International Journal of Mining Science and Technology 23 (2013) 287–292

Contents lists available at SciVerse ScienceDirect

International Journal of Mining Science and Technology

journal homepage: www.elsevier .com/locate / i jmst

Rapid iterative incremental model of the intermittent chaos of deep holedeveloping in coal-gas outburst

Pan Yue a,⇑, Li Aiwu b

a School of Civil Engineering, Qingdao Technological University, Qingdao 266520, Chinab School of Science, Qingdao Technological University, Qingdao 266520, China

a r t i c l e i n f o

Article history:Received 5 July 2012Received in revised form 5 August 2012Accepted 14 September 2012Available online 2 June 2013

Keywords:Coal-gas outburstHoleCoal shell failureComplexityChaosLogistic mappings

2095-2686/$ - see front matter � 2013 Published byhttp://dx.doi.org/10.1016/j.ijmst.2013.04.005

⇑ Corresponding author. Tel.: +86 532 86879624.E-mail address: panyue@qtech.edu.cn (Y. Pan).

a b s t r a c t

In view of the occurrence of the coal and gas, outburst coal body separates in series of layer form, andtosses in a series of coal shell, and the morphological characteristics of the holes that formed in the coallayers are very similar to some iterative morphological characteristics of the system state under highlynonlinear condition in chaos theory. Two kinds of morphology as well as their starting and end statesare comparatively studied in this paper. The research results indicate that the outburst coal and rock sys-tem is in a chaotic state of lower nested hierarchy before outburst, and the process that lots of holes formowing to continuous outburst of a series of coal shells in a short time is in a rhythmical fast iterative stageof intermittent chaos state. And the state of the coal-gas system is in a stable equilibrium state after out-burst. The behaviors of outburst occurrence, development and termination, based on the universal prop-erties of various nonlinear mappings in describing complex problems, can be described by iterativeoperation in mathematics which uses the Logistic function f ðx;lÞ ¼ lxð1� xÞ and the composite functionF(3, x) = f(3)(x, l) as kernel function. The primary equation of relative hole depth x and outburst parameterl in kernel function are given in this paper. The given results can deepen and enrich the understanding ofphysical essence of outburst.

� 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology.

1. Introduction

Coal-gas outburst is the most complicated mine disaster trig-gered by underground mining, and it is highly nonlinear behaviorof the coal-gas system being far from equilibrium state. China isthe most serious country in aspect of coal-gas outburst [1]. Whenoutburst occurs, plenty of gas-pulverized coal streams with thecharacter of wind storm pour into air shaft or goaf in a short time.It leads to the damage of pit facilities, miners’ casualties and out-burst of gas, and seriously threatens pit safety, also limits the exer-tion of productivity. In June 1995, the accident of coal-gas outbursthappened in Limin Coal Mine of Lianshao Mineral Bureau, Hunanprovince, in which there were 3200 ton coal and 300 thousandcubic meter gas bursting into streams, and 19 persons died [2,3].In addition, another accident of coal-gas outburst happened in Lul-ing Coal Mine of of Huaibei Coal Industry in April, 2002, and therewere 10,000 ton coal and 600 thousand cubic meter gas burstinginto streams, 13 persons died. In order to prevent and cure coal-gas outburst, people have done a lot of research.

Elsevier B.V. on behalf of China Un

2. Experimental data

Jiang and others did tens of times about simulation test of coal-gas outburst [3]. After outburst, they sawed the coal body whichwas not outburst in the inner edge of test equipment, and foundthat in the residual coal body there had been embryonic form ofspherical shell layers formed by the way of self-organization, inter-mittent crack being boundary between layers, as shown in Fig. 1.They plotted the sustainable outburst of the coal shells throughavalanching burst and forming process of the holes, as noted inFigs. 2 and 3. Therefore, Jiang proposed the theory of coal shell fail-ure [4].

