Tyto poznámky jsou verzí poznámek

• Dalibora Pražáka,
• Martina Křepely,
• Michala Kozáka,

a rukopisných poznámek Tomáše Bárty, viz stránky Dalibora Pražáka, včetně videa přednášek na studentském úložišti.

Sbírka úloh z ODR Tomáše Bárty a Dalibora Pražáka je tady a tady.

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Sylabus

• Kapitola 13,v3_5.10.2023: dynamické systémy; staré verze: v2_5.10, v1

  Změny ve verzi “kapitola13_v4_9.10.2023.pdf” ve srovnání s verzí “kapitola13_v3_5.10.2023.pdf”:

  • v definici dynamického systému přidána vlastnost II.b;
  • zaveden pojem maximality dynamického systému (označení níže definice)
• Kapitola 14: La Sallého princip invariance
• Kapitola 15: Poincaré-Bendixsonova teorie
• Van der Pol oscillator (added 8.11.2023)
• Kapitola 16: Carathéodoryho teorie
• Kapitola 18: Optimální regulace

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Requirements for the exam

Written part contains 3 problems and lasts 120 minutes.

Typical tasks: local/global controllability, Poincaré-Bendixson theory: (non)existence of periodic solutions, necessary optimality conditions (application of the Pontrjagin maximum principle), stability investigation using the approximation of the central variety, bifurcation.

Problems from the year 2020/21: [pdf1], [pdf2]

For the oral part, five questions are chosen randomly:

1. key concept (its knowledge is a necessary condition for continuing in the exam)
2. 2x formulation of a theorem (without proof)
3. 1x easy and 1x hard theorem (formulation and the proof)

List of questions for the WS 2023/24 (not final version: please check for the updates): [draft 11.1.2024]

Please subscribe for the exam in SIS, and write down your name in this spredsheet, so that there are approximately 3 people coming each hour.

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Week 1 (5.10.2023)

Lecture: file “kapitola13_v3_5.10.2023” till topological conjugation (not including).

Comments/issues:

• Lemma 13.1. In the “\(\subset\)” part of the proof we use property V. of dynamical systems, while in the “\(\supset\)” part we use property V.’ instead. Indeed,

  • ”\(\subset\)” is written accurately in syllabus.

  • ”\(\supset\)” is written inaccurately. In fact, we have

    \(y\in\overline{\gamma^+(\varphi(t_k, x_0))}\cap\Omega = \overline{\left\{\varphi(s, \varphi(t_k, x_0)):\ s\in [0,t^+(\varphi(t_k, x_0))\right\}}\cap\Omega.\)

    Using property V’, the latter is a subset of

    \(\overline{\left\{\varphi(s+t_k, x_0):\ t_k\leq s+t_k < t^+(x_0)\right\}}\cap\Omega\) and the argument from syllabus follows.

  So we either need to require both V and V’, or add the condition that \(J(\varphi(t, x_0)) = J(x_0) - t,\) under which V is equivalent to V’. Without that one can try to play with artificially restricting time domain for \(\varphi\) and try to construct dynamical systems for which V and V’ are not equivalent.

• There is no maximality condition so far in the definition of dynamical system. One can try to add it in the following form: \(\forall x_0\in\Omega:\ \neg\left(\exists\lim\limits_{t\to t^+(x_0)-0}\varphi(t, x_0) \in \Omega\right)\) and similarly for \(t\to t^-(x_0)+0.\)

• \(\omega(x_0)\) is indeed compact under requirement that \(\gamma^+(x_0)\) is relatively compact in \(\Omega.\) Indeed, we need to establish closedness (in \(\mathbb{R}^n\)) and boundedness:

  • \(\omega(x_0)\) is closed in \(\Omega,\) i.e. \(\overline{\omega(x_0)} \cap \Omega = \omega(x_0).\)
  • \(\gamma^+(x_0)\) is relatively compact in \(\Omega,\) i.e. \(\overline{\gamma^+(x_0)}\cap\Omega =:K\) is compact. Then \(\gamma^+(x_0) \subset K,\) hence \(\omega(x_0)\subset K.\) Then \(\overline{\omega(x_0)}\subset K\) and therefore \(\overline{\omega(x_0)}\subset\Omega,\) and hence \(\overline{\omega(x_0)}\cap\Omega = \overline{\omega(x_0)},\) and hence \(\omega(x_0)\) is closed in \(\mathbb{R}^n.\)
  • \(\omega(x_0)\subset K,\) hence it is bounded.

