(Lecture notes for the M2R-DET course, D. Fabre, nov. 2018)

    Work in progress !

    This documents gives mathematical support for the study of linear dynamical systems. The notions will be used throughout the course, especially in lecture 1 and lecture 9.

    Paragraphs in green can be skipped at first lecture.

    Introduction and General definitions

    • A dynamical system (DS) of order N is by definition a system of N order-one ordinary differential equations (ODEs) written as follows : \displaystyle \frac{d X}{dt} = F(X,t) where X(t) = [x_1(t);x_2(t);x_3(t); ... x_N(t)] is a N-dimensional state variable belonging to the phase space

    • Remark: Dynamical systems are a particular cases of “algebraic differential systems” with the form E \frac{d X}{dt} = F(X,t) where E is a non-invertible matrix. Such systems are encountered in cases where we have both dynamical equations and constraints. In most cases algebraic differential systems can actually be treated as dynamical ones.

    • The dynamical system is said to be autonomous if F(X,t)=F(X).

    • The function F(X,t) is called the flow of the DS.

    • A trajectory in the phase space is defined by the curve X(t) corresponding to the solution of the system corresponding to a given initial condition (i.e. the speficifation of X(t=0) = X_{init}).

    • A fixed point or equilibrium point of an autonomous DS is a point X=X^0 such as F(X^0) = 0. It is thus a particular type of trajectory reduced to a single point.

    • A limit cycle is a closed trajectory in the phase-space.

    • A phase portrait is a graphical representation of a selection of trajectories of the DS (including the most significant ones such as fixed points, limit cycles, etc…).

    Rather than focusing on individual trajectories corresponding to specific initial conditions, the general idea of DS theory is to consider the dynamics of the system from a global point of view by investigating the flow in the phase space in a geometrical and topological way.

    Illustration : the pendulum

    ( use program PhasePortrait_NonLinear.m

    Conservative vs. non-conservative systems

    In the framework of DS theory, a system is said to be conservative if the flow verifies div(F) = \partial f_i / \partial x_i = 0.

    On the other hand, a system is said to be dissipative if div(F) <0 (and anti-dissipative if div(F) > 0)

    Conservative systems are usually encountered in idealised cases. Realistic cases (especially in the field of fluid mechanics) are generally dissipative.

    A geometrical interpretation of the property div(F) <0 is that the flow is contracting; as a consequence, in dissipative systems the trajectories converge towards specific regions of the phase-space called atractors. The most simple kinds of attractors are fixed points and limit cycles, but other kind can me encountered, including strange ones…

    Note : Dynamical systems vs. Hamiltonian systems

    The formalism of Dynamical systems is close to the one of Hamiltonian mechanics. However, there is a important difference. In Hamiltonian mechanics, a Hamiltonian system is defined by the state-vector X = [x_1(t), x_2(t), ... x_N(t) ; \dot{x}_1(t), \dot{x}_2(t), ... \dot{x}_N(t)]. Hence, a hamiltonian system of order N corresponds to a dynamical system of order 2N.

    The definitions of conservative and dissipative systems are also rather different in both theories. A hamiltonian system is said to be conservative if it is time-reversible ; a consequence is the existence of an invariant of the movement, i.e. a positive function E(X) (identified with the total energy) such as dE/dt = 0.

    On the other hand a hamiltonian system is said to be dissipative if dE/dt \le 0 along each trajectory.

    For a general differential system, it is generally not possible to introduce an energy function verifying these properties However there are many cases where a dynamical system origniating from a physical model has an underlying structure and where defining an energy is possible.

    The two definitions also turn out to be equivalent in the very important case of dynamical systems of dimension 2.

    Stability of fixed-points


    • A fixed point X^0 is said to be stable if there exists a neighborhood \Omega_0 of X^0 such that X(t) \rightarrow X_0 for all initial conditions X(t=0) \in \Omega_0 .

    - A fixed point is said to be exponentially stable (or linearly stable) if it is stable and verifies the additional condition : ||X(t)-X^0|| \leq C e^{-\gamma t} for some positive constants \gamma and C.

    - A fixed point is said to be monotonically stable if it is stable and verifies the additional condition : d||X-X^0||/dt \leq 0.

    • A fixed point is said to be unstable if there exists at least one trajectory starting from an initial condition aribrarily close to X^0 and diverging from this point.

    Linearization of a dynamical system in the neighborhood of a fixed point.

    The general method for studying the stability properties of a fixed-point consists of Linearizing the DS in the vicinity of the fixed-point.

