A feedback loop to a 3D chaotic system with only six-terms on the right-hand of the equations and only two nonlinearities is applied to intentionally build a minimalist novel 4D chaotic system. Simulations depict the coexistence of strange attractors not by the modification of an unique or several parameters but surprisingly by a slight modification of the intial conditions. It is acknowledged that a strange attractor is locally unstable but globally stable. Our experiementation displays that strange attractors could be unstable at all scales.Une boucle de rétroaction à un système chaotique 3D à six termes est appliquée pour construire intentionnellement un nouveau système chaotique minimaliste 4D. Les simulations décrivent la coexistence d'attracteurs étranges non par la modification d'un ou de plusieurs paramètres mais de manière surprenante par une légère modification des conditions initiales. Il est reconnu qu'un attracteur étrange est localement instable mais globalement stable. Notre expérimentation montre que les attracteurs étranges pourraient être instables à toute échelle

Bouali, Safieddine

English

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Strange Attractor Morphogenesis by Sensitive
Dependence on Initial Conditions
Safieddine Bouali
To cite this version:
Safieddine Bouali. Strange Attractor Morphogenesis by Sensitive Dependence on Initial Conditions.
2019. ￿hal-02276579￿
 1 
 
 
Strange Attractor Morphogenesis by Sensitive Dependence on 
Initial Conditions 
 
 
Safieddine Bouali 
University of Tunis, Management Institute, 
Department of Quantitative Methods & Economics 
41, rue de la Liberté, 2000, Le Bardo, Tunisia 
Safieddine.Bouali@isg.rnu.tn 
http://chaos-3d.e-monsite.com 
 
 
Abstract : 
 
A feedback loop to a 3D chaotic system with only six-terms on the right-hand of the equations 
and only two nonlinearities is applied to intentionally build a minimalist novel 4D chaotic 
system.  Simulations depict the coexistence of strange attractors not by the modification of an 
unique or several parameters but surprisingly by a slight modification of the intial conditions. 
It is acknowledged that a strange attractor is locally unstable but globally stable. Our 
experiementation displays that strange attractors could be unstable at all scales. 
 
 
I. Introduction 
 
 
Morphology of the strange attractors as mathematical "objects" could have relevance to report 
the whole envelop of amplitudes and  frequencies reached by the chaotic oscillations  in the 
phase space. 
One could well argue that their shapes are just the condensed summary of long listings of time 
series. However, the morphological plasticity have a significance introduced by the sensitive 
dependence on parameters (SDP) of the given attractor and helps the modeler to configurate 
adaquately its system. Such pertinence arises more intensely when the chaotic behaviors 
become captured by the sensitive dependance on initial conditions (SDIC) as displayed in a 
previous 3D chaotic system (Bouali, 2013). Its properties are in full contradiction to the 
findings of the Butterfly Effect (Lorenz, 1993), indicating that small gaps between the initial 
conditions lead the trajectories to different dynamics, but converge asymptotically toward an 
unique attractor. Locally unstable but globally stable, a strange attractor.. 
The primary objective of this paper is to experiment the findings of an interactive perturbation 
of this system of 2013 violating the SDIC scopes. 
To this end, a feedback  loop altering the structural stability of the model and extending the 
model to the fourth dimension is applied. We expect the emergence of a morphogenesis at the 
sense of Thom (2018 for the last edition).  
 2 
 
The experimentation represents a preliminary step to the topological study of strange 
attractors (Gilmore and Lefranc, 2011; Letellier and Gilmore, 2013).  
The secondary objective is represented by the self-selected constraint to develop a 4D version 
with the minimum number of terms in the right-hand of the equations.  
In light of the coexisting attractors computed from such unique 4D dynamical system only 
with SDIC, we indicate its suitability as a chaotic generator for secure communication 
schemes. 
In section III, the morphological variety of the phase portraits are described as well as the 
feature of the (planar) rotated copy for any phase portrait found.  Section IV presents remarks 
on the origin of the attractor plasticity and the high suitability of the 4D model as a chaotic 
generator for cryptology. 
 
