open systems in non-equilibrium: complexity, dynamics

453RASTOGI & MATHUR: OPEN SYSTEMS IN NON-EQUILIBRIUMJournal of Scientific & Industrial ResearchVol. 71, July 2012, pp. 453-467

*Author for correspondenceE-mail: [email protected]

Open systems in non-equilibrium: complexity, dynamics, modeling andmechanism

R P Rastogi1* and P Mathur2

1Chemistry Department, Gorakhpur University, Gorakhpur 273 009, India2Department of Mathematics and Astronomy, Lucknow University, Lucknow 226 003, India

Received 05 December 2011; revised 15 May 2012; accepted 21 May 2012

This communication presents modeling of chaos in the context of mechanism and predictability of dynamic evolution. Basicreasons of temporal chaos on the basis of Lorenz, Rössler’s attractor, Parallel autocatalytic attractors and Chua’s circuit has beenanalyzed, and observed that primary factors responsible for chaos are (i) parallel auto-catalytic reaction or (ii) cross-catalysis(generation of x by y and generation of y by x) or (iii) combination of (i) and (ii). Applicability of modified Lorenz and Rösslermodels by combining (i) and (ii) has been discussed for socio-political and socio-economic systems. A bifurcation phenomenon inchaotic social systems has also been analyzed.

Keywords: Chaos modeling, Cross-catalysis, Open systems in non-equilibrium, Parallel auto-catalytic reaction

IntroductionPhysico-chemical systems are complex systems

involving multiple components and multi-processes suchas tug of war between forces and counter forces whereinteraction between positive and negative feedback isinvolved. The concepts yield acceptable mechanism fortime-invariant steady states, oscillations and chaos, andalso hold good for similar phenomena in socio-politicaland socio-economic systems. Dynamics of natural andopen systems in non-equilibrium is of considerableinterest in this age of globalization and growth ofinformation technology1-6. Open systems cannot be stablefor a long time, and become more and more complexdue to induction of new forces (causes), and theirinteraction with earlier forces (causes) along withinteraction with components and processes. Suchphenomena have been examined in terms of causalityprinciple and causal relations involving interactionbetween causes and effects7.

Advances in dynamics of open systems in non-equilibrium from theoretical and experimental angles2,8,9,have potential for application to social sciences. In farfrom equilibrium region, chaos can occur. There has beengood deal of impact of Chaos theory1,4,10,11 in non-linear

science. For dynamics of chaos in complex systems,various models (Lorenz model12, Rössler model13, Chua‘scircuit14 and three-step parallel autocatalytic reactionmodel15) have been studied. Chaos theory also stimulatedgood deal of interest in social sciences. However, it simplydealt with metaphor analysis, sociology of knowledgegeneration through various disciplines16-19 ,historiography20 and time-series21 . Philosophicaldiscussion related to following two aspects of theory:i) Order out of chaos (Ilya Prigogine); and ii) Hidden-order within chaos (Lorenz, Menderbolt). Althoughefforts to develop appropriate models for chaos inaeronautical engineering and earthquake dynamics havebeen made, no serious effort has been made to developsuitable model for chaos in social system for typical cases.Dynamics of chaos in social systems has been studiedby using modified Lorenz model and modified Rösslermodel22-24.

This communication presents dynamics of opensystems including social systems in non-equilibrium in awide range in the context of modeling and mechanism,role of autocatalytic reactions and cross catalysis indifferent situations when new processes appear.

Nature of SystemsIn nature, systems range from very simple to highly

complex systems. Basic sciences deal with simple

454 J SCI IND RES VOL 71 JULY 2012

systems, which can be analyzed by simple techniquesand include: i) Physical systems involving fewcomponents, and fewer processes (fluid flow and heatflux); ii) Chemical systems involving supramolecules andmolecular networks25-30, and can be designed to performcomplex functions, which might yield useful informationregarding origins of life; iii) Physico-chemical systemsinvolving multi-component multi-processes and multipleinteractions between atoms and molecules; iv) Biologicalsystems having subsystems containing bio-molecules anddifferent species, multi-components and multi-processes,deeper study of which involves multi-disciplinaryapproach; v) Social, socio-political, socio-economicsystems comprising many subsystems (family, tribes,nations, and universe) involving a very large number ofcomponents, processes, subsystems and variables notalways easy to identify and define; and vi) Under openand closed systems31, a system is considered an opensystem if it interacts with its environment, and closedsystem if it does not (e.g., an automobile plant is moreopen system than a prison).

