Lyapunov Indices and the Poincar\'e Mapping in a Study of the Stability of the Krebs Cycle

0.55 MB·

by  V. I. Grytsay

NONLINEAR PROCESSES
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1 doi: 10.15407/ujpe60.06.0561
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V.I. GRYTSAY
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BogolyubovInstitute forTheoretical Physics,Nat. Acad. ofSci. ofUkraine
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(14b, Metrolohichna Str., Kyiv03680, Ukraine; e-mail: [email protected])
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J LYAPUNOV INDICES AND THE POINCARE´ MAPPING

8 PACS05.45.-a,05.45.Pq, IN A STUDY OF THE STABILITY OF THE KREBS CYCLE
2 05.65.+b

]
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On the basis of a mathematical model, we continue the study of the metabolic Krebs cycle
M
(or the tricarboxylic acid cycle). For the first time, we consider its consistency and stability,
. whichdepend onthe dissipation of a transmembrane potential formedby the respiratory chain
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in the plasmatic membrane of a cell. The phase-parametric characteristic of the dynamics of
i
b the ATP level depending on a given parameter is constructed. The scenario of formation of
- multipleautoperiodicandchaoticmodesispresented. Poincar´esectionsandmappingsarecon-
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structed. Thestabilityof modesandthe fractalityofthe obtained bifurcationsare studied. The
[
full spectra of Lyapunov indices, divergences, KS-entropies, horizons of predictability, and

1 Lyapunov dimensionalities of strange attractors are calculated. Some conclusions about the
v structural-functional connections determining the dependence of the cell respiration cyclicity
4 onthesynchronization ofthefunctioningofthetricarboxylicacidcycleandtheelectrontrans-
5 port chain are presented.
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Keywords: Krebscycle,metabolicprocess,self-organization, strangeattractor,bifurcation,
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0 Feigenbaum scenario.
.
2 The most important task of synergetics is the search atoms released in this case are transferred into the
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for the general physical laws explaining the natural respiratory chain, where the main source of the en-
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regularityoftheformationofalifeontheEarth. The ergy for a cell, ATP, is produced in the course of
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: first descriptive experiment demonstrating the possi- the reaction of oxidative phosphorylation.With the
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bility for a cyclic metabolic process to exist was ex- help of NADH, there arise the negative feedbacks,
i
X ecuted by B.P. Belousov in 1951 [1]. With the help due to which the synchronization of the process of
r of some chemical substances (citric acid, potassium catabolism and the respiration of a cell occurs. The
a bromate, cerium, etc.), he successfully constructed a jointexistenceofthesemetabolicprocessesispossible
model of autooscillatory metabolic process involving onlyundertheirself-organizationinasinglecycle. In
the Krebs cycle [2]. By this, he showed for the first addition, the Krebs cycle is a source of molecules-
time that the vital metabolic processes in a cell can precursors, which are used in the synthesis of com-
be supporteddue to their self-organizationinthe au- pounds important for the vital activity of cells in
toperiodic mode. other biochemical reactions.
The tricarboxylic acid cycle occupies a particular Studying the functioning of the tricarboxylic acid
place in the vital activity of aerobic cells. As a re- cyclewascarriedoutinexperimentsandtheoretically
sultofthecyclicmetabolicprocess,theacetylgroups [3–11]. In particular, the given process was analyzed
formed in the decay of carbohydrates, fats, and pro- on the basis of the mathematical model of growth
teins are oxidized there to carbon dioxide. Hydrogen of Candida utilis cells on ethanol. This model was
developed by Professor V.P. Gachok [12, 13]. The
(cid:13)c V.I.GRYTSAY,2015 analogous approaches to the modeling of a growth
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V.I.Grytsay
of cells were considered by J. Monod, V.S. Pod- permanent self-organization between each other and
gorskii, L.N. Drozdov-Tikhomirov, N.T. Rakhimova, dependontheintensityofadissipationoftheelectro-
G.Yu. Riznichenko, and others [14–18]. Within such chemical gradient of the potential in metabolic pro-
models, the unstable modes in the cultivation of cesses.
