian decomposition of strain, tilt and positioning data

Weak long-lived ground deformation associated with the volcanic unrest of Mt. Iwate, northeastern Japan in 1998–2002 is detected from borehole strainmeter, tiltmeter and global positioning system (GPS) data. This deformation is relatively weak (0.01 μ/day strain, 0.01 μrad/day tilt, and 0.01 cm/day displacement), but persists for a period of 5 years. To detect such weak long-term deformation, a new decomposition method based on Bayesian theory is developed and applied to the recorded data. This method is effective for extracting secular drift from the strain and tilt data, and seasonal changes from the GPS data. From a selection of possible models, a composite source model consisting of a dike series and a Mogi source is identified as the best fit to the observed data. The deformation is divided into 8 phases based on the characteristic variations in tilt and strain. Based on this model, the deformation sequence is considered to have initiated with the intrusion of a dike at intermediated depth below the summit region between February and April 1998, with short periods of shallow dikes intrusion toward the west in March and April. Shallow dike intrusion resumed in September, reaching a final volume of 5.9 × 10 6 m 3. The Mogi source is inferred to have been located 3 km beneath the western flank of the volcano, slowly inflating to a maximum volume 1.2 × 10 7 m 3 by the end of 2000, then deflating by 49% to 2002. This ground deformation was accompanied by seismic migration, from the summit region in the vicinity of the intruding dike to the west following the dike extension. This dike-correlating seismic activity ceased simultaneously with deflation of the Mogi source, suggesting that the Mogi source was a hydrothermal reservoir that was breached by the intruding dike. The Bayesian decomposition method developed in this study makes it possible to clarify the detailed characteristics of dike intrusions and the long-lived activity of the Mogi source utilizing the high sensitivity of strain and tiltmeters coupled with the stability of GPS.

Light-induced deformation and instability of a liquid interface. I. Statics

We study in detail the deformations of a liquid-liquid interface induced by the electromagnetic radiation pressure of a focused cw laser beam. Using a simple linear model of static equilibrium of the interface under the effect of radiation pressure, buoyancy, and Laplace pressure, we explain the observed hump height variations for any value of the optical Bond number Bo=(omega0/lc)2 (lc is the capillary length and omega0 is the waist of the beam) in the regime of weak deformations and show that the deformations are independent of the direction of propagation of the laser. By increasing the beam power, we observe an instability of the interface leading to the formation of a long jet when the laser propagates from the more refringent phase to the less refringent one. We propose that the total internal reflection of the incident light on the highly deformed interface could be at the origin of this instability. Using a nonlinear model of static equilibrium of the interface taking account of the angular dependance of radiation pressure, we explain the measured beam power threshold of the instability P, as well as the shape of the interface deformations observed at large waists just below the instability onset. According to this model, the instability should occur when the interface slope reaches the angle of total reflection, theta(TR). We find experimentally that, just below the instability threshold, the maximum incidence angle along the interface, theta(imax), is significantly smaller than theta(TR) and that our nonlinear model does not present any instability up to theta(imax)=theta(TR). Thus, although the proposed instability model correctly predicts the instability threshold P, it fails to describe the actual instability mechanism. We finally discuss possible additional effects that could explain the instability.

Metamorphic textures and geochemistry of the Cyprus-type massive sulfide lenses at Zurabad, Khoy, Iran

The Upper Cretaceous volcanogenic-exhalative Fe-Cu massive sulfide lenses at Zurabad, Iran occur within spilitized basalts of the Khoy ophiolite. Both the host rocks and the sulfide lenses are metamorphosed to greenschist facies and have been subjected to weak deformation causing textural changes in the ore. The presence of preserved banded pyrite, pyrrhotite, chalcopyrite and sphalerite in a matrix rich in chlorite indicates that the ore was a primary facies of synvolcanic hydrothermal massive sulfide mineralization, similar to those formed in modern seafloor environments and in Cyprus-type ophiolite deposits. Coarse-grained pyrite porphyroblasts or crystalloblasts, recrystallized pyrrhotite and chalcopyrite, together with regional silicate-sulfide foliation were produced by sub-seafloor metamorphism followed by regional metamorphism and ophiolite obduction. Brittle deformation and shearing produced cataclastic textures in pyrite, which were filled by chalcopyrite and pyrrhotite. Post-obduction and or post-metamorphic events produced late veinlets of pyrite-chalcopyrite and gangue minerals. Supergene processes related to gossan formation converted pyrite, pyrrhotite and chalcopyrite to marcasite and birdseye pyrite. Primary sphalerite contains an average of 14 mole % FeS, which shows equilibration at temperatures below 240 °C, and a pressure less than 1 kilobar. Based on its equilibrated textures with other sulfides and silicates in the foliation, pyrrhotite was a primary mineral in the Zurabad volcanogenic massive sulfide lenses. Therefore, pyrrhotite can be primary, and with pyrite forms a buffer that constrains sulphur fugacity during regional metamorphism. On the evidence of textures, mineralogy and geochemistry, pyrite is stable in the greenschist facies, provided that the activity of sulphur remains high. Of prime importance in exploration of the massive sulfide lenses in the Zurabad area has been the recognition of banded gossans and coarse pyrite porphyroblasts, which may prove to be a successful exploration tool in the search for further massive sulfide mineralization in the district.

Spatiotemporal filtering using principal component analysis and Karhunen‐Loeve expansion approaches for regional GPS network analysis

Spatial filtering is an effective way to improve the precision of coordinate time series for regional GPS networks by reducing so‐called common mode errors, thereby providing better resolution for detecting weak or transient deformation signals. The commonly used approach to regional filtering assumes that the common mode error is spatially uniform, which is a good approximation for networks of hundreds of kilometers extent, but breaks down as the spatial extent increases. A more rigorous approach should remove the assumption of spatially uniform distribution and let the data themselves reveal the spatial distribution of the common mode error. The principal component analysis (PCA) and the Karhunen‐Loeve expansion (KLE) both decompose network time series into a set of temporally varying modes and their spatial responses. Therefore they provide a mathematical framework to perform spatiotemporal filtering. We apply the combination of PCA and KLE to daily station coordinate time series of the Southern California Integrated GPS Network (SCIGN) for the period 2000 to 2004. We demonstrate that spatially and temporally correlated common mode errors are the dominant error source in daily GPS solutions. The spatial characteristics of the common mode errors are close to uniform for all east, north, and vertical components, which implies a very long wavelength source for the common mode errors, compared to the spatial extent of the GPS network in southern California. Furthermore, the common mode errors exhibit temporally nonrandom patterns.

Drift of spiral waves controlled by a polarized electric field

The drift behavior of spiral waves under the influence of a polarized electric field is investigated in the light that both the polarized electric field and the spiral waves possess rotation symmetry. Numerical simulations of a reaction-diffusion model show that the drift velocity of the spiral tip can be controlled by changing the polarization mode of the polarized electric field and some interesting drift phenomena are observed. When the electric field is circularly polarized and its rotation follows that of the spiral, the drift speed of the spiral tip reaches its maximal value. On the contrary, opposite rotation between the spiral and electric field locks the drift of the spiral tip. Analytical results based on the weak deformation approximation are consistent with the numerical results. We hope that our theoretical results will be observed in experiments, such as the Belousov-Zhabotinsky reaction.