Each system has a boundary that separates it fromenvironment. In a closed system, system boundary isrigid whereas in an open system boundary is more flexible.A system has flows of information materials and energy(including human energy). These enter from theenvironment as inputs (raw materials). Feedback is thekey to systems control. As operations of a systemproceed, information is feedback to appropriate personin industrial establishment. Analysis of complex systemsrequires systems thinking and capability to dissectcomplex system into subsystem in order to developmacro-level thinking. Set of subsystems can be formalmodels of selected natural system, in which exchange ofmatter, energy and information can take place. Sinceformal system usually shows changes in the configurationof subsystems over time, then these can be used as atemporal process law for deeper over all changes insubsystem itself. Subsystems have to be open systemsand the closed ones.

Physico-chemical studies show2 that open systemsincluding natural systems can be of two types:i) discontinuous systems that occur in membranephenomena like thermo-osmosis, in which case twosubsystems are separated by a membrane, through thepores of which fluid transport takes place; andii) continuous systems such as thermal diffusion and soreteffect2, in which case no barrier between two subsystemsexists. Formal system can again be of two types:

i) conservative; and ii) dissipative system. Conservativesystem (Lorenz model) preserves the volumes in phase-space of dimension3 or more during volume flow. On theother hand in dissipative systems of physical interest,the solution flows create overall contraction in volumesof the phase space although there might be localexpansion. Typical example of non-linear conservativesystem is Lorenz attractor. Thermodynamic analysis ofsystem in non-equilibrium is made by characterizingbehavior of the system in terms of stimulus (cause) andcorresponding response of the system. If response ofthe system is a linear function of stimulus, then system islinear. On the other hand, when this is not so, system isnon-linear.

VariablesIn physico-chemical systems, broad range of

variables32 can be categorized as extensive (when valueis additive such as volume, mass, energy, entropy,enthalpy) and intensive (referring to points in the systemsuch as molar volume, density, pressure and temperature).On the other hand, in socio-political and socio-economicsystems, number of variables1 is quite large as materialvariables (infrastructure, capital, natural resources),monetary variables (prices, salaries, interest rate etc.),variables linked to knowledge (level of education, skillsand technology), statistical variables (birth and death rate,population structure in terms of age, sex etc.), and socio-cultural variables (legal provisions, tax system, habits,traditions and ethical values).

Interaction between ComponentsEnvironment plays a predominant role in influencing

behaviour of components in chemical and social systems.In chemical systems, molecules can be classified in asolution in three groups: i) free molecules whose rotationsand vibrations are not altered by the presence ofneighbouring molecules; ii) associated molecules whosestates of vibrations or rotations are altered by neighbouringmolecules such as alcohols; and iii) Molecules whosestate of vibration or rotation is altered by unlike molecules(these are supposed to form association complexes).Such complexes are formed by dipole-dipole interaction,hydrogen-bond interaction and charge-transferinteraction. Complex formed between dioxane andchloroform is a typical example 33,34. Interaction can takeplace between two solid species in powdered state35 viavapour phase diffusion, grain boundary diffusion andsurface migration. Detailed geometry of moleculesinfluences the course of interaction, on account of which

455RASTOGI & MATHUR: OPEN SYSTEMS IN NON-EQUILIBRIUM

bioinformatics (Metabonomics and Metabolomics) hasattained great importance in drug industry. Morecomplexity arises because enzymes catalyze one or morereactions, and on the other hand, a reaction is catalyzedby one or more enzymes. Thus complexity in a system isenhanced by increase in complexity in interaction onchemical systems. Similar situations occur in natural andsocial systems.

Correlation CoefficientsIn a similar manner, complexity increases as one

move from one end from simple and ideal systems tomore and more complex systems. In former case,correlation coefficients involved in Poiseulle’s flow arevery well defined as parameters (viscosity, length andradius of capillary). With increasingly complex systems,task becomes increasingly difficult and maximumdifficulty arises in case of social systems.

Statistical Analysis: Research Methods in Social PsychologyFor identifying variables in socio-political and socio-

economic systems, one has to use research methodsfollowed in social sciences36 [systematic observation,survey method, correlation methods, hypothesis,experimentation for detection (dependent variables &independent variables), inferential statistics (that allowsone to evaluate the likelihood that a given pattern ofresearch occurred by chance alone), and meta-analysis(a statistical technique for combining data fromindependent studies in order to determine whetherspecific variables or interaction between variables havesignificant effect)]. Statistical analysis (preparation ofquestionnaire, rating of relative significance of questions,subjectivity of questioning, subjectivity of respondents,choice of sample areas or regions, and choice of samplepersonnel/ respondents) is the most preferred tool forgenerating perception index.

Coupling of Time Derivatives of VariablesTime-derivatives of variables corresponding to flows

involved in irreversible thermodynamics and couplingphenomena is also not easy to comprehend in complexsystems. The rules are clear within the framework ofnon-equilibrium thermodynamics. Following Curie’stheorem, scalar forces cannot be coupled with vectorialforces7.