cells, which were observed in experiments, were stu- We will study the structural-functional connec-
died. The results of computational experiments con- tions, according to which the Krebs cycle and the
cerning the chaotic dynamics described well the ex- respiratory chain are self-organized and operate as
perimental characteristics [19]. a single complex providing a cell with the necessary
Then the Gachok model was improved in [20, 21], energy store for its life. The limits of stability of the
where the influence of the concentration of CO2 on cycle depending on the dissipation of the proton po-
thecellrespirationintensityandthecyclicityofares- tential in various processes of the vital activity of a
piratory process was taken into account. The struc- cell are considered as well.
tural-functionalconnections of the metabolic process
1.Mathematical Model
running in a cell, according to which the appearance
ofcomplicatedoscillationsinthemetabolicprocessin The general scheme of the process is presented in
acellbecomespossible,werefound. Itwasconcluded Fig. 1. According to it with regard for the mass bal-
that those oscillations arise at the level of redox re- ance, we have constructed the mathematical model
actions of the Krebs cycle, reflect the cyclicity of the given by Eqs. (1)–(19).
process, and characterize the self-organization inside
a cell. For some modes, the fractality of bifurcations dS K
=S0 −
wasstudied,andtheindicatorscharacterizingthesta- dt K+S+γψ
bility of strange attractors were established. N
Analogous oscillatory modes in the processes of −k1V(E1)K1+NV(S)−α1S, (1)
photosynthesis and glycolysis, variations in the con- dS1 N
centration of calcium in a cell, and oscillations in a dt =k1V(E1)K1+NV(S)−
heartmuscle andin some biochemicalprocesseswere N
found in [22–26]. −k2V(E2) V(S1), (2)
K1+N
The distinction of the present work from other
dS2 N
ones consists in the modeling of such significantphe- =k2V(E2) V(S1)−
dt K1+N
ntoonmepnootnenatsiatlhoenintflhueenKcereobfsthcyecdleis.sBipyattiohne oMf iatcphreol-l −k3V(S22)V(S3)−k4V(S2)V(S8), (3)
cphoetmenotoiaslmaoritsiceshuynpdoetrhtehseisre[d2u7c],ingtheequtirvaanlesmntetmrabnrsafneer ddSt3 =k4V(S2)V(S8)−k5V(N2)V(S32)−
alongthe respiratorychainoninternalmitochondrial −k3V(S22)V(S3), (4)
membranes. Then the transport of ions H+ inward a
membraneoccursundertheactionoftheformedelec- dS4 =k5V(N2)V(S32)−k7V(N)V(S4)−
trochemical gradient of the potential. This results in dt
