U.P.B. Sci. Bull., Series A, Vol. 77, Iss.4, 2015 ISSN 1223-7027 [613149]

U.P.B. Sci. Bull., Series A, Vol. 77, Iss.4, 2015 ISSN 1223-7027
ON THE NON-LINEAR DY NAMICS IN BIOLOGICAL
STRUCTURES. COMPLEME NTARY MATHEMATICAL
ASPECTS
Viorel-Puiu P ĂUN1,*Cipriana STEF ĂNESCU2, Vlad GHIZDOV ĂȚ2, 3,
Diana SAB ĂU-POPA4, Maricel AGOP 5

Assuming that the biological systems are fractal systems, a few aspects
ofnatural dynamics in biological structures are studied. The “non-linear dynamics”
analysis in an arbitrary space with constant fractal dimension , using an extended
version of the Scale Relativity Theory, has been performed. Additionally, a dedicated mathematical model of biological non-linear system by association with
stochastic Levi type processes was developed.

Keywords: biological structures, non-linear dy namics, self-structuring, chaos,
fractals

1. Introduction
In the most general representation, the biological systems can be divided
into three different categories such as the open, dissipative and non-linear
systems. In our opinion, the “specialization” process of any biological structure
(for instance differentiation process) is based on the legitimate alternation
between chaos and order of mutual states. This behavior is defined by the living matter multivalent logic and its communication codes.
Further to the presentation, we can say now that this visible
interdisciplinary work aims to explain how mathematical knowledge can be used
to describe, predict and control the phenomena observed in some biological
systems [1]. From a functional point of view and not only, we understand here the biological systems in the particular sense of the discipline known under a variety
of names [2] such as: “complexity theory”, “self-organization theory”, “chaos
theory”, or “non-linear dynamics”. Our choice was not easy but in what follows
we will use the latter name, non-linear dynamics respectively. Before complete
presentation of the publication reason of this paper, we will define the notion most

1 University POLITEHNICA of Bucharest, 313 SplaiulIndependentei, Bucharest, 060042,
Romania
2”Gr. T. Popa” University of Medicine and Pharmacy, 16 University Str., Ia și – 700115, Romania
3 “AlexandruIoanCuza” University, 11 Carol I Blvd., Ia și 700506, Romania
4 Oradea University, Fac Econ Sci, Finance Accounting Department, Oradea 410087, Romania
5“Gheorghe Asachi” Technical University, 59A Dimitrie Mangeron Blvd., 700050, Iasi, Romania * Corresponding author’s e-mail: [anonimizat]

308 Viorel-Puiu P ăun, Cipriana Ștefănescu, Vlad Ghizdov ăț, Diana Sab ău-Popa, Maricel Agop
often used here. In this context, we can say that a real system is linear if it can be
adequately modeled by a linear transforma tion or a linear dynamical system. By
consequence, if any linear model appears in adequate, according to binary logic, it
results that this real system is non-linear.
Many biological structures assimilated with complex systems [3, 4]
(circulatory system, respiratory system, brain etc.) are, from a morphological
point of view, fractals. Moreover, their own space (the one generated by these structures) is structurally a fractal spac e, in its most general sense given by
Mandelbrot. In a fractal space, the on ly possible functionalities (which are
compatible with the previously mentioned structures) are achieved by the motions
of the structural units of the biological structures assimilated to complex systems
on continuous but non-differentiable curves. Then, the dynamics of such structures can be analysed using Scale Relativity Theory (SRT) in an arbitrary
constant fractal dimension [5-7] (on the standard SRT see [8, 9]). In our opinion,
these biological structural units can take the form of cells, cell organelles (mainly
those responsible with cell division), m acromolecules (such as proteins) etc.,
depending on the scale resolution.
The present paper is composed from Introduction, three extended chapters
and the main conclusions of this work. The first chapter contains our view of
differential dynamics in biological structur es and the second details the oscillatory
behaviour of biological systems, non-linear ity attestation and the road to chaos. In
the last chapter, we rebuilt the premises and have commented the results.
2. Differential dynamics in biological structures
Let us now admit the following functionalities of the scale covariance
principle: the “laws” associated to biophy sical processes are invariant with regard
to scale resolutions. We can implement this principle by substituting the standard
time derivative operator,
dd t , specific to the classical (differentiable) biophysics
with the complex operator, ˆdd t , specific to the non-standard (non-differentiable)
biophysics. Then ˆdd t becomes not only a motion operator in the “new”
biophysics, but also a “covariant derivative” [12, 13]. Consequently, the
“biological geodesics” of the ar bitrary biological fractal fluid Qcan thus be
written in the form:
23 ˆˆ 0li l i l k
li l i l kdQ Q Q Q QVD Ddt t X X X X X X∂∂ ∂ ∂=+ + + =∂∂ ∂ ∂ ∂ ∂ ∂  (1)
where