ModellingConceptual basis and intuition based on keen and

analytical observation provide fundamental basis for

modeling a phenomenon. In practice, typical models basedon increasing order of complexity can be classified asfollows: i) simple physical models (linear models); ii)chemical systems involving supramolecules and molecularnetworks; iii) physical non-linear models (based onadvanced and computational mathematics); iv) physico-chemical models involving multiple components andmultiple processes; v) in vitro biological models usingsome components of biological system; vi) biologicalmodels involving living biological systems and knownchemical and biological species; vii) socio-economicsystems involving complex variables and not well definedcorrelation coefficients but primarily governed bystatistical considerations; and viii) social and socio-political systems involving variables, coefficients andcausal relations governed by cautious analysis. In reallife situations, open systems in non-equilibrium aresometimes in steady state and sometimes in non-steadystate. Steady state is when forces and counter forcesinteract. However, when multiple forces are operativeinvolving autocatalysis (positive feedback) and inhibitorysteps (negative feedback) exotic nonequilibriumphenomena are observed2. Extensive theoretical studiesbased on non-equilibrium thermodynamics and non-lineardynamics and experimental studies related to simple andcomplex non-equilibrium systems are reported2. Casuallaws can be represented as , where X is force,J is flow and L is constant. When two types of forces

and fluxes are coupled then relationsobtained are and

, where linear case and are cross coefficients, between which Onsagerreciprocity relation holds.

Forces can produce more than one type of effect: a)forces generate same effect, reinforcement of the effectoccurs; b) one or more forces produces opposite effect,in course of time, steady state is observed under specificconditions; and c) effect itself can act as it happens inthe case of autocatalytic reactions or in the growth ofpopulation (prey-predator model)2. For non-linear rangeand for the case with more than two forces,

,

where n is number of

variables. Two examples of appropriate non-equilibrium

are and


,

where are constants.In any cyclic (periodic) phenomena, forces and

counter forces operate as2: a) simple harmonic motion incase of pendulum; and b) self-excited oscillations in livingsystems. Normally there is a tendency of balance, whichleads to periodicity, but when cycle is disturbed by somestep, chaos is developed. Lorenz attractor and Rosslerattractor were discovered but no effect was earlier madeto ascertain the nature of operating forces and counterforces.

Thus for analysis of any non-equilibrium phenomena,fluxes and forces need to be identified in order to haveappropriate non-linear relation between fluxes and forces.Attempts have been made to investigate chaotic dynamics,but little attention has been paid to investigate basicphysical mechanism leading to chaos. This communicationattempts to specify such mechanism in different models.Cyclic and elliptic trajectories give an indication ofautocatalysis, whereas hyperbolic trajectories give anindication of cross-catalysis.

Cause-Effect SequenceIn non-equilibrium phenomena in living systems and

social systems, different processes involved are governedby cause-effect sequence6. When one moves away fromequilibrium in non-linear region, one of the simple

situations corresponds to steady state when two opposingforces balance each other. In many cases6, a particularflow (cause) may generate force (effect), which itselfmay generate opposing flow in reverse direction. Inthermo-osmotic phenomena in membranes,hydrodynamic fluid flow J1 occurs from high pressureside to low pressure side (Fig. 1a), while duringthermo-osmotic flow, fluid flow occurs from lowtemperature compartment to higher temperaturecompartment on the left hand side (Fig. 1b). Thus in asystem, where temperature T2 and pressure P2 are higherthan the corresponding values in compartment II, therewould be opposing, which would be balanced in steadystate (Fig. 1c). Thus flow J1 would depend on two forces(X1 =∆P = P2-P1 and X2=∆T= T2-T1), so that net flow atany moment would be given as J1 = L11 ∆P + L12 ∆T,where L11 and L12 are phenomenological coefficients.Thus in steady state, thermo-osmotic pressure per unitpressure difference would be given as

11

12

0 LL

TP

J

−=

∆∆

=. …(1)

Electro-kinetic phenomena in membranes providemodel experiments for understanding occurrence of non-equilibrium steady state. In this case, both fluid flow(J1 = L11 ∆P + L12 ∆φ) and flow of current (J2 = L21 ∆P +L22 ∆φ)

Fig. 1—Opposing flows and steady state in thermo-osmosis: a) hydrodynamic fluid flow from higher pressure to lower pressure; b)thermo-osmotic flow; and c) attainment of steady state

a) b)

c)


occurs in both direct and reverse direction, where ∆P =pressure difference, and ∆φ =potential difference. In onecase, initially imposed ∆φ causes electro-osmotic flowof fluid from left compartment to the right compartment,on account of which pressure difference is generatedand reverse hydrodynamic flow is generated. Ultimatelysteady state is established and steady electro-osmoticpressure is developed (Fig. 2) as

…(2)

