# Introduction to the Qualitative Theory of Differential by Jaume Llibre

By Jaume Llibre

The booklet offers with non-stop piecewise linear differential structures within the airplane with 3 items separated by means of a couple of parallel directly strains. in addition, those differential structures are symmetric with admire to the beginning of coordinates. This classification of structures pushed through concrete purposes is of curiosity in engineering, specifically up to speed conception and the layout of electrical circuits. via learning those specific differential platforms we'll introduce the fundamental instruments of the qualitative idea of standard differential equations, which permit us to explain the worldwide dynamics of those platforms together with the infinity. The habit in their ideas, their parametric balance or instability and their bifurcations are defined. The e-book is especially acceptable for a primary direction within the qualitative conception of differential equations or dynamical structures, frequently for engineers, mathematicians, and physicists.

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When t2 − 4d > 0, the matrix A has two real eigenvalues with the same sign, λ1 > λ2 . Therefore if t < 0, then E s = R2 and E u = E c = {0}; and if t > 0, then E u = R2 and E s = E c = {0}. 7, depending on t. The corresponding phase portrait of the system x˙ = Ax is obtained by a linear transformation. The origin is called an asymptotically stable node if t < 0, and an asymptotically unstable node if t > 0. When t2 − 4d = 0, there exists a unique eigenvalue λ, which is real, and the real Jordan normal form of A is λ 0 λ 1 J= or J = .

Therefore, Φ(s, x) = P −1 Ψ(s, P x). 8) From this we obtain the expressions of the ﬂow of any planar linear system. 6. Consider the ﬂow Φ(t, x) of the linear system x˙ = Ax, with A ∈ L(R2 ), d = det(A) and t = trace(A). Let J be the real Jordan normal form of A and P be the matrix such that J = P AP −1 . (a) If t2 − 4d > 0, then esλ1 0 Φ (s, x) = P −1 0 esλ2 P x. (b) If t2 − 4d = 0, then either Φ (s, x) = esλ x or Φ (s, x) = P −1 esλ 0 s esλ P x, depending on whether J is diagonal or not. (c) If t2 − 4d < 0, then Φ (s, x) = esα P −1 cos (βs) − sin (βs) sin (βs) cos (βs) P x.

We see that the origin, which is a focus type singular point, is a global attractor. Also, two separatrices born at the two saddles at inﬁnity are observed. The remaining singular points at inﬁnity are two nodes. 326 KΩ one has T = 0. 6(h). Now the global attractor, namely, the singular point at the origin, is replaced by a central region foliated by periodic orbits (bounded period annulus). The remainder of the phase portrait persists without changes. 326 KΩ we have that T > 0. 9(g). As it can be observed, the period annulus disappears and only one periodic solution persists.