A system tends to convert higher position potential into lowerposition potential at any time. The opening of stone door and thestripping of hard coal in working face create a chance to removepart of coal and make the whole coal-gas system at the lower.Coal-gas system has not the capacity to remove the huge coal bodyin a time once. In order to approach the total goal of removing thehuge coal body in a short time, it needs repeatedly operation onlyby which it removes partial coal body and unleashes partial energyevery time. The accident of outburst occurred in ZhongliangshanCoal Mine, Chongqing. It lasted 39 s [5]. Simultaneously, the coalbody to be tossed forms a series of structure with the stable andthe most resistant trait by the way of self-organization to resist

iversity of Mining & Technology.

Outburst entrance

Fig. 1. Profile photo of the not thrown coal in outburst hole [3].

Residual coal piece Residual coal piece

CrackOutburst holeFreshlyexposed surface

Expansion stage Shallow shrink stage

Fig. 2. Diagramatic sketch of forming course of outburst hole [5].

Crack

Roa

dway

Gas streamline

Fig. 3. Dynamic course of coal-gas outburst [4].

288 Y. Pan, A. Li / International Journal of Mining Science and Technology 23 (2013) 287–292

being tossed out. The structure is toward the loading direction andcan be regarded as solid spherical shell. Under the huge groundstress and the gradient of gas pressure, the entity of three-dimen-sional coal layers through the fracture of two-dimensional coalshell form throws into the goaf at one side of exposed face likeavalanche, which makes the coal rock deform and makes the gaspressure reduce. Using this marvelous way is to reduce theposition potential of coal-gas system.

3. Assumption for describing the coal-gas outburst

Mathematical model is the mathematical abstraction of identi-cal phenomena in essence in the course of scientific practice ofhumanity. Through the efforts of many scientists, it is found thatmany natural phenomena have their corresponding mathematicalmodels. For example, the mathematical model of wave propaga-tion phenomenon is hyperbolic type equation; the mathematical

models of heat conduction phenomenon and gas diffusion phe-nomenon are parabolic type equations; the mathematical modelof position potential distribution law is elliptic type equation.There are several catastrophe models in catastrophe theory, andthe research indicates that different catastrophe models corre-spond to different mathematical abstraction of failure phenomenawith distinguishable essence in nature. As far as the sustainableoutburst of the coal shells is concerned, as shown in Figs. 1–3,we want to know if the marvelous process that the outburst ceasesafter forming deep hole of tens of meters has a mathematical mod-el to correspond with yet.

Numerous studies have focused on the coal-gas outburst[6–13]. Some researchers puzzled over the problem of coal-gasoutburst, and tried to establish the model of coal-gas outburst.However, Li as well as the Fushun Branch of China Coal ResearchInstitute proposed that it is an overwhelmingly difficult study ofoutburst mechanism to describe the generation, development,end of outburst by using mathematics [8,9]. Furthermore, Qideemed that it is too early to describe the outburst of gas and coalrock with engineering background quantitatively by the method ofanalytical theory because of the limit of the present developinglevel in mathematical mechanics [10–13].

In the course of outburst of coal-gas system, a series of similarprocesses of coal shells is a self-organization behavior under thecondition of highly nonlinear situation physically. Every outburstof coal shell does evoke the elastic wave in coal rock system andrefraction, reflection of the elastic wave. Meanwhile, at every out-burst instant of coal shell, a sudden increase of partial space at therear of this coal shell and a partial decrease of gas pressure can beproduced. Those two actions not only influence on the next out-burst of coal shell directly but also impact the gas desorptionand gas seepage intensity of the whole coal rock system. Accordingto mathematics, this is a typical iterative process. Thus, it is impos-sible to integrally describe the outburst process only via a differen-tial equation, with a series of coal shells, which has manyaccompanying phenomena. But, from a certain view such as apply-ing iterative computation of Logistic difference equation of chaostheory, we can partly describe the importantly apparent featureof coal-gas outburst which includes hole depth developing andopening formation of a series of coal shells outburst in a short time,describe the state characteristics of outburst before occurrence andend.