II. A Minimalist 4D chaotic system 
 
To achieve our goal to intentionally construct a 4D chaotic model with the minimum extent 
possible set of terms on the right-hand side, we propose the following system: 
{
 
 
?̇? =  𝜶 𝒙 ( 𝟏 − 𝒚 ) −  𝜷𝒛
?̇? =  −( 𝟏 − 𝒙𝟐) 𝒚           
?̇? =  − 𝝁 𝒗                         
?̇? =  − 𝝆 𝒛                          
 
where the overdot denotes a time derivative, x, y, z, and v, the four state variables of the 
model, and α, β, µ, ρ assumed to be positive parameters. 
Such seven-term system embeding only two nonlinearity, a quadratic term, and one cubic 
item, respectively, xy, and x2y, expresses the minimalist extension of a 3D system having an 
exclusive silhouette (Bouali, idem, 2013) with a fourth state variable.  
Requirement to investigate the patterns of its phase portrait lead us to retain the set of 
parameters P (α, β, µ, ρ) = (3, 0.1, 1, 0.2, 0.01). 
 
II.1. Basic mathematical properties 
 
The equilibrium coordinates could be found by setting:  ?̇? =  ?̇? =  ?̇? = ?̇? = 𝟎. 
Thus, the solutions of the following system : 
{
 
 
𝟎 =  𝜶 𝒙 ( 𝟏 − 𝒚 )–  𝜷𝒛   
𝟎 =  −( 𝟏 − 𝒙𝟐) 𝒚           
𝟎 =  − 𝝁 𝒗                         
𝟎 =  − 𝝆 𝒛                          
 
give the coordinates of equilibria. 
In the first equation, and substituting z = 0 from the last relation, we obtain x = 0 or y = 1. 
Firstly, the second equation gives for x= 0, y= 0, and secondly for y =1,  x = ±1. 
The elementary attributes of these three equilibria given by the corresponding eigenvalues λi 
are found by solving the characteristic equation |J − λI | = 0 where J, the Jacobian of the 
model, and I, the unit matrix. This kind of calculation is easy to achieve since the non-zero 
 3 
 
terms of the Jacobian are the least numerous (7 out of 16). In Table 1, coordinates, 
eigenvalues and features of the stability related to the equilibria are reported.  
 
Table 1. Index and Stability of the Equilibria  
Coordinates of the equilibia Corresponding characteristic 
equation,  
|J − λI | = 0, and eigenvalues 
Index and 
stability 
 
 
 
 
E0 (x0, y0, z0, v0) = (0, 0, 0, 0) 
 
(λ2 + 0.002)(1+λ)(3-λ)= 0 
 
λ1 = 
− 𝑖
10√5
              λ2 = 
 𝑖
10√5
 
 
          λ3 = -1                  λ4 = 3 
 
Index-1 
Unstable: Spiral 
saddle point 
 
 
E1 (x1, y1, z1, v1) = (1, 1, 0, 0) 
 
& 
 
E2 (x2, y2, z2, v2) = (-1, 1, 0, 0) 
 
 
          (λ2 + 0.002)(6+λ2)= 0 
 
λ1 = 
− 𝑖
10√5
              λ2 = 
 𝑖
10√5
 
 
          λ3 = −𝑖√6            λ4 = 𝑖√6 
 
Index-0 
Neutrally stable: 
centers  
(1) Index reports the number of eigenvalues with real parts Re(λ) > 0. From 1 to 4, it indicates the degree of 
instability. Index-0: null or negative real parts of all eigenvalues of the equilibrium characterize its stability. 
 
II.2.  A nonuniformly dissipativity 
To qualify the dissipative nature of the 4D dynamical system, the divergence of the whole vector 
field could be derived from the following formulation: 
 
 
𝒅𝒊𝒗. (𝑽𝒐𝒍𝒖𝒎𝒆) =
𝝏𝑽𝒐𝒍𝒖𝒎𝒆
𝝏𝒕⁄
𝑽𝒐𝒍𝒖𝒎𝒆
 = 𝑻𝒓( 𝑱 ) =  
𝝏?̇?
𝝏𝒙
 +
𝝏?̇?
𝝏𝒚
+
𝝏?̇?
𝝏𝒛
+
𝝏?̇?
𝝏𝒗
 
 
To this end, Tr (J), the sum of the diagonal terms of the Jacobian should be negative to attest 
the dissipativity of the flow: 
 