theproductionofafreeenergy,whosesignificantpart −k8V(N)V(S4), (5)
is stored in ATP. A part of this energy is used also
dS5
for other purposes, namely: transport of phosphate, =k7V(N)V(S4)−2k9V(L1−T)V(S5), (6)
transport of ions Ca2+, transformation of ADP to dt
ATP, generation of heat, operation of the “proton- dS6 =2k9V(L1−T)V(S5)−
driventurbine” ofbacterialflagella,etc. Thetransfer dt
tohfeiongisvHen+pthortoenutgihalF,0wFh1i−chATaffPeacstesmthoelecturliecsarcbhoaxnygleics −k10V(N)S62+1S+62M1S8, (7)
apclaidsmcaytcilce.mAenmbarnaanloegoofusaeprroobciecssceilslsrCunannidnigdainuttihlies ddSt7 =k10V(N)S62+1S+62M1S8 −k11V(N)V(S7)−
considered in the present work. Thus, the Krebs cy- S2
cle and the respiratorychain are functioning under a −k12S72+1+7 M2S9V(ψ2)+k3V(S22)V(S3), (8)
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LyapunovIndices andthe Poincar´eMapping
dS8 dC
=k11V(N)V(S7)−k4V(S2)V(S8)+ =k8V(N)V(S4)−α7C, (19)
dt dt
+k6V(T2)S2S+2β1N1+(SN51+S7)2, (9) swchriebreesVth(Xe )ad=soXrp/t(io1n+oXf t)hies etnhzeyfmunecitnionthtehraetgdioen-
ddSt9 =k12S72+1S+72M2S9V(ψ2)− odfimaenloscioanllceossup[1li2n,g1.3T].he variables of the system are
XTS9 The internal parameters of the system are as fol-
−k14(µ1+T)[(µ2+S9+X+M3(1+µ3ψ)]S, (10) lows:
dX XTS9
=k14 − k1 =0.3; k2 =0.3; k3 =0.2; k4 =0.6;
dt (µ1+T)[(µ2+S9+X +M3(1+µ3ψ)]S
−α2X, (11) k5 =0.16; k6 =0.7; k7 =0.08; k8 =0.022;
k9 =0.1; k10 =0.08; k11 =0.08; k12 =0.1;
dQ
dt =−k15V(Q)V(L2−N)+ k14 =0.7; k15 =0.27; k16 =0.18;
+4k16V(L3−Q)V(O2)1+1γ1ψ2, (12) k17 =0.14; k18 =1; k19 =10; n1 =0.07;
dO2 K2 1 n2 =0.07; L=2; L1 =2; L2 =2.5; L3 =2;
dt =O20K2+O2 −k16(L3−Q)V(O2)1+γ1ψ − K =2.5; K1 =0.35; K2 =2; M1 =1;
−k8V(N)V(S4)−α3O2, (13) M2 =0.35; M3 =1; N1 =0.6; N2 =0.03;
dN S2 N3 =0.01; µ1 =1.37; µ2 =0.3; µ3 =0.01;
dt =−k7V(N)V(S4)−k10V(N)S62+1+6 M1S8 − γ =0.7; γ1 =0.7; β1 =0.5; β2 =0.4;
−k11V(N)V(S7)−k5V(N2)V(S32)+ β3 =0.4; E10 =2; E20 =2.
N
+k15V(Q)V(L2−N)−k2V(E2) V(S1)− The external parameters determining the flow-type
K1+N conditions are chosen as
N
−k1V(E1)K1+NV(S), (14) S0 =0.05055; O20 =0.06; α=0.002;
dT
=k17V(L1−T)V(ψ2)+k9V(L−T)V(S3)−α4T − α1 =0.02; α2 =0.004; α3 =0.01;
dt
−k18k6V(T2)S2S+2β1N1+(SN51+S7)2 − αT4he=m0o.0d1e;l coαv5er=st0h.0e1p;rocαe6ss=es0o.f01su;bsαtr7at=e-0e.n0z0y0m1.a-
XTS9 ticoxidationofethanoltoacetate,thecycleinvolving
−k19k14 , (15)
(µ1+T)[µ2+S9+X+M3(1+µ3ψ)S] tri- and dicarboxylic acids, glyoxylate cycle, and res-
dψ piratory chain.