4. Period 3 window of chaos dynamics

In chaos theory, the time variable t in dynamics equation:

dxdt¼ f ðx;lÞ ¼ lxð1� xÞ ð1Þ

In Eq. (1), the time variable t is usually dispersed, then the dif-ference form of dynamics Eq. (1) is obtained, i.e., Logistic mapping.

xnþ1 ¼ f ðxn;lÞ ¼ lxnð1� xnÞ; n ¼ 1;2; . . . ð2Þ

The iterative result of logistic mapping is used to express thedeductive process of a discrete dynamics system state. In Eq. (1),x is the system state variable; t the time; and l (>0) the systemparameter. In Eq. (2), n corresponds to tn; xn is the variable valueof system state at time tn; l (>0) the system parameter; and n in-creases when time t increases.

In Eq. (2), nonlinear degree of the system increases with the in-crease of l. When l = l1 = 3, the single value state of the system isinstable. From l1, let l increase gradually, and let x appear as mul-tiple period point form of 1 ? 2, 2 ? 4, . . ., 2n�1 ? 2n, . . . whenl = l1 = 3.569 945 672 iterative result of the system state valuegoes into chaos state, as shown in Fig. 4.

x

*x

1 4o

Cha

os a

rea

μ3μ2μ1μ*μ μ∞

Fig. 4. From period-double bifurcation to chaos �l3 [14,15].

x

µ1+ 8

Cha

os

Cha

os

Fig. 5. Amplified period 3 window [14,15].

xn

F(3, xn)

Fig. 6. F(3, xn) at bifurcation point �l3 [14,15].

F(3, xn)

xn

Fig. 7. Rapid iterative process of intermittent chaos state.

Y. Pan, A. Li / International Journal of Mining Science and Technology 23 (2013) 287–292 289

In chaos area, system state is not completely out of order. It hasa series of small transparent windows, and the evolution of x iter-ative value is periodical in those windows. Period 3 window whichis in the lower left side, as shown in Fig. 5, is the biggest windowamong these windows. It occurs at

l ¼ �l3 ¼ 1þffiffiffi8p¼ 3:828 ð3Þ

When l increases from �l3, and let x appear as multiple periodpoint form of 3 ? 6, 6 ? 12, . . ., 3 � 2n�1 ? 3 � 2n, . . . till n ?1,iterative value x will go into right chaos area from period 3 win-dow. However, iterative value x will directly go into left chaos area,not through the process of period-double bifurcation, if ldecreases and goes back from l ¼ �l3 ¼ 1þ

ffiffiffi8p

.

5. Rapid iterative incremental model of the intermittent chaosof hole depth developing in coal rock outburst

In the small adjacent region of chaos area where l is slightlysmaller than �l3 ¼ 1þ

ffiffiffi8p

determined by Eq. (3), the evolution ofiterative value takes on very strange characteristics, i.e., it is rhyth-mically rapid iterative in some time, but this rhythmical change isbroken in other time. At the same time, some irregular movementsuddenly appears. After chaos erupts, a rhythmical behavior ap-pears again. The movement will appear again. Chaos to appearby this way is so-called intermittent chaos.

In Fig. 6, the fourth peak curve is that the critical value�l3 ¼ 1þ

ffiffiffi8p

of period 3 window is equal to l. Generating functionf(3)(x, l) = f{f[f(x, l)]} in Eq. (2) is operated 3 times of iterationwhich derives from the function f(x, l) = lx(1 � x) [14,15]. Here,the function is as follows:

Fð3; xnÞ ¼ f ð3Þðxn;lÞ ð4Þ

In Fig. 6, there are three pointcuts between the oblique line y = xand Fð3; xnÞ ¼ f ð3Þðxn; �l3Þ.

The dotted-line square in the bottom left of Fig. 6 is amplifiedand gets Fig. 7 when l is slightly smaller than �l3 [14,15]. In thecourse of iteration, when a point, or a system state, is limited inchannel x does rhythmically rapid iteration. After it comes out ofchannel, x does irregular iteration (namely chaos state), until itgoes into another channel.