𝑻𝒓( 𝑱 ) = 𝛂 (𝟏 −  𝐲) − (𝟏 − 𝐱𝟐)  < 0    
where α, positive value.  
Dissipativity and volume contraction of the flow are accurately identified when these state 
variables of the flow, x, and y (and not including either z or v) fulfil the required condition:  
 
𝛂 (𝟏 −  𝐲) < (𝟏 − 𝐱𝟐)  
 
The system arises nonuniformly dissipative. However,  in the precise case of the above condition 
is met, orbits are ultimately limited in a specific fractal-dimensional subspace of zero volume.  
 4 
 
The set of (x, y) that verifies this inequality marks a domain bounded by an separatrix where the 
flow of the dynamics is dissipative. Besides, the trajectories are not short-term transient dynamics 
and report the zones of chaotic behavior -and not that regular- within the phase space (Joglekar 
and al., 2014; Lai and Tél, 2011; Motter et al., 2005; Kantz and Grassberger, 1985). 
On the other hand, focusing  the 4D model with its P parameters, it is pertinent to compute the 
Lyapunov exponents spectrum, the most useful measure to estimate the chaos. The system 
displays a chaotic nature since for β, varying from 0.01 to  0.21, the dominant Lyapunov exponent 
reaches a positive value (Fig. 1). 
 
 
Fig. 1. Lyapunov exponent spectrum for the 4D chaotic system keeping unchanged the 
specification of parameters P, except β which varies in the range [0.01, 0.21] 
 
III.  A Collection of Attractors with Distinct Morphologies 
We simulate the 4D system keeping unchanged the P parameters and all the graphical 
representations of the trajectories will be projected within the (x, y, z) space. This option will 
assist the analysis of morphological forms and their basins of attraction.  
 
III.1. A non-standard SDIC 
 
Surprisingly, the representation of these normalized 3D projections of the space phases 
depicts a noteworthy morphological variety of the portraits even if the system and the values 
of its parameters are kept unchanged. A non-standard butterfly effect is highlighted. 
The first two projections (fig. 2a, and 2b) display asymmetrical portraits and identifiable 
saddle-focus homoclinic bifurcations leading to a very complex dynamics. Furthermore, the 
third projection (fig. 2c) depicts embedded rolls without, here also, any axis of symmetry. 
Figure 2d shows a lemon-like shape of attractor whose initial conditions seem to belong to a 
zone of regular motion within the basin of attraction. 
 5 
 
 
 
 
Fig. 2. Simulation of the 4D chaotic system with the (unchanged) set of parameters P.  The 4D 
trajectories projected on the space (x, y, z) do not converge to an unique attractor whereas only 
the initial conditions have been modified. The Initial Conditions (a) ICa=( xa, ya, za, va)= (2, 2, 2, 
2), (b) ICb=( xb, yb, zb, vb)= (0.5, 0.5, 0.5, 0.5),  (c) ICc=( xc, yc, zc, vc)= (0.05, 0.05, 0.05, 0.05), and 
(d) ICd=( xd, yd, zd, vd)= (1, 1, 1, 1). 
 
III.2. A rotated copy by π radians for any attractor  
On the other hands, simulations have showed that another type of SDIC operates, albeit with a 
null impact on morphology. Indeed, for a small variation of the initial conditions, an identical  
portrait of the given strange attractor could be created, however with a planar half-turn. This 
is in the same line of the Chua model (Chua and al., 1986; Chua, 1992), or the Bouali model 
(Bouali, 1999) since under a peculiar specification of parameters, each system creates two 
strange attractors obviously with a weak difference of initial conditions. Although located in 
two sub-basin of attraction, the attractors are in reversed position. Nevertheless, this 
duplication is only the effect of the least spectacular SDIC 
The present 4D system creates a pair of portraits for any particular morphology detected in the 
entire basin of attraction (Fig. 3). To note that figure 3a is similar of that of figure 2a. Another 
example of a pair of portraits is given in Figure 4. Here, the figure 4b is similar of that of 
figure 2b. 
 
 6 
 
 
Fig. 3. A pair of 4D trajectories projected on the space (x, y, z) simulated with the (unchanged) set of 
parameters P and (a) ICa=( xa, ya, za, va)= (2, 2, 2, 2), and (b) ICb=( xb, yb, zb, vb)= (2, 2, -2, -2). 
 
