Abstract
The third order explicit autonomous differential equations named as jerk equations represent an interesting subclass of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we show that an algebraically simple system, the Genesio system can be recast into a jerky dynamics and its jerk equation can be derived from one-dimensional Newtonian equation. We also investigate the global dynamical properties of the corresponding jerk system.
Highlights
The term jerk [1], i.e., the third derivative of displacement, x, has attracted some attention because of its relevance to the theory of chaos [2,3,4,5,6,7,8,9,10,11]
We show that an algebraically simple system, the Genesio system can be recast into a jerky dynamics and its jerk equation can be derived from one-dimensional Newtonian equation
We show that the Genesio system can be recast into a jerky dynamics by an affine transformation and the resulting form belongs to class JD2
Summary
The term jerk [1], i.e., the third derivative of displacement, x , has attracted some attention because of its relevance to the theory of chaos [2,3,4,5,6,7,8,9,10,11]. Linz [5,6] introduced the idea and conditions for Newtonian jerky dynamics, derivable by differentiation of a (one-space dimension) Newtonian equation of motion for x , and analyzed the jerky dynamics for onevariable obtained from several familiar autonomous systems of three simultaneous first-order ordinary differential equations which are known to have chaotic solutions He allowed for the possibility of a memory or temporal history integral term in the force function. Such a classification provides simple means to compare the functional complexity of different systems and demonstrate the equivalence of cases not otherwise apparent. We investigate the global dynamics of that jerk equation and show that it shares the common route to chaos as systems in class JD2
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