From Fig. 7, we know that the value of state variable x monot-onously increases in the iterative course, and every iterative ampli-tude value Dxn = (xn � xn�1) of x changes from big to small, thenfrom small to big. The increase of x value and the process thatthe iterative amplitude value Dxn changes from big to small is justcorresponding to the outburst intensified phase in which the thick-ness of outburst coal shell, as noted in Fig. 3, gradually decreasesand the hole radius dilates. On the contrary, the process that theiterative amplitude value Dxn changes from small to big is just cor-responding to the outburst decay phase in which the thickness ofoutburst coal shell gradually increases and the hole radius con-tracts, as shown in Fig. 3. The narrowest portion of the channelneeding the shortest iterative time is corresponding to the coalshell outburst of which the thickness of coal shell is the thinnestand the radius is the biggest. Because the coal grain is granulatedby gas dilation in the coal grain after fracturing, near the narrowportion of the channel whose iteration corresponds to the outburstpeak state, the granularity of coal body fracture is the smallest, andat the beginning and end of the outburst the granularity of frac-tured coal is bigger. It is pointed out that state variable x monoto-nously increases when it is in rapid iterative course along thechannel in Fig. 7. At this stage, characteristics that x monotonouslyincreases corresponds to the trend that hole depth increases con-tinuously with the outburst of coal shell.

In Fig. 7, the width of the channel is related to the distance be-tween l and the critical value �l3 ¼ 1þ

ffiffiffi8p

. The closer l is to �l3,the narrower the channel is, and the more the operating iterativetimes before leaving the channel. And the higher the position

290 Y. Pan, A. Li / International Journal of Mining Science and Technology 23 (2013) 287–292

potential of coal-gas system which is composed of coal layer, topand base plate is (the higher ground stress is, the bigger gas pres-sure is), the larger the outburst scale is. The more the coal-gas out-burst is, the deeper the longitudinal depth formed by the outburstand the larger abdominal cavity is.

When the parameter l is far away from �l3and the channel inFig. 7 is wider, x has an opportunity to fall to the right side ofthe channel’s narrow place at the 1st point before x enters thechannel iteration. Thus, the subsequent iterative process may cor-respond to the outburst in the decay phase, namely, extrusion.Extrusion is another form that the parent population of the coalrock throws the coal body out, which forms hole with big openingsmall abdominal cavity. In the course of extrusion, the granularityof the pulverizing coal shell is bigger than that of the pulverizedcoal in outburst. What’s more, the number of the coal shell inextrusion is far less than that in outburst.

The main apparent feature, reflecting evolution process of coal-gas outburst that a series of coal shells form holes during shorttime continuous outburst in Figs. 2 and 3, can be extremely appro-priately described by rhythmically rapid iterative configurationand process occurring in channel with intermittent chaos statephase in Fig. 7. But, if we want to describe coal-gas outburstwholly, we still have following several questions to be determined.

� Which state should the coal rock system be in before the systemis faced with outburst at stone door opening or at hard coalstripping from the working face or at time-delay outburst?� Accompanying with lots of coal-gas outburst, there are corre-

spondingly longitudinal deep holes forming. After outburststops, which state should the coal rack system be in? What isthe state after outburst related to the rhythmical rapid iterativestage in intermittent chaos state?� In coal-gas outburst, whether the mathematical model of hole

depth developing stage can be expressed by F(3, x) = f(3)(x, l).� What are the concrete formation and physical meaning of state

variable x and parameter l which are used to describe coal-gasoutburst phenomenon in Eq. (4)?

6. Data fitting formation of mathematical model in coal-gasoutburst

6.1. State of the coal-gas system before outburst

In course of stone door opening or hard coal stripping from theworking face, the first quasi-outburst coal shell is coming closerand closer to the free surface of coal layer. And it forms the gradi-ents of high ground stress and high gas along outburst direction.Though the coal body is not in outburst at this time, coal-gassystem is already in highly nonlinear state. Parameter l and statevariable x are all in chaos area on the left side ofl1 < l << �l3 ¼ 1þ

ffiffiffi8p

in Fig. 5. Displacement x of free surfacein coal layer changes irregularly in a region which is on the left bot-tom square of chaos area, with very low nested hierarchy. Becauseof low nested hierarchy, the change of x is very small before out-burst whether the outburst is instantaneous or time-delay.