On the non-linear dynamics in biological systems. Complementary mathematical aspects 309
()()() ()2/ 1 1
4FD i l il il il ilDd t i λλ λλ λλ λλ−
++ −− ++ −−⎡ ⎤ =+ − +⎣ ⎦ (2)

()()() ()3/ 1 1
12FD ilk i l k i l k i l k i l kDd t i λλλ λλλ λλλ λλλ−
+++ −−− +++ −−−⎡⎤ =+ − +⎣⎦ (3)
The biological structures whose functionalities can be associated to a
special class of stochastic Levi type pr ocesses [8, 9] allow for the following:
2il il i lλλλ λ λ δ++ −−== (4)

6ilk ilk i l kλλλ λλλ μ δ+++ −−−=− = (5)
with
1
0
1
0 il
ilkil
il
ilk
ilkδ
δ=⎧⎫=⎨⎬≠⎩⎭
==⎧ ⎫=⎨ ⎬≠≠⎩⎭
and ,λμ structural parameters.
Then
()() 2/ 10,FD il il ildd d t λ δ−== (6)

()() 3/ 1,0FD ilk ilk ilkdd t dμδ−== (7)
so that (1), after seve ral calculations become:

()()
()()()
()23
2/ 1 3/ 1
23ˆˆ 0FFDD l
llilidQ Q Q Q QVd t d tdt t X XXλμ−− ∂∂ ∂ ∂=+ + + =∂∂ ∂∂∑∑ (8)

()()
()3
3/ 1
30FD l
D liiQQ QVd ttX Xμ− ∂∂ ∂++ =∂∂ ∂∑ (9)

310 Viorel-Puiu P ăun, Cipriana Ștefănescu, Vlad Ghizdov ăț, Diana Sab ău-Popa, Maricel Agop
()()
()2
2/ 1
20FD l
F lllQQVd tX Xλ− ∂∂+=∂ ∂∑ (10)
Let us observe the one-dimensional form of equation (9)
()()
()3
3/ 1 1
3 110FD
D
iQQ QVd ttX Xμ− ∂∂ ∂++ =∂∂ ∂∑ (11)
assuming that 1const.DVQ≡ . Therefore, at differential scale, the dynamics of Q is
dictated by fractal differentialKorteweg- de Vries type equations. For details on
standard differential Korteweg-de Vries equations see Ref. [10].
An explicit solution of the above mentioned differential equation, obtained
by adequate normalization in dimensionless variables,
11 1
0,, , ,Qtk X k X MQωτξ θ τ φ== = − ≡ (12a-d)
implies using the method from [13]. It results:
()2
0()21 2 ;()Esaa c n sKsφφ α θθ⎡⎤=+ −+ − ⎡ ⎤⎢⎥ ⎣ ⎦⎣⎦ (13)
where ω is a pulsation specific to the biological structure, k is the inverse of a
characteristic biological structure length, Mis the biological equivalent of the
Mach number, φis the average value of φ, 0Q is the equilibrium value of the
biological fractal field Q, ais the amplitude, ()Ks and ()Es are the complete
elliptical integrals of the first and second kind of modulus s (a measure of the
non-linearity degree)and cn is the Jacobi cnoidal elliptical function with modulus
sand argument ()0αθθ− with0const.θ= [11]. Definitions for ,sλetc. are
presented in [5].

3. Oscillatory behaviour of biological systems. Non-linearity attestation
and the road to chaos
In good accordance with the foregoing, we can immediately transfer the
mathematical results previously obtained to appropriate biological systems. This
means that the biological structures “dynamics” are given through cnoidal space-

On the non-linear dynamics in biological systems. Complementary mathematical aspects 311
time oscillation modes of Q – see the three dimensional dependence (Fig.1), and
the contour curves, respectively (Fig.2a-f).