In second case, when flow of electric current is takenalone, current initially flows from left to right compartmentand pressure difference generates streaming potential,which in turn is responsible for flow of streaming current

in opposite direction. Here X2 represents ∆φ and J2represents flow of net current. In steady state, steadypotential difference is built up and given as

22

21

02LL

P J

−=

∆∆

=

φ…(3)

Non-equilibrium thermodynamics2,8 providetheoretical and mathematical modeling. Phenomenologyof non-equilibrium thermodynamics also providesmodeling of similar non-equilibrium steady statephenomena (thermal diffusion, Dufour effect,electrophoresis and streaming potential and thermo-electricity). From causes (force), effects (flow), directand reverse flows in different phenomena (Table 1),steady state is quite different than equilibrium state. In

Table 1—Typical phenomena showing generation of cause by the effect of earlier cause

S. No. Phenomenon Initial Corresponding flow Subsequent Corresponding flow in cause (effect) cause opposite direction

1 Thermo-osmotic ∆T Thermo-osmotic ∆P Hydrodynamic flowpressure difference Fluid flow

2 Mechanic-caloric effect ∆P Heat flow ∆T Heat flow3 Electro-osmotic ∆T Mass flow ∆C Mass flow

concentration difference4 Electroosmotic pressure ∆ϕ Electro-osmotic ∆T Hydrodynamic flow

difference Fluid flow5 Streaming potential over ∆P Streaming current ∆ϕ Ohmic current

unit pressure difference6 Thermal diffusion Grad. T Mass flow Grad. C Mass flux7 Dufour effect Grad. C Heat flux Grad. T Heat flow8 Electrophoresis ∆ϕ Mass flux against g Sedimentation

gravity9 Sedimentation potential g Sedimentation ∆ϕ Ohmic current

current10 Thermoelectricity ∆T Electric current ∆ϕ Ohmic current11 Peltier effect ∆ϕ Heat flow ∆T Heat flow

∆P= Pressure difference, ∆T = Temperature difference, ∆ϕ = Potential difference, Grad. C = Concentration gradient, Grad. T = Temperaturegradient, g = Force due to gravity

Fig. 2—Cause-effect sequence in cross-phenomena; a) steady state streaming potential when starting cause is X1; and b) steady state electroosmotic pressure difference when starting cause is X1

11

12

01LLP

J

−=

∆∆

=φ

a) b)


steady state, balanced opposing flows continue to takeplace, but in equilibrium, no flow takes place and causesdisappear, as happens in chemical equilibrium where bothforward and back reactions stop. Steady state do occurin suitable chemical systems, which involve positivefeedback (auto-catalytic reaction) and negative feedback(inhibitory reaction), which are balanced. When balancecontinues to be disturbed, several time-invariant statesare observed.

BistabilityA system can exist in two stable steady states

depending on external circumstances and the nature ofcontrol mechanism2. For a typical simple chemicalsystem exhibiting bistability, mechanism has beenproposed2 as A + 2X ↔ 3X and X ↔ B. Thus as onemoves into non-equilibrium region, complexity isenhanced by the presence of autocatalytic reaction aswell as inhibitory reaction accompanied by externalenvironment.

Time-variant Non-equilibrium StatesAs one moves away from equilibrium states,

periodic changes are observed in the system (oscillatorychemical reactions, electro-kinetic oscillations2, weatherand stock market). Here major role is played by auto-catalytic and inhibitory reactions. A similar positive andnegative feedback process is found in social system asauto-catalytic growth of corruption, countered by goodgovernance. This can be illustrated by using oscillatoryreaction model (Brusselator) as

AB+X

2X+YX

3XE

Y+DX

a. Scheme A + B

X + B

Y + D

Y + 2X 3X

EX

Positive feedback

Negative feedback

Autocatalysis

Inhibition

b. Reaction network

Reaction network involves hard core of controlmechanism, containing auto-catalytic reaction andinhibitory reaction. Similarly, hard core of reaction

mechanism can be identified for Belousov-Zhabotinskiioscillatory reaction. Considering simplified version ofField-Koros-Noyes mechanism, reaction network withhard core of reaction mechanism can be represented as

