Bifurcation diagram for the damping coefficient γ.

<p><i>P</i>_<i>m</i> = 0.5. In the figure, from right to left: the transition scenario is from fixed point → periodic motion → chaos → periodic motion → period-doubling cascade → chaos → collapse. The parameter regions are indicated by the arrows here.</p

Dynamics for system collapse within area II.

<p>The parameters are <i>P</i>_<i>m</i> = 0.5 [(a), (b), and (c)] and <i>P</i>_<i>m</i> = 1.4 [(d), (e), and (f)]; γ = −0.35. Similarly the rotor-angle instability is dominant and the voltage remains stable with its magnitude keeping within a finite value region.</p

Dynamics and Collapse in a Power System Model with Voltage Variation: The Damping Effect - Fig 13

<p><b>Phase diagram on the <i>M</i>- γ parameter plane; <i>P</i></b>_{<b><i>m</i></b>}<b>= 0.5 and γ<0 (a) and bifurcation diagram with the change of damping coefficient γ; <i>M</i> = 2.0 (b) in the third-order power system.</b> Again in (a), area <b>I</b> indicates stable fixed point, <b>II</b> represents explosive solutions for system collapse, and <b>III</b> shows very rich dynamics, such as periodic orbits (orange-yellow), chaos (yellow). In (b), from right to left: the transition scenario is similar, from fixed point → periodic motion → chaos → periodic motion → period-doubling cascade → chaos → collapse at γ = −0.2087. We can find that opposite to the case for positive damping in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165943#pone.0165943.g004" target="_blank">Fig 4</a>, now larger inertia constant <i>M</i> enlarges the stable fixed point area <b>I</b>.</p

Modal analysis showing the loci of eigenvalues of fixed point with the change of γ; <i>P</i>_<i>m</i> = 0.5 <i>and M</i> = 1.0 are fixed.

<p>We can find that with decrease of γ from γ = 2.0 to γ = −0.5 with Δγ = −0.1, a pair of leading characteristic values (λ₁ and λ₂) comes across the imaginary axis from the left (stable region) to right (unstable region) at γ = −0.0151, indicating that the fixed point becomes unstable due to a super-critical Hopf bifurcation. This is coincident with our observation in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165943#pone.0165943.g005" target="_blank">Fig 5</a> for the occurrence of stable fixed point region even when the negative damping is considered. However, more complicated behavior within area <b>III</b> can only be available with nonlinear analysis.</p

Different dynamical behaviors with the change of Pm.

<p>From (a) to (d), the parameter <i>P</i>_<i>m</i>'s are 0.01, 0.024, 0.035, and 0.0375, respectively; <i>γ</i> = −0.1 γ = −0.117 is unchanged. From (a) to (c), the route from period-doubling bifurcation to chaos is clear. In (d), an unusual periodic chaos appears, showing chaotic motion in each rotation but nearly periodic motion between these big rotations. This indicates that different with the boundary crisis shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165943#pone.0165943.g009" target="_blank">Fig 9</a>, an un-collapse periodic chaos behavior, similar to chaotic scroll attractors with infinite scrolls, may still appear.</p

Coexistence dynamics for different initial conditions.

<p>They are stable fixed point [(a) and (c)] and stable limit cycle [(b) and (d)] with the parameters (<i>P</i>_<i>m</i> = 0.5 and γ = 0.2) [denoted by the letter A within area <b>II</b> in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165943#pone.0165943.g002" target="_blank">Fig 2(B)</a> for the third-order system]. They are simulated with the same parameters but two different initial conditions. This coexistence can also be extensively found in the second-order system under the condition of positive damping. In (d), the angle is treated as mod 2π.</p

Dynamical behavior of the stable equilibrium state.

<p>The time-series diagrams [(a)-(c)] and trajectory diagram in the phase space (d) for parameters within area <b>I</b> in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0165943#pone.0165943.g005" target="_blank">Fig 5(B)</a>; <i>P</i>_<i>m</i> = 1.2 and γ = −0.117. Clearly here the equilibrium state persists for negative damping; this is one of key differences with the second-order system.</p

siRNA duplexes used in this study

<p>siRNA duplexes used in this study</p

Experimental design in this study.

<p>Experimental design in this study.</p

LPS increased but baicalin reduced LPS-mediated TLR4 and PPARγ overexpression and phosphorylation.

<p>A: Representative gels for TLR4, PPARγ and p-PPARγ expression. B: Semiquantitation of TLR4, PPARγ and p-PPARγ in each group. ^*<i>P</i><0.05, ^**<i>P</i><0.01 vs. saline group; ^#<i>P</i><0.05, ^##<i>P</i><0.01 vs. LPS group.</p