dt =4k15V(Q)V(L2−N)+4k17V((L1−T)V(ψ2)− The incoming ethanol S is oxidized by the alcohol
S2 dehydrogenaseenzymeE1 toacetaldehydeS1 (1)and
−ddE2t1k1=2SE721+0β12+S+72MS22SN92VN+(2ψS21)−−αψ, (16) itetnhhtaeettnheenebSvy2ciret(olh2lne)mm,a(eec3nte)att..baTolTlhdihseeemhfmyordaomnrdodeegdlecanaacacncesoetbaueetnnetezscxyafcmonhreapntaEhgri2etsdictsoiiwptuaaitatche--
N
−n1V(E1) V(S)−α5E1, (17) tion by the change of acetate by acetyl-CoA. On the
K1+N first stage of the Krebs cycle due to the citrate syn-
ddEt2 =E20β3S+12S12N3N+3S2 − ftohramseedreiancttihoen,Karceebtsylc-yCcoleAcjroeianttelycwitirtahteoxSa3la(c4e).taTtehSen8
N substances S4−S8 are createdsuccessivelyon stages
−n2V(E2)K1+NV(S1)−α6E2, (18) (5)–(9). In the model, the Krebs cycle is represented
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V.I.Grytsay
Fig.1. GeneralschemeofthemetabolicprocessofgrowthofcellsCandidautilis onethanol
byonlythosesubstratesthatparticipateinthereduc- with glyoxylate and the formation of malate. This
tionofNADHandthephosphorylationADT→ATP. glyoxylate-linked way is shown in Fig. 1 as an en-
Acetyl-CoA passes along the chain to malate repre- zymatic reaction with the consumption of S2 and S3
sentedinthe modelas intramitochondrialS7 (8)and and the formation of S7. The parameter k3 controls
cytosolicS9(10)ones. Malatecanbealsosynthesized the activityofthe glyoxylate-linkedway(3), (4), (8).
inanotherwayrelatedtotheactivityoftwoenzymes: The yield of S7 into cytosol is controlled by its con-
isocitrate lyase and malate synthetase. The former centration, which can increase due to S9, by causing
catalyzes the splitting of isocitrate to succinate, and the inhibition of its transport with the participation
the latter catalyzes the condensation of acetyl-CoA of protons of the mitochondrial membrane.
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LyapunovIndices andthe Poincar´eMapping
The formed malate S9 is used by a cell for its Equation (19) is related to the formation of car-
growth,namelyforthebiosynthesisofproteinX (11). bon dioxide. Its removal from the solution into the
The energy consumption of the given process is sup- environment(α7)istakenintoaccount. Carbondiox-
ported by the process ATP → ADP. The presence of ide is produced inthe Krebs cycle (5). Inaddition, it
ethanol in the external solution causes the “ageing” squeezes out oxygen from the solution (13), by de-
ofexternalmembranesofcells,whichleadsto the in- creasing the activity of the respiratory chain.
hibition of this process. The inhibition of the process The study of solutions of the given mathematical
also happens due to the enhanced levelof the kinetic model (1)–(19) was performed with the help of the
membrane potential ψ. The parameter µ0 is related theoryofnonlineardifferentialequations[28,29]and
to the lysis and the washout of cells. the methods of mathematical modeling of biochemi-
In the model, the respiratory chain of a cell is cal systems applied and developed in [30–47].
represented in two forms: oxidized, Q, (12) and re- To solve this autonomous system of nonlinear dif-
duced, q, ones. They obey the integral of motion ferential equations, we applied the Runge—Kutta–
Q(t)+q(t)=L3. Merson method. The accuracy of solutions was set
Achangeofthe concentrationofoxygenintheres- to be 10−8. To get the reliable results, we took the
piratory chain is determined by Eq. (13). duration of calculations to be 106. For this time in-
terval,thesystem,beingintheinitialtransientstate,
The activity of the respiratory chain is affected by
approaches the asymptotic attractor mode, i.e., its
the level of NADH (14). Its high concentration leads
trajectory “sticks” the corresponding attractor.
totheenhancedendogenicrespirationinthereducing
processintherespiratorychain(parameterk15). The
accumulation of NADH occurs as a result of the re- 2.Results of Studies
duction of NAD+ at the transformation of ethanol
With theuse ofthe Mitchellhypothesis,we nowcon-
and in the Krebs cycle. These variables obey the in-
sider the chain of formation of the proton poten-
tegral of motion NAD+(t)+NADH(t)=L2.