Malonic acid

Bromomalonic acid

Y

A Reservoir

2X X 2P

P + A

Autocatalysis Inhibition

+A +Y

A = BrO3-

X = HBrO2

Y = Br -

Z = 2Ce4+

P = Productsf = Stoichiometric factor

X

2PA + X 2X2X P + A

A + Y

X + Y

Z fYa. Reaction Scheme

b. Reaction Network

Complex PeriodicityIn chemical and natural systems, occurrence of

periodic complex oscillations is quite common2,6. Ithappens in living systems and social systems on accountof complex reaction network or complex interrelationshipof socio-political and socio-economic processes. This ismore frequent on account of effect of globalization(systems getting more and more open) in the latter case.However, primary factor responsible is one or morecontrol37 along with complexity in reaction network.Prey-predator interaction is a good model forunderstanding auto-catalysis. Consider a situation wherea certain habitat is shared by predators and their preys.Predators feed only on preys whereas the latter feed onvegetation. Subsequently, it multiplies, then it dies or iseaten up by more powerful animal. The first is auto-catalytic increase in animal population, while in the secondprocess, stage (inhibitory process) provides a control ondisproportionate increase in population. Similar phenomenaoccur in social system such as corruption breeds morecorruption, which can be countered by good governance.Similarly, inefficiency in administration leads to greaterinefficiency, leading to development of unpredictablesituations, which can be countered by good leadership.Thus, in social systems, one can have easily more thanone auto-catalytic processes and more than one controlmechanism. In underdeveloped countries, both corruptionand inefficiency prevails, hence very often, chaoticconditions prevail. Different types of a periodicity have


been reported2. One type is the one where oscillationshaving specific frequency of oscillation are followed byoscillations of different frequency but separated by atime-pause, where each type is controlled by a specifictype of control mechanism38.

Interaction of Control MechanismsIn B-Z oscillators, in general either of the following

types of control mechanisms is found to be operative: i)Br- control mechanism; and ii) Free-radical (BrO2

•)control mechanism. It involves different types ofauto-catalytic and inhibitory reactions. Dual frequencyoscillations separated by a time-pause involving B-Zoscillator with fructose and oxalic acid39 and xylose +oxalic acid40 are used as organic substrate. The casewhen glucose +oxalic acid oscillator is used as organicsubstrate reveal that initially oscillations controlled byfirst type of control mechanism occur, while after a certaintime-interval, second type of oscillations controlled bysecond type of control mechanism are observed38. Propermodeling and computational study is needed to get deeperinsight into nature and dynamics of chaos41,42.

Dissipative Structure and Self-OrganizationNormally, when internal production of entropy (diS>0)

disorder prevails, order can be restored, if there isdissipation of entropy to the surrounding in some manner.This happens during the formation of chemical wavesand Leisegang rings through diffusion2. Detailed modelsproposed of structure based on specific reaction onmechanism also involve similar cause-effect sequence.In chemical waves, again autocatalytic and inhibitoryreactions play significant role. However, involvement ofan additional process (diffusion) creates a new situationand coloured bands are formed. For more complexstructure (fractals) having very complex geometry,diffusion limited-aggregation model have been used2. Forstill complex structure (dendritic structure) formed duringpolymerization of Polypyrrole, DLA models along withcomplex reaction mechanism (Diaz’s mechanism) hasbeen employed43.

Deterministic ChaosChaos is deterministic if it obeys some laws that

completely specify its motion governed by specificmathematical equations. 2D and 3D plots under differentconditions yield different type of geometrical figures,which are called attractors. Chaotic attractor is a curve(orbit) of infinite length, because evolution of dynamical

system may take infinite time. It is an open curve ofinfinite length contained in a finite volume. The curvefolds back on itself. Specific feature of a chaotic attractoris its sensitive dependence on initial conditions. In manycases, an attractor, called Strange attractor, is a subsetof points in the phase space that is fundamentally differentas compared to that of objects belonging to Euclideangeometry and has fractal dimension. Strange attractorand deterministic chaos are yielded by some typicalmathematical models (Lorentz’s attractor, Chua’s circuit,Rossler attractor and Three-step parallel auto-catalyticreaction model14]). In first two cases, cross-catalysis[XàY (X generating Y) & YàX (Y generating X)] isinvolved. Side reactions create disturbance, related withgeneration and decay of X and Y.

Lorentz AttractorOscillator is concerned with stability analysis and

onset of convection or turbulent motion in a fluid heatedfrom the bottom and cooled from the top. In this idealizedmodel of a fluid, warm fluid rises and cool fluid alonesinks setting up clockwise or counter-clockwise current.Non-linear equations involved describing the phenomenaare

yxdtdx

σσ +−= …(4)

yrxxzdtdy

−+−= …(5)

bzxydtdz

−= …(6)

where x is amplitude of convective motion, y istemperature difference between descending andascending current, z is distortion of vertical profile fromlinearity, s is Prantl number, r is Rayleigh number and bare specific parameters of the system constants. Here,x promotes y and in turn y promotes x. Specific featuresof this model become evident when one uses it to describea reaction system involving similar species and steps asfollows:

(R3) ............................. ;y x Generation xrdtdy

= →

............... ; Decaysx

.......... ;x y Generation

xdtdx

ydtdx

σ

σ

=→

= →

Decay) (Self .....(R2)

.....(R1)

I.


II.

III.