Fig.1. Three-dimensional representation of the cnoidal oscillation mode as a function of the
biological field via normalized space-time coordinates and non-linear degree

Fig.2. a-f Two-dimensional representation of the cnoidal oscillation modes as a function of
the biological field for various non-linear degrees (contour curves)
The cnoidal oscillation modes ha ve the following characteristic
parameters:

312 Viorel-Puiu P ăun, Cipriana Ștefănescu, Vlad Ghizdov ăț, Diana Sab ău-Popa, Maricel Agop
i) Wave number
()12aksKsπ= (14)
ii) Phase velocity
()
()2
23 164Es sUaKs s⎡ ⎤+=+ −⎢
⎣Φ ⎥
⎦ (15)
iii) Quasi-period (see fig. 3a, b)
()()()
()3 12 22
231
32 1 Es aa s
s Ks s Ks Ks sT
⎡ ⎤++−⎢ ⎥
⎣=
⎦Φ
(16)

Fig.3. The quasi-period dynamics of the biological normalized field via amplitude and non-linear
degree, with respect to the average value of the biological normalized field

On the non-linear dynamics in biological systems. Complementary mathematical aspects 313
In Fig. 3, the three dimensional representation of the quasi-period T with
respect to the amplitude a and the non-linear degree s is provided.
The structure of these oscillation mode s is obtained by explicating their
degeneration with respect to the s parameter:
i) For 0s→ , (13) reduces to a harmonic packagetype sequence, while for
0s≡(12) reduces to a harmonic type sequence;
ii) For s→1, (13) reduces to a soliton package type sequence, while for 1s≡
(13) reduces to a soliton type sequence.
Eliminating the variable „a” in (14) and (15) the following results:
()2 261 6 ( ) ,UA s kπλλ−Φ = = (17a,b)
where
()22 2() 3 () () 1 ()Ass K s E s s K s=− + (18)
It can be observed – see Fig. 4 that the nonlinearity s generates three
distinct dynamics regimes in biological structures: non-quasi-autonomous regime
(by harmonic type sequences, harmonic package type sequence or harmonic–
harmonic package type sequence), quasi-autonomous regime (by soliton type
sequences, soliton package type sequences, soliton – soliton package type
sequence), and transient regime (by mixtures), respectively.

Fig. 4. Dynamic regimes in biological structures

The real dynamics regimes of the biological structures are mixtures of the
previous pure sequences (mixed modes): harmonic package – soliton, harmonic

314 Viorel-Puiu P ăun, Cipriana Ștefănescu, Vlad Ghizdov ăț, Diana Sab ău-Popa, Maricel Agop
package – solitonic package etc. [10]. Such situations can be accomplished only
by the fractalization of the movement va riables, which mathematically involves a
Wick rotation (2ieπ× ) in the variables space (),sθ. Then, all the θ attributes are
transferred to s and vice versa. For details a bout fractalization (methodology,
implications, examples, etc.) please see [9, 10]. Taking into account the above
observations, the mixtures will be felt as chaoticity of the biological structures,
according to different scenarios of chaos transition (intermittency, Ruelle-Takens, sub-harmonic bifurcations, etc.). Indeed, the routes to chaos through intermittency
and quasi-periodicity (Ruelle-Takens scenar io) can be assimilated to the sections
πand δ, respectively of the cnoidal oscillation modes (Fig. 5), while the route to
chaos through sub-harmonic bifurcations can be assimilated to the section σ of the
same modes (Fig. 6).

Fig. 5. Routes to chaos through intermittency and quasi-periodicity (Ruelle-Takens), as sections
π and δ of the cnoidaloscillation modes, respectively

On the non-linear dynamics in biological systems. Complementary mathematical aspects 315

Fig. 6. Route to chaos by sub-harmonics bifurcations as section σ of the cnoidal oscillation modes

Finally, it is worth to be mentioned that, although in most cases it is
impossible to obtain accurate mathematical models of biological systems, their
physical and chemical properties comply well with the general properties of
dynamical systems in which self-oscillations are possible.

