Codimension-three bifurcations: Explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps

Many physical and engineering systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied. Such systems have been found to yield piecewise smooth maps, and in some cases the obtained map is discontinuous. By investigating the normal form of such maps, we have detected a type of codimension-three bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space. The results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering.

Figure 1

Typical bifurcation scenarios observable in piecewise smooth maps under variation of one parameter: pure period increment (a), period increment with coexistence of attractors (b), period increment with period adding inclusions (c), and period increment with chaotic inclusions (d). Top row: one-parameter bifurcation diagrams; bottom row: the corresponding period diagrams. The scenarios are shown for system (1) at $a=1$, $l=-1$, and $b=-0.3$ (a); $b=0$ (b); $b=0.3$ (c). The scenario (d) occurs in the same system at $b=0.46$, $\mu =-50$, and $l=-1$.

Figure 2

Numerically calculated bifurcation structures of the $\Sigma ∕\Delta$ modulator in the parameter plane ${\Pi }_{\Sigma ∕\Delta }$ caused by border collision bifurcations. Marked are areas corresponding to periodic dynamics with different periods as well as chaotic and divergent behavior. The numerically calculated insets show the bifurcation diagrams along the marked lines.

Figure 3

Numerically determined bifurcation structures formed by codimension-one border collision bifurcation surfaces of system (1) for the case $l=-1$ induced by the two codimension-three big bang bifurcations at the points $B_1^{-}=(0,0,0)$ and $B_2^{-}=(0,0,1)$. The dashed lines mark the plane ${\Pi }_{\Sigma ∕\Delta }$ to demonstrate the connection to Fig. 2.

Figure 4

Left: two-parameter bifurcation diagrams showing numerically determined unfoldings of the three types of codimension-two big bang bifurcations, generating (a) a period increment scenario with pairwise coexistence of periodic orbits for $b=-1$, (b) a pure period increment scenario for $b=0$, and (c) a period adding scenario for $b=1$. Right: bifurcation scenarios along the arc around the codimension-two bifurcation points marked in each figure in the left-hand column.

Figure 5

Numerically determined bifurcation structures formed by codimension-one border collision bifurcation surfaces of system (1) for the case $l=1$ induced by the two codimension-three big bang bifurcations at the points $B_1^{+}=(0,-1,0)$ and $B_2^{+}=(0,-1,\infty )$. Note that the coordinate transformation $\eta \to \arctan \left(\eta \right)$ for $\eta ∊\{a,b,\mu \}$ has been used to locate the codimension-three bifurcation point within finite parameter range. $B_3^{+}$ and $B_4^{+}$ are not shown for the sake of clarity of the diagram, but their existence can be inferred because of the symmetry of the parameter space.

Figure 6

(a) Analytically determined structure of the plane $a\times b$ in the region of periodic and chaotic dynamics for $l=0$ and a sufficiently large value of $\mu$. (b) Bifurcation scenario along the arc marked in (a) calculated numerically. (c) Analytically calculated structure of the 3D parameter space $a\times b\times \mu$ in the case $l=0$ for $\mu >0$. Note that the same parameter scaling as in Fig. 5 is used here.

Figure 7

Bifurcation structures corresponding to the influence region of the codimension-three bifurcation $B_1^{-}$ and an embedding of the plane ${\Pi }_{\Sigma ∕\Delta }$ in this structure.