Chua’s CircuitA rather simple electronic circuit, proposed by Chua13

during 1980’s, allows almost all of the dynamicalbehaviour seen in computer simulation, which could beimplemented in an electronic lab. Chua’s circuit is anRLC circuit with 4 linear elements (2 capacitors, 1 resistorand 1 inductor) and a non linear diode. The experimentcan be modeled as

( ){ }xgxycdtdx

−−= 1 …(7)

{ }zyxcdtdy

−−= 2 …(8)

zcdtdz

1−= …(9)

where, similar to the former case, x generates y and ygenerates x.

Non-isothermal first-order reactions with Arrheniustemperature dependency display chaos when two or morereactions proceed independently. Chaotic behaviour isdemonstrated for: 1) two parallel exothermic reactions14;2) two consecutive exothermic reactions; and3) exothermic-endothermic consecutive reactionsinvolving thermal feedback. The third one essentiallyinvolves in two oscillators.

Rössler AttractorOtto Rössler model, featuring a chaotic attractor2,

has a set of differential equations as

)( zydtdx

+−= …(10)

ayxdtdy

+= …(11)

czxzbdtdz

−+= …(12)

where a = 0.2, b = 0.2 and c = 5.7 are constants, withinitial condition (-1, 0, 0). A chaotic time series in x, y, z isgenerated on solution.

Here again, if one considers model from the angleof a reacting system, it is found that it involves auto-catalysis of y and z and a complex reaction network as

DecayzbticAutocatalyzx

ticAutocataly

→→⇔+

⇒→→

yzxyx

Simple Mathematical Reaction ModelsA model of a reaction system consists of two parallel

isothermal autocatalytic reactions in CSTR14 . Self-sustained chaotic behaviour can occur in the system,when parameters are changed. Parallel autocatalyticreaction model involving dual control mechanism is givenas

A + 2BD + 2B B

3B3BC

Chaos can occur over wide ranges of values ofcertain parameters with very complex transitions betweenchaos and periodicity, which occur as parameters arevaried. But there are following two basic mathematicalproblems with the model: 1) Examination of chaoticbehaviour in systems of ordinary differential equation ishampered by the lack of techniques for determining aperiod, when chaos is possible, and if so, region ofparameter space where it occurs; and 2) Anotherdifficulty arises while examining the possibility of chaoticbehaviour involves in discrimination of long-range periodicbehaviour from chaotic behaviour. In this context,computation of Lyapunov exponent2 leads to straight-forward conclusion, since presence of positive exponentindicates the presence of chaos.

In all the four above cases, 3D non linear differentialequation containing three dynamical variables areinvolved. In case of Lorenz oscillator and Chua’s circuit,

Decay) (Self ....(R7).......... ; Decaysz bzdtdz

−=→

Decay) (Self .....(R4).................... ; Decaysy ydtdy

−=→

( ) (R5) ....... ; z andy ofDecay zx zxdtdy

+−=→+

( )[ ] ).......(R6yx..or ; zyx Generation zxydtdz

+⇔= →+


cross-catalysis (generator of x by y and generation of yby x) is the basic cause of development of chaos. Whilein other three cases, two autocatalytic steps are primarilyresponsible for generation of chaos.

Modified Lorenz ModelModified Lorenz model (Fig. 3; 3 variables & 5

parameters) is obtained by considering autocatalysis ofvariables x and y, which yield cross-catalysis among twovariables x and y as

xcxydtdx

+−= )(σ …(13)

yeyzxdtdy

+−−= )(ρ …(14)

zxydtdz

β−= …(15)

where c and e are additional constants than in originalLorenz model. In this system, following processes wereexpressed: a) Eq. (13) indicates generation of x by y andautocatalytic production of x when ; b) Eq. (14)indicates generation of y and x, and autocatalyticproduction of y when e >1; and c) Eq. (15) indicatesgeneration of z by y and x, and decay of z.

Modified Rössler ModelTaking cross-catalysis into account, Rössler’s model

was modified22 by considering the system of non-lineardifferential equations as

zqypdtdx

−= …(16)

ayxdtdy

+= …(17)

zcxzrbdtdz

−+= …(18)

where p, q, r are additional constants as compared tooriginal Rössler model and b denotes public opinion andlevel of moral and ethical values. Also, p = k – 1, wherek is a constant involved in the additional term added toEq. (4) so as to obtain cross-catalysis between x and y.Thus modified Rössler model (Fig. 4; 3 variables & 6parameters) where following processes were expressed:a) Eq. (16) indicates generation of x by y and decay of xby z; b) Eq. (17) indicates generation of y by x andautocatalytic production of y; and c) Eq. (18) indicatesconstant source of inspiration, autocatalytic productionof z and decay of z.