4. Results and discussion
In the real world, the biological syst ems are obviously dissipative vs. mass
and energy, but profoundly non-linear. In this sense, to function properly their living structures need to consume “food” from outside firstly, and to get rid of the
decay products, secondly.
Assuming that, from a morphologic and/or functional point of view, any
biological structure is a fractal in the most general sense provided by Mandelbrot
[1], to describe the dynamics of such systems in the afore mentioned perspective,
a mathematical model is obtained.This mathematical model is based on an
extended version of the Scale Relativity Theoryin the sense of Nottale hypothesis
[14, 15], namely the one in which the motions of the complex system’s structural units, assimilated to the biological struct ures particles, take place on continuous
and non-differentiable curves in a fractal arbitrary constant dimension. For further
details also see other excellent results on the same topic [16].According to the
new horizons accessible today in nonlinear dynamics [17], the classical
mathematical calculation for gauge field theories [18] and elliptic functions [19], follows the classical books, cited in the text.

316 Viorel-Puiu P ăun, Cipriana Ștefănescu, Vlad Ghizdov ăț, Diana Sab ău-Popa, Maricel Agop
By employing this mathematical model the motion operator is defined,
which is a complex operator that, based on a scale covariance principle, gains the
status of covariant scale derivative. In such a conjecture the “geodesics”
associated to an arbitrary biological fiel d are obtained, namely the “global” ones
and the ones induced by the separation of motions on resolution scales
(differential scale and fractal scale).
The differentiable “dynamics” obtained by integrating the differential
equation associated to the “biological geodesics” at differential scale resolution is
induced, in the one-dimensional case, by space-time cnoidal oscillation modes of
the biological field. Depending on the strength of interactions between the
structural units of the complex system assimilated to a biological structure, these
cnoidal oscillation modes degenerate, e ither in a harmonic sequence and a
harmonic package type sequence in the case of a null, 0s=, or “weak” , 0 s→ ,
interstructural “coupling” , or in a solitonic sequence and a solitonic package type
sequence in the case of a very “strong”, 1s=, or strong, 1s→, interstructural
“coupling”. From this point on the various chaos transition scenarios
(intermittency, Ruelle-Takens, sub-harmonic bifurcations etc.) can be “simulated”
through the above mentioned sequences mixt ures. In our opinion, the presence of
chaos in biological structures “dynamics” can induce, taking into account both the resolution scale dependence (cell, tissue, organ etc) and the “external medium”
feedback dependence, either disorder (for example an “uncontrolled” cell
proliferation process that leads to cancer tumors), or order (for example a
“controlled” cell proliferation process that leads to pattern generation such as
tissues, organs etc.). Moreover, a primne ss of chaos transition scenarios exists,
and it is unique, either in the case of disorder, or in the case of order.

It is well known that a one-dimensional Toda type network of non-linear
oscillators can be attributed to cnoidal oscillation modes. Furthermore, by mapping it, a neural network can be induced [7, 17]. Since the “identity” of any
biological structure is dictated by the morphological-functional “compatibility”, in
this status the “coherence” duplication of tw o neural networks is involved, namely
the structural morphological specific neural network, and the spectral functional
specific neural network. In such a framework the communication code between the structural units of the complex system assimilate to a biological structure is
also generated, a code of algebraic nature, taking into account the Elliptic
Functions Equivalency Theorem [19].
As an assumed objective, in the future, we plan to expand some
mathematical results obtained on the ma terial non-linear systems (engineering)
[20] at biological non-linear structures [21, 22] together within formational non-
differentiable entropy [23, 24], and indubitably, to propose the same fractal analysis using the time series method [25].

On the non-linear dynamics in biological systems. Complementary mathematical aspects 317
5. Conclusions
The previous results specify the fact that in the differentiation and
specialization process of any biological structure “software”, “multivalent laws”,
“communication codes” etc. are self-generated through morphological-functional compatibility. Understanding these logical elements employed by the living
matter and its biological structures, and the way in which they are interconnected,
can prove to be extremely valuable with respect to future medical engineering
projects, and in particularly, to the simula tion of cell and tissue behavior at the
time of injury or during healing.
It was demonstrated that the cnoidal oscillation mode is a function of the
biological field via normalized space- time coordinates and non-linear degree.
Moreover, we have offered a two-dime nsional representation of the cnoidal
oscillation modes as a function of the biological field for various non-linear
degrees. Also, were highlighted in a clear manner, the routes to chaos through intermittency and quasi-periodicity respectively, as sections of the cnoidal
oscillation modes.

Acknowledgements
The current work was supported by the strategic grant
POSDRU/159/1.5/S/133652, Project “Integrated support system for improving
doctoral and post-doctoral research quality in Romania and for advocating the role
of science in society” co-financed by the European Social Found within the Sectorial Operational Program Human Resources Development 2007-2013.

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