• misprint in definition of dynamical system: \(s\) and \(t\) are exchanged in the semigroup equality (in V and V’).

• bad formulation of property II of dynamical system: it is the projection of that set to the \(t\)-axis which is an interval containing zero.

Practice: online practice book, chapter 13, 1(i), 3(i, ii), 4.

Comments/issues:

• problem 3: just using qualitative analysis it is hard to conclude about \(\omega\)-limit set without Dulac criteria. However, for \(-1<x<1,\) consider the function

  \[V(x,y) = (x+y)^2 - \log(1-x^2) = (x+y)^2 + |\log(1-x^2)|.\]

  Taking the derivative along a solution, one finds \(\frac{d}{dt}V(x(t), y(t)) = 2x(t)^2.\) Hence, there are no periodic orbits, and the trajectory spirals outward. It will then approach the lines \(x=\pm1,\) hence those lines form the \(\omega\)-limit set.

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Week 2 (12.10.2023)

Practice:

Problems; Solutions of problems 1- 7.

Please take a look at solutions of problems 1-7 and also at problems 10-14, we did not look at them during class. If something is not clear, please ask.

Comments/issues:

• Regarding problem 9. Indeed, two properties are violated: \(G\) is not open, and \(\varphi\) is not continuous at a point \((t, (0, y)),\) when the time \(t\) is negative.

  • Indeed, taking a sequence \((t_k, x_k, y_k)\to(t, 0, y)\) with \(x_k<0,\) the limit of the function \(\varphi\) is \(\lim\limits_{k\to\infty}\varphi(t_k, (x_k, y_k)) = \lim\limits_{k\to\infty}(t_k + x_k, y_k) = (t, y).\)

  • On the other hand, taking a sequence \((t_k, x_k, y_k)\to(t, 0, y)\) with \(x_k>0,\) the limit of the function \(\varphi\) is \(\lim\limits_{k\to\infty}\varphi(t_k, (x_k, y_k)) = \lim\limits_{k\to\infty}(x_k, t_k + y_k) = (0, t + y).\)

  • We can also translate these calculations into an intuitive understanding. Indeed, the continuity of \(\varphi\) means, that if we take two points \(\widehat x_{0}\) and \(\widetilde x_{0}\) near each other, and travel from them for more or less the same amount of time, we must arrive at two points near each other. In our example it is violated: take a point \((0, y)\) and \((-\varepsilon, y)\) and travel from them for the time \(t<0\) (i.e. we are traveling backward). We will then arrive at two completely far away points: from the point \((-\varepsilon, y)\) we will arrive at the point \((-\varepsilon+t, y)\), while from the point \((0, y)\) we will arrive at the point \((0, y+t).\)

• If we exclude the points \((x_0=0, y_0)\) from consideration in problem 9, i.e. if we take \(\Omega=\left\{(x,y): x\neq0\right\},\) we obtain a dynamical system, all the conditions are satisfied. Note that there is no contradiction with the conclusion of problem 1: the limit \(\varphi(t, (x_0<0, y_0))\) as \(t\to t^+(x_0)=-x_0\), which is equal to \((0, y_0),\) does not lie in \(\Omega.\)

Lecture:

• file “kapitola13_v3_5.10.2023”: topological conjugation.
• file “kapitola14.pdf”: formulation of La Salle theorem, did not prove it at the lecture. Please take a look at the proof by yourself, if everything is clear we will skip the proof in the next lecture, if not, we will prove it.

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Week 3 (19.10.2023)

Lecture:

• file “kapitola14.pdf”: La Salle theorem.
• file “kapitola15.pdf”: formulation of Poincaré-Bendixson theorem, proved 3 auxiliary lemmas. On practice: proved Poincaré-Dulac criteria of non-existence of periodic solutions.

