Sliding Hopf bifurcation in interval systems

Figure 1.  (a) Given an $ \alpha $ -parametrized family of characteristic polynomials with unknown coefficients that are limited to some intervals, the dashed lines bound a corridor for the real part $ \tau(\alpha) = {\rm Re} \mu(\alpha) $ of an eigenvalue, while the solid line indicates a sliding scenario for some selector of the family.(b) Possible complex behavior of the branch of periodic solutions

Figure 2.  (a) The dark grey domain that consists of two connected components is the set $ \mathfrak{R} $ of purely imaginary characteristic roots $ i\beta $ of the interval polynomial (17) for Example 4.1.The two solid black curves inside the two components of $ \mathfrak{R} $ show the set of purely imaginary roots for a representative polynomial $ P(\alpha)(\cdot) $ that belongs to the family (17).This representative has two purely imaginary roots $ i\beta_1(\alpha) $ , $ i\beta_2(\alpha) $ for some interval of $ \alpha $ values $ [\alpha_1, \alpha_2]\subset (\alpha_-, \alpha_+) = (-0.4, 0.8) $ .The light grey domains are the sets $ \mathfrak{S}_2 $ and $ \mathfrak{S}_3 $ obtained from the dark grey domain $ \mathfrak{R} $ by the transformations $ (\alpha, \beta) \mapsto (\alpha, \beta/2) $ and $ (\alpha, \beta) \mapsto (\alpha, \beta/3) $ , respectively; the dashed curves inside $ \mathfrak{S}_2 $ are the images of the solid black curves in $ \mathfrak{R} $ under this transformation.The intersection of the solid curve and the dashed curve inside the smaller component of $ \mathfrak{R} $ corresponds to the $ 2:1 $ resonance $ i\beta_1(\alpha) = 2 i\beta_2(\alpha) $ .The dashed quadrangle $ \mathcal{D}_1 $ contains the set $ \mathfrak{R} $ ; its boundary does not intersect $ \mathfrak{S}_i $ in accordance with (R5′).(b) The real parts $ \tau_1(\alpha) $ , $ \tau_2(\alpha) $ of the roots of the representative polynomial $ P(\alpha, \cdot) $ that belongs to the family (17) (schematic).The sliding intervals $ \tau_1(\alpha) = 0 $ , $ \tau_2(\alpha) = 0 $ correspond to the black curves $ \beta_1(\alpha) $ , $ \beta_2(\alpha) $ shown inside the dark grey domain $ \mathfrak{R} $ on panel (a)

Figure 3.  (a) The nonintersecting sets $ \mathfrak{R} $ (dark grey) and $ \mathfrak{S} $ (light grey) for Example 4.2 with $ [\alpha_1, \alpha_2] = [-0.5, 1.2] $ .The black curve is the set of purely imaginary roots of a representative polynomial $ P(\alpha)(\cdot) $ that belongs to the family (17).At the corner point of this curve, $ P(\alpha)(\cdot) $ has a purely imaginary root of multiplicity 2.The real parts of the roots of $ P(\alpha)(\cdot) $ behave as shown on panel (b) of Figure 2. (b) Curves $ (w_1(\alpha_\pm, \cdot), w_2(\alpha_\pm, \cdot)) $ (thick lines) for Example 4.3. The thin curve $ (w_1(\alpha, \cdot), w_2(\alpha, \cdot)) $ with $ \alpha = 0.075 $ from the interior of the interval $ [\alpha_-, \alpha_+] $ intersects the negative cone $ \{(w_1, w_2) : w_1\le 0, w_2\le0\} $