 
 
Fig.4. Another pair of portraits on the space (x, y, z) simulated with the (unchanged) set of parameters P 
and (a) ICa=( xa, ya, za, va)= (-0.5, 0.5, -0.5, -0.5), and (b) ICb=( xb, yb, zb, vb)= (0.5, 0.5, 0.5, 0.5). 
 
 7 
 
In fact, the powerful SDIC able to drive trajectories toward distinct strange attractors stemms 
from the original 3D system (Bouali, idem, 2013). Embedded attractors phenomenon had 
been also reported. However, no identifiable morphogenesis could be noticed despite the light 
dissimilarity of rolls and wings of the attractors in the three dimensional space. 
Comparatively, therefore, it is the feedback of the fourth state variable by means of a 
supplementary dimension that triggered a complex mutation of the attractor morphology. 
 
IV. Concluding Remarks 
The Butterfly Effect as the most explicit feature of a strange attractor constitutes a blazing 
legacy of the wide scientific contribution Edward Norton Lorenz (1963). The SDIC reports 
that a small gap between two initial conditions in its idealized meteorological system deflects 
their paths in a way that no prediction of their drift could be done. Indeed, their gap grows at 
exponentially distance. Such singular phenomenon states that a strange attractor is locally 
unstable. However,  it is globally stable since the trajectories converge to the same attractor.  
Our 4D system  moves beyond this result. 
The main outcome of our paper is precisely that a strange attractor is not only locally unstable 
but also it could be globally unstable. Such   novel SDIC allows a marked volatility of the 
chaotic dynamics at two levels. On the one hand, the bifurcation to a different strange 
attractor by a small variation of the initial conditions.  On the other hand, the jump toward a 
rotated copy of any strange attractor by a slight change of the previous initial conditions even 
if this result had been observed elsewhese. 
Such model could be used to preserve cryptography against hacker attacks of the 
communication systems based on the synchronization of chaotic oscillators (Frasca and al., 
2018; Huang, 2004; Kocarev and Parlitz, 1995; Pecora and Carroll, 1990). Even if the 
oscillator parameters usually used as secret keys are recovered, the chaotic system remains 
unknow due to its drastic SDIC associating both volatility of morphology or rotation. 
Indeed, capture of both parameters and initial conditions appears costly in lenght time of 
computation, and highly uncertain in case of jump from a chaotic generator to an other. In this 
case, the predictability horizon stands far beyond the computation time and the physical 
latences of the (hacking) circuit receiver.   
 
References 
 
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cycles, Int. J. Bifurcation and Chaos, 9, 4, pp. 745–756.  
Bouali S. (2013), A 3D Strange Attractor with a Distinctive Silhouette. The Butterfly Effect Revisited, 
e-print, arXiv:1311.6128v2 [nlin.CD] 
Chua L.O. (1992), The genesis of Chua’s circuit. AEÜ, Int. J. Electron. Commun., 46, 4, pp. 250–257.  
Chua L., Komura M., and Matsumoto T. (1986), The double scroll family, IEEE Trans. Circuits Syst., 
33, pp. 1072-1118. 
Frasca M., Gambuzza L. V., Buscarino A., Fortuna L. (2018), Synchronization in Networks of 
Nonlinear Circuits: Essential Topics with MATLAB® Code, SpringerBriefs in Nonlinear Circuits, 
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 8 
 
Gilmore R. and Lefranc M. (2011), The Topology of Chaos: Alice in Stretch and Squeezeland, Wiley 
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Huang D. (2004), Synchronization-based estimation of all parameters of chaotic systems from time 
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Joglekar M., Ott E., and Yorke J. A. (2014), Scaling of Chaos Versus Periodicity: How Certain Is It 
that an Attractor Is Chaotic?, Phys. Rev. Lett., 113, 8, pp. 084101-1, -4. 
Kantz H., and Grassberger P. (1985), Repellers, Semi-Attractors, and Long-Lived Chaotic Transients, 
Physica D: Nonlinear Phenomena, Vol.17,  1, pp. 75-86. 
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communication, Physical Review Letters, 74, pp. 5028–5031. 
Lai Y.-C., and Tél T. (2011), Transient Chaos, Complex Dynamics on Finite-Time Scales, Springer, 
New York.  
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Strange Attractor Morphogenesis by Sensitive Dependence on Initial Conditions

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Strange Attractor Morphogenesis by Sensitive Dependence on Initial Conditions

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