6.2. State of the coal-gas system after outburst termination

Every elastic wave caused by coal shell outburst results in thevibration of the coal rock system. Outburst has an intensifiedphase, and it then gradually goes into decay phase after reachingthe peak state. At the intensified phase of outburst, the contractionand expansion of the coal rock system forms a powerful dynamicforce potential, and dynamic inertia can maintain the tendencyof the coal rock system to continuously outburst until the hole be-comes big enough in the outburst coal layer. Coal rock system can

deform and unload, and gas pressure dramatically reduces. Dy-namic process cannot maintain yet, and outburst ceases ultimately.In course of outburst, a large number of gas desorption occurs incoal body, and gas desorption needs to absorb energy. Many exam-ples indicate that the temperature of the coal body reduces in out-burst field, and all those phenomena indicate that the coal rocksystem is already in low position potential state with energy over-drawing after outburst termination [9].

In Fig. 7, the system that carries out continuous and rapid itera-tion must be supplied with energy continuously. Actually, in the pro-cess of coal-gas outburst or hole forming, outburst parameter l thatrepresents nonlinear degree of the coal rock system is in constantdecrease, not in an unchanging value as shown in Fig. 7. The decreaseof l makes the channel in Fig. 7 widen, and state variable x is to stopiteration after it quickly goes out of the channel. Because of energyoverdrawing, the value of parameter l obviously decreases, andthe coal rock system goes back to be the stable equilibrium state. Be-sides, the outburst parameter l falls down at l⁄ position among theinterval (1 < l < 3) in Fig. 4. When the state variable x is at single va-lue immovable point x⁄ corresponding tol⁄in Fig. 4, ordinate x⁄ is therelative depth of the hole after outburst termination.

6.3. Description of hole depth developing in the use ofF(3, x) = f(3)(x, l)

From Fig. 4, we know that the system is at different states whenparameter l takes different value scopes. Different states can bedescribed by sectional function. When l < l1 � 3.57, the functiondescribing system characteristic is Logistic function f(x, l) =lx(1 � x), considering the stage iterating from arbitrary initial va-lue and having stable periodical solution. From the statement inSection 5, it is observed that hole depth developing phase incoal-gas outburst can be extremely appropriately described byrapid iterative configuration and iterative process occurring inchannel in Fig. 7.

Feigenbaum’s research indicates that the function of Logisticparabolic mapping is different from that of exponential mappingor sine mapping, but they all take on period-double bifurcation se-quence and chaos belt period-double inverse bifurcation sequence,and move towards chaos in the use of identical Feigenbaum con-stant d as convergence speed rate. They all have infinitely nestedself-similar structure, and each of adjacent hierarchies has identi-cal scale transformation factor a. Those properties are generallyapplicable to various nonlinear mapping. Franceschini convertedhigh-dimension Navier–Stokes hydrokinetics equation into a equa-tion group composed of five first-order ordinary differential equa-tions, and he similarly obtained the process from period-doublebifurcation to chaos, and from this he obtained identical Feigen-baum constant through this equation group [16]. There is differ-ence between an equation and an equation group composed offive equations, only general applicability is invariable in this infi-nite change. This general applicability has nothing to do with sim-ilarities and differences of equations, highness and lowness ofphase space dimensionality, and subject territory. It only hassomething to do with complexity of problem, and it is a generallyapplicable law of complexity.

Due to the general applicability, using the relative easy solutioncan replace a relative hard solution to solve the similar complexityproblems. Because of the general applicability, the answers of thetwo problems are identical in the aspect of complexity. Hence,using the form of mapping function is not so important for describ-ing the complexity of problem. As for and for the process that con-tinuous outburst of a series of coal shells ultimately leads to hole toform, the compound function F(3, x) = f(3)(x, l) of the parent func-tion f(x, l) = lx(1 � x) can be used to describe the process of finallyforming hole.

Y. Pan, A. Li / International Journal of Mining Science and Technology 23 (2013) 287–292 291

6.4. Initial forms of state variable x and parameter l

When coal-gas outburst is described by using Logistic mapping,state variable x in mapping function is the relative hole depth, andl is outburst parameter.