    For this we consider small-amplitude perturbations by assuming:

    \displaystyle X = X^0 + \epsilon X'(t)

    Then a Taylor expansion leads to :

    \displaystyle \frac{ d X'}{dt } + A X' + \mathcal{O}( \epsilon)

    Wnere A is the Jacobian matrix or the DS :

    \displaystyle A = \left[ \nabla F \right]_{X^0} \qquad ( i.e. A_{ij} = [ \partial f_i / \partial x_j ])

    Assuming \epsilon \ll1, we can neglect the higer-order terms (nonlinear terms). We can thus replace the system by a linear one which is much easier to study.

    Eigenvalue analysis

    In the two next section we assume that the fixed point is at the origin, i.e. X^0 = 0 and X \equiv X'.

    Except in “exceptional cases”, the solution of the problem can be written as follows:

    \displaystyle X(t) = \sum_{\lambda_n \in {\mathcal Sp}} c_n \hat X_n e^{\lambda_n t} \qquad \mathrm{ (Eq. 1)}

    where \lambda_n are the eigenvalues and the corresponding \hat{X}_n the eigenvectors. The latter are solutions of the eigenvalue problem: \displaystyle \lambda \hat{X} = A \hat{X}

    We supose that the eigenvalues are sorted in the order of decaying real part, namely \displaystyle Re(\lambda_1) \geq Re(\lambda_2) \geq ... \geq Re(\lambda_N)

    According on the eigenvalues we are in one of the three following cases:

    • The system is linearly unstable if there is at least one eigenvalue with positive real part, namely : if Re(\lambda_1)>0

    • Reversely, the system is linearly stable if all eigenvalues have negative real part, namely : if Re(\lambda_1)<0.

    • If Re(\lambda_1)=0 the linear system is said to be marginally stable. In this case it is necessary to consider the nonlinear terms to decide the stability.

    Note that if A is symmetric (or hermitian), the condition for stability implies monotonous stability. If A is nonsymmetric, on the other hand, the system may be exponentially stable but not monotonously stable. This situation corresponds to “transient growth” and will be reviewed in lecture 9. <>

    A more complete document on linear systems of order N can be found here. In the sequel of this document we restrict to the case of 2-dimensional dynamical systems (N=2).

    Classification of fixed points in two dimensions

    A 2x2 system can be written as follows :

    \displaystyle \frac{d}{dt} \left[ \begin{array}{c} x_{1} \\ x_{2} \end{array} \right] = \left[ \begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \end{array} \right]

    We look for the eigenvalues/eigenvectors by solving the eigenvalue problem : A \hat{X} = \lambda \hat{X}.

    If \lambda_1 and \lambda_2 are distinct, the solution of this problem can be expressed in terms of the eigenvalues/eigenvectors, as follows :

    \displaystyle X(t) = c_1 \hat X_1 e^{\lambda_1 t} + c_2 \hat X_2 e^{\lambda_2 t}

    \lambda_1 and \lambda_2 are the solutions of the characteristic polynomial $ det (A - I) =0$ which can also be written as follows :

    \displaystyle \lambda^2 - tr (A) \lambda + det(A) = 0

    Classification of fixed points : node, focus, saddle, center.

    Classification of fixed-points in 2-dimensional systems

    Classification of fixed-points in 2-dimensional systems

    Practical work :

    Study the possible types of fixed points in 2D using the program PhasePortrait_Linear.m



    We consider an autonomous system depending upon a control parameter r, as follows:

    \displaystyle \frac{d X}{dt} = F(X,r)

    Definitions :

    • A bifurcation is a qualitative modification of the behaviour of a system occuring for a specific value r=r_c, generally associated to a modification of the number of fixed points and./or limit cycles, and a change of their stability properties.

    • A fixed-point bifurcation occurs if for r=r_c the linearized system at a fixed point admits (at least) one eigenvalue with zero real part.

    • We call a bifurcation diagram a representation of the amplitude (with a convenient definition) of the fixed-point solutions (and possibly limit cycles) as function of the control parameter. By convention, stable solutions will be represented with full lines, and unstable ones with dashed lines.

    Illustrations (practical work)

    Using programs PhasePortrait_NonLinear.m and PhasePortrait_NonLinear.m, build the bifurcation diagrams of the following problems: - Rotating pendulum - Inverted pendulum - Brusselator

    Mathematical analysis of fixed-point bifurcations


    • The dimension n_b of a bifurcation is the number of eigenvalues which cross the real axis for r=r_c. The generic cases are n_b = 1 (one real eigenvalue) and n_b = 2 (a pair of complex eigenvalues). All other cases are exceptional (or “codimension-2”) and not considered here.