tial of a cell. For each turnover of the cycle of citric
In the respiratory chain and the Krebs cycle, the
acid,thespecificdehydrogenasessplitofffourpairsof
substrate-linked phosphorylation of ADP with the
hydrogen atoms from isocitrate (5), α-ketoglutarate
formation of ATP (15) is also realized. The energy
(6), succinate (7), and malate (8). Their separation
consumption due to the process ATP → ADP in- into ions H+ and electrons occurs in the internal
duces the biosynthesis of components of the Krebs membrane through three H+-transferring loops con-
cycle (parameter k18) and the growth of cells on the sisting of ubiquinone and three cytochromes. Each
substrate(parameterk19). Forthesevariables,thein- loop transfers two ions H+ outward the membrane,
tegralofmotionATP(t)+ADP(t)=L1 holds. Thus, which leads to the appearance of a transmembrane
the level of ATP produced in the redox processes in
electrochemical potential (16). The acceptor of elec-
the respiratory chain ADP → ATP determines the
tronsintherespiratorychain(12)isoxygen(13). Ions
intensity of the Krebs cycle and the biosynthesis of H+, which are accumulated on the external side of
proteins.
the membrane, move again inward along the elec-
Inthe respiratorychain,the kinetic membrane po- trochemical gradient through molecules of F0F1 −
tentialψ(16)iscreatedundertherunningofreducing ATPase. This transition of ions H+ from the zone
processes Q → q. It is consumed at the substrate- withtheirhighconcentrationtothezonewithalower
linked phosphorylation ADP → ATP in the respi- one is accompanied by the free energy release. This
ratory chain and the Krebs cycle. Its enhanced level results in the synthesis of ATP from ADP (15) by
inhibits the biosynthesis of proteins and the process the reaction of oxidative phosphorylation. In other
of reduction of the respiratory chain. words, the continuous turnover of ions H+ through
Equations (17) and (18) describe the activity of the membrane occurs. Its driving force is the trans-
enzymes E1 and E2, respectively. We consider their ferofelectronsalongtherespiratorychain. Thus,the
biosynthesis (E10 and E20), the inactivation in the joint self-organizationofthe Krebscycle andthe res-
courseofthe enzymatic reaction(n1 andn2), andall piratorychaindepends onthe dynamicsofformation
possible irreversible inactivations (α5 and α6). oftheprotonpotential. Thevariationofthepotential
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V.I.Grytsay
Fig. 2. Phase-parametricdiagramforthevariableT(t): a –α∈(0.0018,0.025);b –α∈(0.011,0.014);c –α∈(0.0114,0.0117);
d –α∈(0.0068,0.0073)
is also affected by its dissipation in other metabolic this point in a lot of earlier calculated modes. When
processes in a cell, besides the current of ions H+ the trajectory approaches the attractor, we observe
through the membrane. In the present work, we will the intersection of the plane by the trajectory in a
studythe changesinthe dynamicsofthe Krebscycle single direction for every given value of α. On the
depending on the dissipation of the protonpotential. phase-parametric diagram, we indicate the value of
We now construct the phase-parametric diagram T(t). If a multiple periodic limiting cycle arises, we
for the multiplicity of autooscillations of the ATP observe a number of points on the plane, which coin-
level as a function of the dissipation of the pro- cide in the period. If a deterministic chaos appears,
ton potential α (16) (see Fig. 2) by the method of the points, where the trajectory intersects the plane,
cutting. are located chaotically.