The form that the dimensionless state variable is correspondingto relative hole depth x is provided as follows:

x ¼ l1

Lð5Þ

where l1 is the distance from outburst front surface to the hole en-trance; and L the total length of longitudinal damaged area measur-ing from hole entrance along outburst coal layer. Finite elementanalysis, made by Chongqing Branch of Chinese Academy of Sci-ences, indicates there still exists considerable long damaged areafrom the bottom of hole to the deep coal layer after outburst termi-nation. So, L should be many times of l1.

The bigger the outburst parameter l value is, the stronger thenonlinearity of the system is, and the easier outburst occurs. Theaccidents of outburst mainly occur in sub layer with high groundstress, higher gas pressure and thicker deteriorated soft coal. Thisis because that deteriorated coal has higher gas content, elasticmodulus of soft coal is lower and elastic energy released by coalbody fracture is higher. Higher gas content in higher ground stresscan only produce higher pressure gas and also form the higherpressure gradient under the condition of worse emission for thepurpose of satisfying the condition of outburst occurrence. In viewof the above, under consideration for the releasing of deformationenergy in the roof and floor of coal layer influencing on the out-burst, primary form of dimensionless outburst parameter l of coaland rock system is given as follow:

l ¼ r� pt

l2� kEI

L4 d� ��

ðEcrcÞ ð6Þ

where rc is the compressive strength of outburst coal; and Ec theelastic modulus. It is beneficial to outburst that rc and Ec are small.Ecrc should be at the denominator position. (r � pt)/l2 is the pres-sure gradient nearby the outburst front surface, and the bigger pres-sure gradient is beneficial to outburst, so (r � pt)/l2 should be at thenumerator position. r = r

0+ p is the total stress in coal layer along

outburst direction; p the highest gas pressure at rear of the outburstfront surface; r

0the frame stress of coal body of corresponding po-

sition; pt the gas pressure at exposed surface; l2 the distance fromoutburst front surface to the site corresponding to the highest gaspressure p; r

0the frame stress of the coal body owing to high

ground stress exerting on corresponding site; d the thickness ofthe outburst coal layer, the thicker coal layer is beneficial to out-burst; d should be at the numerator position of outburst parameter.kEI/L4 the calculated rigidity of roof and floor of outburst coal layerunder the condition of loading distrusting on average; and k a con-stant. At the same deformation, the bigger rigidity of roof and floorcan cause more energy released by roof and floor during the holeforming, so kEI/L4 should be at the numerator position.

To sum up foregoing discussions r–u in this section, as far asthe coal-gas outburst phenomenon is concerned, the mathematicalmodels of system state in different phases such as the previousphase of outburst, the phase of hole depth developing and laterphase of outburst can be expressed by discrete dynamics system as

xnþ1 ¼ f ðxn;lÞ ¼ lxnð1� xnÞ; n ¼ 1;2; . . .

l1 < l << �l3; before outburstxnþ1 ¼ Fð3; xnÞ ¼ f ð3Þðxn;lÞ; n ¼ 1;2; . . .

l1 << l 6 �l3; during outburst�x� ¼ f ð�x�;l�Þ ¼ l��x�ð1� �x�Þ;

0 < l� << l1 ¼ 3; after outburst

9>>>>>>>>=>>>>>>>>;

ð7Þ

Eqs. (5) and (6) are the initial form of relative role depth x and out-burst parameter l, and they need to be consummated further. Byusing nonlinear theory, namely, viewpoint and method of chaostheory, recognition to physical essence of outburst can be deepenedand enriched.

7. Conclusions

(1) Hole depth developing in coal-gas outburst is a continuousoutburst process of a series of coal bodies which can be sim-plified to the form of three-dimensional shell of the ball cap,and this process is an iterative one of the discrete dynamicsystem in mathematics which cannot be described by akinetic equation at all.