    • The central manifold is a geometrical manifold of dimension n_b on which all trajectories in the vicinity of the bifurcation point “relax rapidly” and then “slowly evolve” along it. One can justify that this manifold is tangent to the subspaces generated by the nearly-neutral eigenmodes of the fixed points.

    • The normal form of a bifurcation is a generic DS of dimension n_b describing the dynamics along the central manifold, to which each type of bifurcation can be reduced by a series of “elementary” bifurcations.

    One-dimensional bifurcations

    One-dimensional bifurcation are the one associated to a single real eigenvalue. The central manifold is a line and the dynamics can be reduced to a one-dimensional equation in terms of a curvilinear coordinate x along the manifold :

    \displaystyle dx/dt = f(x;r)

    It is convenient to introduce the potential function V(x) such that f(x) = -dV/dx. Hence the fixed points of the function f(x) corresponds to extremums of the potential. One can demonstrate that stable fixed points are minimums and unstable ones are maximums.

    The pitchfork bifurcation

    The normal form of a pitchfork bifurcation occuring at r_c = 0 is :

    \displaystyle dx/dt = f(x;r) = r x - s x^3 \quad \mathrm{ with } \quad s = \pm 1.

    If s=1 the bifurcation is supercritical and for r>0 there exist two stable solutions. On the other hand if s=-1 the bifurcation is subcritical and no stable solution exist for r>0 (on the other hand two unstable solutions exist for r<0.

    The pitchfork bifurcation is generic to systems admitting a spatial (reflexion) symmmetry.

    The transcritical bifurcation

    The normal form of a transcritical bifurcation occuring at r_c = 0 is :

    \displaystyle dx/dt = f(x;r) = r x - x^2.

    This bifurcation corresponds to the exchange of stability of two fixed-point solutions. It is much less comon in fluid dynamics.

    The saddle-node bifurcation

    The normal form of a saddle-node bifurcation occuring at r_c = 0 is :

    \displaystyle dx/dt = f(x;r) = r - x^2

    This bifurcation is a “fold” connecting a stable neutral-point and an unstable neutral point.

    two-dimensional bifurcations

    Two-dimensional bifurcations include the Hopf bifurcation (two purely imaginary and opposite eigenvalues). And the Takens-Bogdanov* bifurcation (two zero eigenvalues). The latter case is exceptionnal (or “codimension-two”) so we only consider the first.

    Hopf bifurcation

    There is no unique definition of the “normal form” of a Hopf bifurcation, but it is generally possible to reduce the problem to the Van der Pol equation

    \displaystyle \ddot x + \omega_0^2 f(x;r) = (r - s x^2) \dot x \quad \mathrm{ with } \quad s = \pm 1.

    • If s=+1 the bifurcation is supercritical and a stable limit cycle exists for r>0.

    • If s=-1 the bifurcation is subcritical and an unstable limit cycle exists for r<0.

    Perturbation methods allow to predict the solution for r\ll 1 (averaging method ; multiple-scale methods).

    Practical work

    Using programs PhasePortrait_NonLinear.m and PhasePortrait_NonLinear.m, build the bifurcation diagrams of the buffalo-wolf system.


    1. Charru, exercices 1.6.4 and 1.6.5 (warning : there is an error in equations 1.62 and 8.4 ! the sign of V(A) must be changed)

    2. (Charru, exercice 11.7.11) Build the bifurcation diagram of the following amplitude equation, where \mu is the bifurcation parameter: \displaystyle \frac{d x}{d t} = x (x^2+\mu)(x^2+\mu^2-1)

    3. (Charru, exercice 11.7.12) Build the bifurcation diagram of the following amplitude equation, where \mu is the bifurcation parameter: \displaystyle \frac{d x}{d t} = -4 x ( (x-1)^2-\mu-1 )

    Verify that the bifurcation diagram is the one given in figure 11.20 of Charru.

    (warning : there is a sign error in the book !)

    1. Charru, exercice 11.7.13. You may use the program PhasePortrait_NonLinear.m to draw phase portraits of the system for several values of the bifurcation parameter \mu.

    2. Consider the dynamical system defined as follows : \displaystyle \frac{d}{d t} \left[ \begin{array}{c} x_1 \\ x_1 \end{array} \right] = \left[ \begin{array}{c} r x_1 - x_1^3 -3 x_2^2 x_1 \\ (r-1) x_2 + x_1^2 x_2 - x_2^3 \end{array} \right]

    where r is a control parameter.

    1. Using the program PhasePortrait_NonLinear.m, draw phase portraits for various values of r.

    2. Study the number of fixed points and their stability as function of r.

    3. Regroup the results under the form of a bifurcation diagram showing the amplitude parameter A= |x_1|+|x_2| of all solutions as function of r.