Abstract
The global phase portrait describes the qualitative behaviour of the solution set for all time. In general, this is as close as we can get to solving nonlinear systems. The question of particular interest is: For what parameter values does the global phase portrait of a dynamical system change its qualitative structure? In this paper, we attempt to answer the above question specifically for the case of certain third order nonlinear differential equations of the form . The linear case where is also considered. Our phase portrait analysis shows that under certain conditions on the coefficients as well as the function , we have asymptotic stability of solutions.
Highlights
A line connecting the plotted points in their chronological order shows temporal evolution more clearly on the graph
We describe the notion of the flow of a system of differential equations
Two autonomous systems of differential equations such as (7) and (8) are said to be topologically equivalent in a neighborhood of the origin or to have the same qualitative structure near the origin if there is a homeomorphism Φ mapping an open set U containing the origin onto an open set V containing the origin which maps trajectories of (7) in U onto trajectories of (8) in V and preserves their orientation by time in the sense that if a trajectory is directed from x1 to x2 in U, its image is directed from Φ ( x1 ) to Φ ( x2 ) in V
Summary
A line connecting the plotted points in their chronological order shows temporal evolution more clearly on the graph. The complete line on the graph (i.e. the sequence of measured values or list of successive iterates plotted on a phase space graph) describes a time path or trajectory [1]. A trajectory that comes back upon itself to form a closed loop in phase space is called an orbit [2]. We note that each plotted point along any trajectory has evolved directly from the preceding point. An orbit or trajectory moves around in the phase space with time. The phase space plot is a world that shows the trajectory and its development. A phase space with plotted trajectories ideally shows the complete set of all possible states that a dynamical system can ever be in
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