In the phase space of a trajectory of the system, Let us considerthe diagramin Fig.2,a fromright
weplacethecuttingplaneforQ=0.9. Suchchoiceis to left. As the value of the coefficient of dissipation
explainedbythesymmetryofoscillationsofalevelof α decreases below 0.025, we see the transition from
the oxidized form of the respiratory chain relative to the 1-foldperiodic autooscillatorymode to the 2-fold
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LyapunovIndices andthe Poincar´eMapping
Fig.3. KineticcurvesofcomponentsoftheKrebscycleS2,S8,N,ψ,andT: a –inthe1-foldperiodicmode1·20forα=0.025;
b –inthechaoticmodeofthestrangeattractor 1·2x forα=0.002
one at the point αj = 0.02468.As a result of the efficient of dissipation α continues to decrease, the
bifurcation, the doubling of the period of oscillations bifurcations arise in the windows of periodicity, and
arises. For αj+1 = 0.01364, we observe the repeated the chaotic modes are seen again(see Fig. 2, d). The
doublingoftheperiod. Then,forαj+2 =0.01196,the self-similarityoftheformationofthewindowsofperi-
period of autooscillations is doubled once more. odicity on large and small scales indicates once more
Let us separate a small section of the diagram for the fractality of the bifurcation diagram.
α ∈ (0.011,0.014) (Fig. 2, a) and represent it in a In Fig. 3,a, b, we showthe kinetic curvesfor some
magnified form (Fig. 2, b). For αj+3 = 0.011645, we components of the metabolic process of cell respira-
see the next bifurcation with the doubling of the pe- tion: in the 1-fold periodic (α = 0.025) and chaotic
riod,andthediagrambecomessimilartotheprevious (α=0.002) modes.
one. The subsequent decrease in the scale of the di- In Fig. 3, a, we see the harmonic interconnec-
agram in Fig. 2, c reveals the next bifurcation with tionoftheautooscillationsofacetyl-CoA(S2), which
the doubling of the period for αj+4 = 0.011524, and is supplied to the cycle of citric acid, and ox-
theself-similarityofthediagramisrepeated. Thisin- alacetate (S8) closing the cycle. Oscillations of the
dicates the fractal nature of the obtained cascade of level of NAD·H(N), which transfers electrons to
bifurcations. Afterthecriticalvalueoftheparameter ubiquinone and cytochromes, occur with the same
αdeterminedbytheaccuracyofcomputer-basedcal- frequency. These hydrogen-transferringand electron-
culations, the deterministic chaos takes place. This transferringproteins alternate in a respiratorychain,
means that any appeared fluctuation under given by forming “three loops” in it. Electrons are trans-
instable modes of the real physical system can in- ferredto oxygen(acceptorof electrons),andions H+
duce a chaotic mode. This scenario of the transition move to the external side of the membrane, by pro-
to a chaos corresponds to the Feigenbaum scenario ducing the gradient of the potential ψ. This gradient
[48]. We calculated the value of universal Feigen- creates the driving force for the return of H+ inward
baumconstantbythedataonbifurcationsandfound the membrane through a complex ATP-synthetase
that it differs from the classical one. This means system. This results in the creation of new covalent
that the dynamics of system (1)–(19) cannot be re- bonds, throughwhichthe terminalphosphategroups
duced completely to a one-dimensional Feigenbaum join ADP with the formation of ATP(T).
