الفهرس 
Basic Concepts of Non-linear DynamicsOnly 14 pages are availabe for public view
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Abstract
The differential equations have been become one of the essential tools in
understanding physics and engineering problems. It will lasts as an important
tool in mathematical science and its different applications. It goes beyond to
economic, social, biological sciences.
A major feature of cholinergic diseases such as Alzheimer’s and
Parkinson’s diseases is the disturbances and abnormalities occurring in the
components of the Acetylcholine (ACh) neurocycle. A fundamental
understanding of the ACh neurocycle is therefore very critical in order to design
drugs that keep the ACh concentrations in the normal physiological range.
In this dissertation, a novel two-enzyme-two-compartment model is
proposed in order to explore the bifurcation, dynamics, and chaotic
characteristics of the ACh neurocycle. The model takes into consideration the
physiological events of the choline uptake into the presynaptic neuron and the
ACh release in the postsynaptic neuron. The disturbances and irregularities
(chaotic attractors) occurring in the ACh cholinergic system may be good
indications of cholinergic diseases such as Alzheimer’s and Parkinson’s
diseases. As there is strong evidence that cholinergic brain diseases like
Alzheimer’s disease and Parkinson’s disease are related to the concentration of
ACh, the present findings are useful for uncovering some of the characteristics
of these diseases and encouraging more physiological research.
In this work, two different mathematical models have been used. The first
model represents a half neurocycle which concentrated on ACh activities only
(hydrolysis). The second one represents a complete neurocycle which include all
the activities of hydrolysis and producing ( from choline ) of ACh.
Many mathematical methods have been used to discover the dynamics
behavior for the above models. We use the mathematical methods such as
numerical in addition to analytical methods. Numerical methods have been used
when the methods (analytical) failed to get or discover the dynamic solutions of
the above systems.
The first model which is composed of four non-linear 1st order ordinary
differential equations, So dealing with it by the analytical methods which
discover a wide spaces of dynamic phenomena’s (periodic-chaotic). The
numerical methods have been used to compete the picture and to ensure the
availability of analytical methods and discovering of static and dynamic
phenomena’s.
As the importance of quantal feeding in neurocycle in general. The effect
of quantal feeding on the dynamics behavior of the system has been studied.
The results show a realistic change in the nature of dynamic solutions.
For the second model; the numerical methods have been used since the
analytical methods failed completely in dealing with higher dimension systems
(Eight-dimension). Continuation techniques especially (two–Parameter
Continuation technique) have been used to discover a wide ranges of rich
dynamic solutions (steady state - Periodic-Chaotic), the results shows that there
exists a various regions of multiplicity of steady-state ; in addition to periodic
solutions and the results shows also that there exists an isolated static solutions
(Isola).
In order to discover the characteristics of some complex attractors
,Lyapunov exponents technique, the most common way of complex attractors
diagnosis has been used. The study handles also the transition from simple
oscillation to bursting oscillation and it is found that the transient time is very
sensitive for the feed conditions (Highly sensitivity).