Chaos in Social SystemsLiving system is made up of number of subsystems,

which are interconnected involving different types ofcontrol mechanism working in a harmonious manner.Once this is disturbed, acute health problems appear.Chaos theory can easily be applied to social systemsafflicted with subsystems, corruption and badgovernance. Corruption promotes bad governance andin turn bad governance promotes corruption. Subsystem,political will or administrative will, counter the effect ofcorruption and bad governance.

CorruptionCorruption is an illegitimate favor for personal gains

for doing an official work. It is illegitimate transaction ofcash or kind and also includes nepotism (parochialism)favoring family, caste, creed, language, place of birth orregion and cronyism (favoring friends).

Fig. 3—Schematic representation of Modified Lorenz model

Fig. 4—Schematic representation of Modified Rössler model


Bad GovernanceBasic features of bad governance are: i)

Unnecessary delaying tactics (red tapism); ii) Digressionof matter; iii) Twisting wrong interpretation of laws, rulesand regulations; iv) Unnecessary referring for opinion /concurrence to other agencies / officers / departments(like law, finance, personnel etc.); v) Making/coveringreal issue with bad intension; vi) Demanding irrelevantpapers / documents for verification; and vii) Directlydemanding cash or kind or share (%) in deal. This isfacilitated by nexus between higher level and lower levelbureaucracy.

Political and Administrative WillPolitical will is governed by political and committed

leadership and supported by social leadership, providinggood leadership. As defined by the World Bank in 1999,“Good governance is epitomized by predictable, open andenlightened policy making (transparent process); abureaucracy imbued with a professional ethos; anexecutive arm of government for its action; and a strongcivil society participating in public affairs and all behavingunder the rule of law”.

Public OpinionPublic opinion at different levels influences political

will to combat corruption. At macro-level, public opinionis reflected through print and electronic media, while atmicro-level, forces at macro level, and forces of socialpsychology influence public opinion. Public opinion alsoinfluences environment, through which flows occurs.

Cyclic Changes in Social SystemIn a corrupt social system, cyclic change occurs

(Fig. 5) due to flow of corruption in one direction (positivefeedback) opposed by the steps taken for prevention andreduction of corruption due to political and administrativewill (negative feedback) when cycle is disturbed bychanges in the parameter (chaos).

Theoretical AnalysisModified Lorenz Model

In this case, Eqs (13) - (15) yield three fixedpoints , where ,

.

For local stability analysis, the system (13) - (15) islinearized around a fixed point and Jacobian matrix M isobtained at each steady state as

where ss represent corresponding fixed point.Characteristic equation corresponding to M is

which reduces to

…(19)

where; ,

The solution of Eq. (19) gives three eigenvalues(EVs) corresponding to each fixed point. There are twopossibilities, either all EVs are real or one EV is real andtwo are complex. From these EVs, it follows that: i) Ifall EVs are real and negative, there is an exponentialdecay; ii) If all EVs have negative real parts then thereis decay with oscillations; iii) If all EVs are real and atleast one positive then there is an exponential increase;iv) If at least one EV has positive real parts, there is anincrease with oscillations; and v) If all EVs have zeroreal parts and non-zero imaginary parts then there is stableoscillations. Let σ = 10, β = 8/3, ρ = 28 be fixed. If c >σ and e > 1, 3 EVs obtained at each fixed points arereal. Two of these EVs are positive and one is negative.Thus by Routh-Hurwitz44 criterion, fixed points are

Fig. 5—Flow and counter flow of corruption

and


unstable saddle points. Similar results are also obtainedwhen c < σ and e is either > or < 1 (Fig. 6).

Modified Rössler ModelIn this case, differential Eqs (16) - (18), yield

fixed points E4 and E5 given as

a n d

, where

For local stability analysis, following modified Lorenzmodel, characteristic equation is given as

…(20)

where, ,

and

The solution of Eq. (20) gives 3 EVs correspondingto each fixed points. Computational results for modifiedRössler model are given as: i) If p varies (q = 0.2, a =0.001, c = 0.2, b= 0.2, r=0.8) then for p ≤ - 0.5, tori isobtained; when -0.5 <p < 0.1, stability is achieved; andwhen p > 0.1, spiralling chaotic attractor is obtained(Fig. 7); ii) If q varies (p = 0.1, a = 0.001, c = 0.2, b=0.03, r=0.8) then trajectory is transformed to a straight

Fig. 6—Attractors in modified Lorenz model ( ) having: i) c = -1.43, e = 0; ii) c = 2.83, e = 0; iii) c = 0, e = 10.11;