mapping. Autooscillations arisen in the given metabolic pro-
For α = 0.0073 and α = 0.01125 (Fig. 2, a, b), cess are regulated by the level of dissipation of the
we see the appearance of the windows of periodici- proton potential α. Its decrease leads to the succes-
ty. The deterministic chaos is broken, and the peri- sivedoublingofthe periodofautooscillationsand,as
odicandquasiperiodicmodesappear. Analogouswin- followsfromcalculations,totheappearanceofchaotic
dows of periodicity are also observed on bifurcation oscillations (Fig. 3, b). The decrease in α means, in
diagrams on a less scale (see Fig. 2, c). As the co- particular, a decrease in the current of ions H+ from
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V.I.Grytsay
Fig. 4. Projectionsofsystem’sphaseportraits: a –regularattractor1·20,α=0.025;b–regularattractor1·21,α=0.015;c–
regularattractor 1·22,α=0.013; d –regularattractor 1·23,α=0.0117; e –regularattractor 1·24,α=0.01158; f –strange
attractor 1·2x,α=0.011
the external side of the membrane to the internal α = 0.025, we observe the coordination of the Krebs
one. Thetimecoordinationbetweenthetricarboxylic cycle and the rate of transfer of charges in the respi-
acid cycle and the transfer of electrons and ions H+ ratory chain. A decrease in α means the deceleration
along the respiratory chain is violated, and the pro- of some metabolic processes related to the dissipa-
cess of oxidative phosphorylation is decelerated. The tion of a membrane potential. The enhanced level of
deceleration of the process of production of ATP in- ψ blocks the respiratory chain (12)–(13), by holding
creases the rate of the metabolic process involving itinthereducedstate. Underthenewconditions,the
tricarboxylic acids (7). Moreover, the frequency of system reveals the self-organization, by coordinating
the givencycle increases,which causes anincrease in the dynamics of the tricarboxylic acid cycle with the
the multiplicity of the period of autooscillations. As transfer of electrons along the respiratory chain. In
theparameterαbecomescritical,allgivenmetabolic the correspondencewith the new appearedcycle,the
processes become desynchronized, which leads to a kinetics ofvariationsinthe protonpotentialgradient
chaotic mode (Fig. 3, b). and the ATP level are formed.
The given sequence demonstrating a growth of We now give the example of a possible test of
the multiplicity of oscillations can be observed in strange attractors for the fractality. Let us consider
Fig. 4. There, we show the sequential appearance of the strange attractor 1 · 2x (Fig. 5, a) formed for
bifurcations and a complication of the projections of α = 0.0078. We separate a small rectangular area of
thephaseportraitsofregularattractors,asthecoeffi- theprojectionofthephasespacet∈(100000,115000)
cientofdissipationαdecreases,untilastrangeattrac- withasinglephasecurveandrepresentitinFig.5,b.
toreventuallyarises(seeFig.4,f). Suchscenariocan The calculation of a phase portrait was executed in
be explained by the existence of positive feedbacks the intervalt∈(100000,320000).As is seen, the geo-
in the given system, which stabilize or intensify the metric structure of the given strange attractor is re-
given metabolic processes. For the optimum value of peated on small and large scales of the projection of
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LyapunovIndices andthe Poincar´eMapping
Fig. 5. Projections of system’s phase portraits: a – strange attractor 1·2x, α = 0.0078, t ∈ (100000,115000); b – strange
attractor 1·2x,α=0.0078 t∈(100000,320000); c –regular attractor 5·20,α=0.0073
Fig. 6. Projections of the phase portraits of the strange attractor 1·2x, α = 0.0078: a – 2-dimensional projection in the
coordinates (S3,ψ);b –3-dimensionalprojectioninthecoordinates (S,S1,Q)
the phase portrait. Each appeared curve of the pro- As an example, Fig. 6, a, b shows 2- and 3-
jection of the chaotic attractor is a source of the for- dimensional projections of the phase portrait of the
mation of new curves. Moreover,the geometric regu- strange attractor for α=0.0078.
