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M. C. Depassier

Publications and source records attributed to M. C. Depassier.

28 records · Page 2Linked to original sources
arXiv2001

Variational approach to a class of nonlinear oscillators with several limit cycles

We study limit cycles of nonlinear oscillators described by the equation $\ddot x + νF(\dot x) + x =0$. Depending on the nonlinearity this equation may exhibit different number of limit cycles. We show that limit cycles correspond to relative extrema of a certain functional. Analytical results in the limits $ν->0$ and $ν-> \infty$ are in agreement with previously known criteria. For intermediate $ν$ numerical determination of the limit cycles can be obtained.

nlin.CD↗
arXiv1997

Phase Space Derivation of a Variational Principle for One Dimensional Hamiltonian Systems

We consider the bifurcation problem u'' + λu = N(u) with two point boundary conditions where N(u) is a general nonlinear term which may also depend on the eigenvalue λ. A new derivation of a variational principle for the lowest eigenvalue λis given. This derivation makes use only of simple algebraic inequalities and leads directly to a more explicit expression for the eigenvalue than what had been given previously.

patt-sol↗
arXiv1996

On a Conjecture of Goriely for the Speed of Fronts of the Reaction--Diffusion Equation

In a recent paper Goriely considers the one--dimensional scalar reaction--diffusion equation $u_t = u_{xx} + f(u)$ with a polynomial reaction term $f(u)$ and conjectures the existence of a relation between a global resonance of the hamiltonian system $ u_{xx} + f(u) = 0$ and the asymptotic speed of propagation of fronts of the reaction diffusion equation. Based on this conjecture an explicit expression for the speed of the front is given. We give a counterexample to this conjecture and conclude that additional restrictions should be placed on the reaction terms for which it may hold.

patt-sol↗
arXiv1996

A Variational Principle for Eigenvalue Problems of Hamiltonian Systems

We consider the bifurcation problem $u'' + λu = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $λ$. We give a variational characterization of the bifurcating branch $λ$ as a function of the amplitude of the solution. As an application we show how it can be used to obtain simple approximate closed formulae for the period of large amplitude oscillations.

patt-sol↗
arXiv1995

The Speed of Fronts of the Reaction Diffusion Equation

We study the speed of propagation of fronts for the scalar reaction-diffusion equation $u_t = u_{xx} + f(u)$\, with $f(0) = f(1) = 0$. We give a new integral variational principle for the speed of the fronts joining the state $u=1$ to $u=0$. No assumptions are made on the reaction term $f(u)$ other than those needed to guarantee the existence of the front. Therefore our results apply to the classical case $f > 0$ in $(0,1)$, to the bistable case and to cases in which $f$ has more than one internal zero in $(0,1)$.

patt-sol↗
arXiv1994

A Variational Principle for the Asymptotic Speed of Fronts of the Density Dependent Diffusion--Reaction Equation

We show that the minimal speed for the existence of monotonic fronts of the equation $u_t = (u^m)_{xx} + f(u)$ with $f(0) = f(1) = 0$, $m >1$ and $f>0$ in $(0,1)$ derives from a variational principle. The variational principle allows to calculate, in principle, the exact speed for arbitrary $f$. The case $m=1$ when $f'(0)=0$ is included as an extension of the results.

patt-sol↗
arXiv1994

New exact fronts for the nonlinear diffusion equation with quintic nonlinearities

We consider travelling wave solutions of the reaction diffusion equation with quintic nonlinearities $u_t = u_{xx} + μu (1 -u ) ( 1 +αu + βu^2 +γu^3)$. If the parameters $α, β$ and $γ$ obey a special relation, then the criterion for the existence of a strong heteroclinic connection can be expressed in terms of two of these parameters. If an additional restriction is imposed, explicit front solutions can be obtained. The approach used can be extended to polynomials whose highest degree is odd.

patt-sol↗
arXiv1994

On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation

We consider the problem of the speed selection mechanism for the one dimensional nonlinear diffusion equation $u_t = u_{xx} + f(u)$. It has been rigorously shown by Aronson and Weinberger that for a wide class of functions $f$, sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed $c^*$ such that $2 \sqrt{f'(0)} \leq c^* < 2 \sqrt{\sup(f(u)/u)}$. The lower value $c_L = 2 \sqrt{f'(0)}$ is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the the speed of the selected front, this bound depends on $f$ and thus enables us to assess the extent to which the linear marginal selection mechanism is valid.

patt-sol↗