(2) The main physical characteristics reflecting coal-gas out-burst is that a series of coal bodies have been in continuousoutburst process to form the hole in a short time, and itsmathematical model can be described appropriately by exe-cuting rapid iteration operation in intermittent chaos stateand iterative incremental process. Because of the universalproperties of nonlinear mappings, outburst evolution pro-cess can be expressed as Eq. (7) of the discrete dynamic sys-tem by the Logistic function and its composite functionF(3, xn) = f(3)(xn, l). In other words, before outburst coal androck system is in a chaos state of lower nested tier, as shownin little square frame in Fig. 5, relative hole depth x fluctu-ates nearby the small value. During outburst coal and rocksystem is in intermittent chaos state, i.e., the stage that xdoes rhythmical rapid iteration in channel. As is shown inFigs. 2, 3 and 7, x increases in one direction and continu-ously, while the outburst parameter l reduces. Because ofthe energy overdraft, the system is in a stable equilibriumstate for some time after outburst. x is at stable fixed point�x� corresponding to l⁄ in Fig. 4. Here, �x� is the relative depthof the hole after outburst stops.

(3) Eqs. (5) and (6) respectively correspond to the relative holedepth x and the outburst parameter l. They are the preli-minary form or linear form.

(4) Coal-gas outburst is the highly nonlinear behavior far fromthe equilibrium state. Through using the viewpoint andmethod of nonlinearity theory, i.e., chaos theory, the under-standing about the physical nature of outburst can be deep-ened and enriched.

(5) By using nonlinear theory, namely, viewpoint and method ofchaos theory, recognition to physical essence of outburst canbe deepened and enriched.

References

[1] Zhao XS, Zou YL. Analysis of accident reasons and countermeasures about thecoal and gas outburst in China in the last two years. Saf Environ Prot MiningInd 2010;37(1):84–7.

[2] Nie BS, He XQ, Wang EY. Present situation and progress trend and condition ofprediction technology of coal and gas outburst. China Saf Sci J2003;13(6):40–4.

[3] Jian CL. Analyses on the developing and mechanical conditions of coal gasoutburst front. J China Univ Mining Technol 1994;23(4):1–9.

[4] Jian CL. Study of forming mechanism of outburst hole. Chin J Rock Mech Eng2000;19(2):225–8.

[5] Zhu LS. Analyses on the mechanics of coal and gas outburst. Saf Environ ProtMining Ind 2002;29(2):23–5.

[6] Cheng WY, Liu XY, Wang KJ. Study on regulation about shock-wave-frontpropagating for coal gas outburst. J China Coal Soc 2004;29(1):57–60.

[7] Cai CG. Experimental study on 3-D simulation of coal and gas outbursts. JChina Coal Soc 2004;29(1):66–9.

[8] Fu S. Branch of china coal research institute development and prospect onforecast and prevention-cure technology of coal and gas outburst. Coal MineSaf 2003;34(Suppl.):14–7.

292 Y. Pan, A. Li / International Journal of Mining Science and Technology 23 (2013) 287–292

[9] Li XJ, Lin BQ. Research situation and the analysis about the coal and gasoutburst mechanism. Coal Geol Explor 2010;38(1):7–9.

[10] Qi WS, Ling BC, Cai SJ. Developing trend and perspective in the research ofpredicting the coal and gas outburst. China Saf Sci J 2003;13(12):1–4.

[11] Zhang TJ, Ren SX, Li SG, Zhang TC, Xu HJ. Application of the catastropheprogression method in predicting coal and gas outburst. J China Univ MiningTechnol 2009;19(4):430–4.

[12] Lei DJ, Li CW, Zhang ZM, Zhang YG. Coal and gas outburst mechanism of the‘‘three soft’’ coal seam in western Henan. Mining Sci Technol2010;20(5):712–7.

[13] Li DQ, Cheng YP, Wang L, Wang HF, Zhou HX. Prediction method for risks ofcoal and gas outbursts based on spatial chaos theory using gas desorptionindex of drill cuttings. Mining Sci Technol 2011;21(3):439–43.

[14] Yi CX. Nonlinear science and its applications in physicalgeography. Beijing: Meteorology Press; 1995. 176–216.

[15] Huang RS. Chaos and its applications. Wuhan: Wuhan University Press; 2000.172–174.

[16] Franceschini VC. Sequences of infinite bifurcation and turbulence in five-modetruncation of the Navier–Stoke equations. Stat Phys 1979;21(4):707–26.