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Application of Jumarie-Stancu collocation series method and multi-Step
possesses a relatively higher dimension of 2.09, a reflection of multiple influences that combine
indicating increased complexity. to affect glucose dynamics and stability overall.
During the regulation of glucose and insulin dy-
3. Bifurcation diagrams namics within the human body, every parameter
from a 1 to a 21 is crucial. Diabetes requires under-
The parameter a 16 could represent the system’s
standing these bifurcations to predict and man-
response to stress or external factors affecting glu-
age the complex dynamics of glucose regulation.
cose regulation. a 16 (-2 to 0) could represent the
Predicting and controlling critical transitions in
system’s response to stress or external factors af-
complex systems, whether biological or physical,
fecting glucose regulation. Bifurcations may indi-
requires understanding bifurcation. Stability may
cate shifting glucose levels due to changing exter-
shift to chaos or vice versa, affecting the stabil-
nal conditions. Related dynamics is bifurcation
ity and behavior of the system. By understand-
in a 16 shows external stressors affect glucose dy-
ing these bifurcations, we can develop strategies
namics. In a bifurcated liver, a parameter a 17
for managing biological processes, designing sta-
could describe the delay in insulin response, with
ble physical systems, and preventing undesirable
bifurcations representing an oscillatory glucose re-
behaviors. Critical transitions can be predicted
sponse. a 17 (0 to 1) may describe insulin response
and controlled by understanding the bifurcation
delay. As response times vary, bifurcations may of parameters in complex systems. During these
indicate changes from stable glucose regulation to
oscillatory or chaotic behavior. The bifurcation transitions, the system can go from stability to
in a 17 might be related to liver response delays. chaos or vice versa, affecting its behavior and sta-
Insulin and other metabolic pathways might be bility significantly. By recognizing these bifurca-
represented by a 18 . There might be interactions tions, we can design stable physical systems and
between insulin and other metabolic pathways in prevent undesirable behaviors like chaotic oscilla-
a 18 . a 18 bifurcations (-1 to 0) may indicate a tions.
change from stable to complex glucose dynam-
ics. A bifurcation in a 18 may give insight into 4. Methods
glucose regulation. An indicator of insulin resis-
tance might be a 19 . There is a possibility that 4.1. Jumarie-Stancu collocation series
the rate at which insulin resistance develops can method
be represented by a 19 (-1 to 0). The point of bi- We apply the JSCSM to approximate the solu-
furcation could result in a transition from stable tions for ˆu(t), ˆv(t), and ˆw(t) in the following steps.
glucose regulation to instability as resistance in- Assume each function can be expanded as a
creases as the point of bifurcation is reached. In series:
the case of an increase in a 19 , this could be a re-
sult of varying rates at which insulin resistance ∞ n
a
ˆ u(t) = Σ n=0 n ι ,
develops as a result of variable insulin resistance
n=0 n ι ,
levels leading to a change in glucose stability. In ˆ v(t) = Σ ∞ b n
the system (1), there might be a parameter called ˆ w(t) = Σ ∞ c n
n=0 n ι ,
a 20 that describes the strength of feedback mech-
anisms that exist. It might be possible to de-
scribe the strength of feedback mechanisms in the where a n , b n , and c n are coefficients to be deter-
Model (1) by the parameter a 20 . It is possible mined for ˆu, ˆv, and ˆw, respectively.
Using the Jumarie fractional derivative, we
that bifurcations in a 20 (-1 to 0) could indicate
approximate the fractional Caputo derivative
that glucose dynamics are changing from stable to
C D α for each series term:
complex due to changes in the strength of feed- 0,ι
back. This could be attributed to the bifurca- ι n−α
C α n
tion of a 20 indicating how variations in feedback D ι ≈ α(n + 1) · α(n + 1 − α).
0,ι
strength can affect the stability of glucose levels.
Then, each derivative can be represented as:
A parameter known as a 21 is used as a potential
C α ∞ C α n
a
indicator of how multi-factor influences glucose D ˆu(t) ≈ Σ n=0 n D (ι ),
0,ι
0,ι
regulation in a system. In order to better under- C D ˆv(t) ≈ Σ ∞ b C α n
α
n=0 n D (ι ),
stand how several factors act on the regulation 0,ι 0,ι
C D w(t) ≈ Σ ∞ c C α n
α
ˆ
0,ι
of glucose, a 21 (-1 to 0) might be a good indica- 0,ι n=0 n D (ι ).
tor. Bifurcations could indicate transitions from
stable to chaotic behavior as these factors vary. Select collocation points {ι 0 , ι 1 , . . . , ι m } within
It is possible that the bifurcation in a 21 could be the time interval of interest.
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