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==== Stability and dynamics ====
[[File:Figure ODE.jpg|thumb|488x488px|'''Fig. 1''' Phase portrait (dashed line) and approximate phase portrait (solid line) of the nonlinear ODE (4.10)-(4.11) computed by the ''LL''2, [[Runge–Kutta methods|''RK''4]], ''LLRK''4 schemes with step size h=1/2 , and p=q=6.]] By construction, the LL and HOLL discretizations inherit the stability and dynamics of the linear ODEs, but it is not the case of the LL schemes in general. With <math>p\leq q\leq p+2</math>, the LL schemes (4.6)-(4.9) are [[Stiff equation|''A''-stable]]<ref name=":3" />. With ''q = p + 1 or q = p + 2'', the LL schemes (4.6)-(4.9) are also [[L-stability|''L''-stable]]<ref name=":3" />. For linear ODEs, the LL schemes (4.6)-(4.9) converge with order ''p + q'' <ref name=":3" /> <ref name= ″:25″ />. In addition, with ''p = q = 6'' and <math>m_{n}</math> ''= d'', all the above described LL schemes yield to the ″exact computation″ (up to the precision of the [[floating-point arithmetic]]) of linear ODEs on the current personal computers <ref name=":3" /> <ref name= ″:25″ /> <ref name=":2" /> <ref name=":9" />. This includes [[Stiff equation|stiff]] and highly oscillatory linear equations. Moreover, the LL schemes (4.6)-(4.9) are regular for linear ODEs and inherit the [[Symplectic geometry|symplectic structure]] of [[Hamiltonian mechanics|Hamiltonian]] [[harmonic oscillator]]s <ref name= ″:25″ /> <ref name=":2" /> <ref name=":9" />. These LL schemes are also linearization preserving, and display a better reproduction of the [[Stable manifold|stable and unstable manifolds]] around [[hyperbolic equilibrium point]]s and [[Limit cycle|periodic orbits]] that [[Numerical methods for ordinary differential equations|other numerical schemes]] with the same stepsize. For instance, Figure 1 shows the [[phase portrait]] of the ODEs
<math>\frac{dx_{1}}{dt} =-2x_{1}+x_{2}+1-\mu f\left( x_{1},\lambda \right)
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