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Dynamical tangles in third-order oscillator with single jump function.

Petr┼żela J, Gotthans T, Guzan M - ScientificWorldJournal (2014)

Bottom Line: This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity.The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems.The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

View Article: PubMed Central - PubMed

Affiliation: Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic.

ABSTRACT
This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

Show MeSH
Trajectory close to the heteroclinic orbit for dynamical system cases A (a) and B (b).
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Related In: Results  -  Collection


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fig6: Trajectory close to the heteroclinic orbit for dynamical system cases A (a) and B (b).

Mentions: The strategic orbits are unstable such that for numerical integration the initial conditions must be chosen carefully. For the strategic orbits, the process of numerical integration must begin in the close neighborhood of the fixed point. Using time-forward integration the state point goes into the direction of unstable manifold towards the opposite equilibrium. The saddle loop is finished by using time-backward numerical integration. The resulting state trajectories for both sets of the parameters are given in Figure 6. Despite being structurally unstable, these trajectories can be constructed and destructed via a manipulation with vector field geometry, that is, by changing the internal system parameters. Recently it has been verified that inverse approach can be used; starting with the satisfaction of one ST the original mathematical model can be derived [21].


Dynamical tangles in third-order oscillator with single jump function.

Petr┼żela J, Gotthans T, Guzan M - ScientificWorldJournal (2014)

Trajectory close to the heteroclinic orbit for dynamical system cases A (a) and B (b).
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4269091&req=5

fig6: Trajectory close to the heteroclinic orbit for dynamical system cases A (a) and B (b).
Mentions: The strategic orbits are unstable such that for numerical integration the initial conditions must be chosen carefully. For the strategic orbits, the process of numerical integration must begin in the close neighborhood of the fixed point. Using time-forward integration the state point goes into the direction of unstable manifold towards the opposite equilibrium. The saddle loop is finished by using time-backward numerical integration. The resulting state trajectories for both sets of the parameters are given in Figure 6. Despite being structurally unstable, these trajectories can be constructed and destructed via a manipulation with vector field geometry, that is, by changing the internal system parameters. Recently it has been verified that inverse approach can be used; starting with the satisfaction of one ST the original mathematical model can be derived [21].

Bottom Line: This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity.The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems.The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

View Article: PubMed Central - PubMed

Affiliation: Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic.

ABSTRACT
This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

Show MeSH