grandes-ecoles 2017 QIV.C.2

grandes-ecoles · France · centrale-maths2__psi Systems of differential equations

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit a non-zero periodic solution of period $T$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system
$$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$
$B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit a non-zero periodic solution of period $T$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QII.D.1 QII.D.2 QII.E.1 QII.E.2 QII.E.3 QIII.A QIII.B.1 QIII.B.2 QIII.C QIII.D.1 QIII.D.2 QIII.E.1 QIII.E.2 QIII.E.3 QIV.A.1 QIV.A.2 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2 QV.A QV.B.1 QV.B.2 QV.B.3 QV.B.4