& iv) c = 0.5, e = 2.8

Fig. 7—Attractors (q = 0.2, a = 0.001, c = 0.2, b= 0.2, r=0.8) with initial condition (0,1,0) for: i) p = 0.1; and ii) p=0.2


line (linear with respect to y and z) (Fig. 8); iii) If avaries (p = 0.2, q = 0.2, c = 0.2, b= 0.08, r=0.8) then if -0.05 < a < 0.01, spiralling chaotic attractor is obtainedwhereas for 0.01 < a < 0.08 distorted strange attractor

is obtained; iv) If b varies (p = 0.1, q = 0.2, c = 0.2, a=0.001, r=0.8) then spiralling attractor occurs if 0 < b <0.15 and if b < 0 chaotic character does not exist,whereas if b > 0.15 stability is achieved (Fig. 9); and iv)

Fig. 8—Attractors (p = 0.1, a = 0.001, c = 0.2, b= 0.08, r=0.8) with initial condition (0,1,0) for: i) q = 0.1; q = 0.3; and ii) q=0.4

Fig. 9—Attractors (p = 0.1, q= 0.1, a = 0.001, c = 0.2, r=0.8) with initial condition (0,1,0) for: i) b = 0.05; ii) b = 0.1; iii) b = 0.16; and iv)b=0.2


If r varies (p = 0.1, q = 0.2, a = 0.001, b =0.03, c = 0.2)then a fixed point attractor for r > 7.8, whereas for r <7.8 spiralling chaotic attractor occurs (Fig. 10).

Results confirm important contribution of cross-catalysis and autocatalysis in controlling chaoticdynamics. Thus two basic regimes (stable & unstable)are obtained. In unstable regime, there are two sub-regimes as predicted by mathematical analysiscorresponding to spiralling attractor. The first occurswhen autocatalysis predominates while hyperbolictrajectory of strange attractor is obtained when cross-catalysis predominates. Thus mathematical analysis andcomputational results both provide information on (i)changes in trajectory of attractor in unstable region, (ii)transition from spiraling attractor to spiraling repeller and(iii) new types of bifurcation. Important features duringcomputational analysis of trajectories are as follows: i)Axis of x shrinks, trajectory of y and z variables (bothspiralling and repelling) become predominant; and ii)Shrinkage occurs to such an extent that a linear relationbetween y and z is obtained, which is characteristic of anew type of dynamic stability (linear variation). Thusregarding dynamics of chaos in social systems based onmodified Rössler model, it is concluded that: i) Instabilitydecreases when rate of cross-catalysis is decreased; ii)Instability decreases when rate of autocatalysis isdecreased; iii) Stability is increased when political will z

is enhanced by public opinion b; and iv) Stability isincreased when r increases.

Application to Social SystemIn underdeveloped societies, corruption and bad

governance develop auto-catalytically. On the other hand,when it is not controlled by inhibiting action of goodgovernment and decision-making, chaotic conditionsprevail. In decision-making, foresight and quick decisionare very important. If it is absent, problems develop auto-catalytically and if appropriate negative feedback is notthere, chaotic conditions can result due to interaction ofseveral control mechanisms, as it happens in physico-chemical systems. Eqs (13-15) related to Lorenzattractor in a way represent chaotic situation44 dominatingsociety, when corruption (x), bad governance (y) andpolitical will (z) are important variables. Even in this casex generates y and y generates x.

Fig. 10—Attractors (p = 0.1, q= 0.2, a = 0.001, b= 0.03, c = 0.2) together with initial condition (0,1,0) for: i) r = 0.08; ii) r = 2.58; andiii) r = 8.8

autocatalytic

autocatalytic

crosscatalysis

y

x

z (controlling parameter)


Similarly, situation like current economic recessioncan prevail when control mechanism in one nationinteracts with control mechanism in another, governedby the process of globalization. In a sense, Rosslerattractor [Eqs (10)-(12)] represents such a situation,where x represents globalization, y and z representsNation I and Nation II respectively. Evidently,autocatalytic processes controlled by specific mechanismare involved in the phenomenon. In real social and socio-political systems, both cross-catalysis (in case of Lorenzoscillator) and parallel auto-catalysis (in case of Rosslerattractor) may be simultaneously operative in specificcases.

ConclusionsNatural systems contain number of subsystems and

are intrinsically complex, governed by complex non linearcausal relations involving correlation coefficient, whichare not always easy to identify. Various types of models[physical, physic-chemical, biological (in vivo and invitro), mathematical, computational and statisticalmodels] have been employed to analyze complexity inthe systems including chaos. Lorenz model, Chua’s circuit,Rössler’s model and parallel autocatalytic reaction modelreveal the role of autocatalytic reactions (positivefeedback), inhibiting reactions (negative feedback) andcross catalysis in generating chaos. Based on these ideas,mathematical modeling of chaos in socio-political andsocio-economic system can give meaningful results ofpractical value (management and decision making).

AcknowledgementsAuthors thank INSA, New Delhi for financial support.

Authors also thank Dr Mukul Das of IITR, Lucknowand Mr Ashish Yadav for helping in manuscriptpreparation.

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