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Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

  • * Corresponding author: Luciano Lopez

    * Corresponding author: Luciano Lopez 
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  • Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

    Mathematics Subject Classification: Primary: 65L05; Secondary: 34L99.


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  • Figure 1.  System (13): Crossing and attractive sliding region for $\lambda = -10$ (up) and $\lambda = 15$ (down)

    Figure 2.  System (13). Up: Behaviour of the the singular sets with respect to $\lambda$ space and for $\lambda = -5$ Down: Projection of the singular sets on the $x_1\lambda$ plane and crossing /sliding region

    Figure 3.  System (15). Crossing and sliding regions for $F = -1$ (up) and $F = 1$ (down)

    Figure 4.  System (15). Localization of the singular sets in the $x_1x_2F$ space for $F = 0$ (up) and $F = 1$ (down)

    Figure 5.  2D continuation method: Computation of the first patch

    Figure 6.  Advancing the front: General step. Incomplete patch centered at a frontal point (A) and completed patch (B) are colored in yellow

    Figure 7.  Boundaries of initial sets for Example 2.1

    Figure 8.  Reconstruction of the separatrices surfaces from some collection of points (blue and red dots on the surfaces). The black line illustrates a piece of a trajectory starting from the initial point $[1,1,5]^{\top}$. The chosen initial condition belongs to the subregion of points in $R_2$ starting from which any trajectory (forward in time) reaches the crossing region on the discontinuity surface $x_2 = 0$

    Figure 9.  Projections of the numerical solution obtained solving (15) from the initial point $[1,1,5]^{\top}$. The red and blue curves on the sliding surface $x_2 = 0$ (the first plot) represent the singular curves depicted in the selected region and obtained using the continuation algorithm

    Figure 10.  Reconstruction of some portions of the separatrix surfaces $m_1$ in $R_1$ (red surface) and $m_2$ in $R_2$ (blue surface) obtained interpolating a collection of points (red and blue dots on the surface, respectively) randomly chosen from trajectories obtained integrating -backward in time- the PWS system

    Figure 11.  Separatrix surface $m_2$ in $R_2$ and exit curves on the discontinuity surface $\Sigma$

    Figure 12.  3D example via 2D continuation, first portion

    Figure 13.  3D example via 2D continuation, second portion

    Table 1.  Values of the interpolated separatrix surface on different points in the interpolation domain

    SeparatrixPoint Interpolared surface value
    $m_1$ $[1,1,5]^{\top}$ $>0$
    $m_1$ $[0.5, -0.5, 10]^{\top}$ $>0$
    $m_1$ $[0,1,-0,8]^{\top}$ $<0$
    $m_2$ $[1,1,5]^{\top}$ $ < 0$
    $m_2$ $[0.5, 0.5, 10]^{\top}$ $ < 0$
    $m_2$ $[0,1,20]^{\top}$ $>0$
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