larity of the construction of trajectories in the phase In Fig. 7, a, b, c, we give the constructed pro-
space is repeated. This fact confirms once more that jection of a section by the plane N = 1.128 and
the phase-parametric diagrams are similar on small Poincar´e maps for the given strange attractor. The
and large scales, which testifies to the fractal nature choice of a cutting surface was such that the phase
of the given strange attractor. trajectory N(t) under a decrease in the given com-
In Fig. 5, c, we present a projection of the phase ponent intersects it the maximally possible number
portrait of the regular attractor 5 · 20 formed in a of times, and the tangency is excluded. Figure 7, a
window of periodicity (Fig. 2) for α = 0.0073.The indicates the chaoticity of the given strange at-
deterministic chaos is destroyed, and the periodic tractor in the plane (Q,O2). The Poincar´e map for
mode is established. The identical windows of peri- (Qn,Qn+1) shows the instability of the phase curve
odicity are observedalsoon smallerscales of the dia- for the given component. Points of the map are lo-
gram. Outside these windows, the chaotic modes are cated randomly on a large part of the area. At the
formed. same time, the Poincar´e map for (O2n,O2n+1) has a
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ISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 6
V.I.Grytsay
Fig. 7. Projection of the section by the plane N = 1.128 (a) and Poincar´e maps (b, c) of the strange attractor 1·2x for
α=0.0078
quasistripform. Theshapeofthegivencurveisinde- allyorthogonalandnormedbyone. Insometime∆t,
pendent of the number of points of the mapping. All the trajectory arrives at a point Y1, and the pertur-
points of the mapping lie on this curve. The chaos bation vectors become b1, b1,...,b1 . Their renormal-
1 2 19
for the given component exists only in the limits of ization and orthogonalization by the Gram–Schmidt
this curve. Along this direction of the phase space, method are performed by the following scheme:
the trajectory of the strange attractor is stable, but
aperiodic. b1 = b1 ,
1
Thechaoticityforeachcomponenthastheownreg- (cid:13)b1(cid:13)
(cid:13) (cid:13)
ularity. WiththehelpofPoincar´emappings,itispos-
b′
sible to study the system and to find the reason for b′ =b0−(b0,b1)b1, b1 = 2 ,
2 2 2 1 1 2
the formation of a specific type of the strange at- (cid:13)b′(cid:13)
(cid:13) 2(cid:13)
tractor of the system. This allows one to investigate (cid:13) (cid:13)
thestructural-functionalconnectionsinthemetabolic b′ =b0−(b0,b1)b1−(b0,b1)b1, b1 = b′3 ,
process and the reasons, for which the appearance of 3 3 3 1 1 3 2 2 3 kb′k
3
chaotic modes becomes possible.
b′
In order to uniquely identify the type of obtained b′4 =b04−(b04,b11)b11−(b04,b12)b12−(b4,b13)b13, b14 = kb′4k,
attractors and to determine their stability, we calcu- 4
lated the full spectra of Lyapunov indices and their .................................................................................
19
sum Λ = λ for the chosen points. The calcu-
lation wasPcaj=rr1iedj out by Benettin’s algorithm with b′19 =b019−(b019,b11)b11−(b019,b12)b12−(b19,b13)b13−...−
the orthogonalization of the vectors of perturbations
by the Gram–Schmidt method [29]. b′
Asaspecific featureofcalculationsofthe givenin- −(b19,b118)b118, b119 = kb′19k,
19
dicators,wementionthecomplexityofthecomputer-
Thenthecalculationsarecontinued,bystartingfrom
based determination of the perturbation vectors pre-
sented by 19×19 matrices. the point Y1 and perturbation vectors b11, b12,...,b119.
After the next time interval ∆t, a new collection of
The algorithm of calculations of the full spectrum
perturbation vectors b2, b2,...,b2 is formed and un-
of Lyapunov indices consisted in the following. Ta- 1 2 19
dergoes again the orthogonalization and the renor-
king some point on the attractor Y0 as the initial
malization by the above-indicated scheme. The de-
one, we traced the trajectory outgoing from it and
scribedsequence ofmanipulationsis repeateda suffi-
the evolution of K perturbation vectors. In our case,
ciently large number of times, M. In this case in the
K =19 (the number of variablesof the system). The
course of calculations, we evaluated the sums
initial equations of the system supplemented by 19
complexes of equations in variations were solved nu-
M
mtheercicoalllelyc.tiAons othfeveicntiotiraslbp0,ebrt0u,.r.b.,abt0io,nwvheiccthoarsr,ewmeutsuet- Z1 =Xln(cid:13)(cid:13)b′1i(cid:13)(cid:13),
1 2 19 i=1
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ISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 6