Practice:

In the file Problems Week 3, we did the following problems:

• 4 (\(B=\frac{1}{x}\)),
• 5 (\(B=1\)),
• 6 (\(B=(1+x^2+y^2)^{-1}\)).

Did not manage to prove existence of periodic solutions in problem 11 (problem 16 in chapter 13: “dynamical systems” in the online practice book) – please think about it and we will try to finish it next time.

In problem 12, there is a misprint: the brackets should be the same.

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Week 4 (26.10.2023)

Lecture:

• finished proof of Poincaré-Bendixson, file “kapitola15.pdf”
• started Caratheodory theory.
  • \(\Omega\subset\mathbb{R}^{n+1}, CAR(\Omega):\) on every closed cylinder \(Q\subset\Omega\) the following conditions are satisfied:
    • for all \(x,\) function \(t\mapsto f(t,x)\) is measurable;
    • for almost all \(t,\) function \(x\mapsto f(t,x)\) is continuous;
    • there exists a function \(m(.)\in L^1\) such that \(\lvert f(t,x)\rvert\leq m(t).\)
  • \(x:I=[a,b]\to\mathbb{R}^n\) is a solution to \(x' = f(t,x)\) in the Caratheodory sense, if
    • we have \(\mathrm{graf}(x)\subset\Omega,\)
    • function \(x\in AC(I)\) is absolutely continuous,
    • \(x'(t) = f(t, x(t))\) almost everywhere (a.e.).
  • Lemma 1:
    • function \(x:I\to\mathbb{R}^n\) is continuous, \(\mathrm{graf}(x)\subset\Omega.\) Then \(t\mapsto f(t, x(t))\) is measurable and integrable.
  • Lemma 2 (integral formulation of ODE):
    • function \(x:I\to\mathbb{R}^n\) is continuous, \(\mathrm{graf}(x)\subset\Omega.\) Then \(x\) solves ODE in Caratheodory sense iff \(x(t_2)-x(t_1) = \in_{t_1}^{t_2}f(s, x(s))ds\) for all \(t_1, t_2.\)
  • Theorem (only formulated, did not proved yet):
    • we have \(f:[0, T]\times\mathbb{R}^n\to\mathbb{R}^n\) is a Caratheodory function, satisfying
    • generalized Lipschitz condition: there exists a function \(m\in L^1\) such that
      • for all \(t, x, y:\) \(\lvert f(t, x) - f(t, y\rvert) \leq m(t)\lvert x - y \rvert.\)
    • Then there exists a unique solution to the ODE satisfying initial condition \(x(0) = x_0,\) and that solution depends continuously on \(x_0.\)

Comments/issues:

• in the proof of Lemma 1,
  • we constructed a sequence of piecewise and hence measurable functions \(x_k(t),\) which converge pointwise (and actually uniformly) to \(x(t).\)
  • then argument was that since \(x_k(t) \to x(t),\) then \(f(t, x_k(t)) \to f(t, x(t))\) for those values of \(t,\) for which the function \(x\mapsto f(t, x)\) is continuous (with respect to the second variable).
  • There is no problem here, since the function \(f(t, .)\) is continuous in \(x\) for all \(x\), so it could not happen that \(x(t)\) is an excluded value from the continuity.

Practice:

• Van der Pol oscillator \(x'' + (x^2-1)x' + x = 0,\) proved existence of a limit cycle: from the qualitative analysis for the system

  \[\begin{cases}x' = y,\\y' = -x - (x^2-1)y,\end{cases}\]

  if we start at a point \(A(-1, x_A > 1),\) we must go subsequently to points

  • \[B(1, x_B)\ \ \to \ \ C(1, x_C<0)\ \ \to \ \ D(-1, x_D<0) \ \ \to \ \ E(x_E<-1, 0) \ \ \to \ \ G(-1, y_G).\]
  • If \(y_G < y_A,\) then the trajectory from \(A\) to \(G\) together with the segment \([A, G]\) form a contour, from which a trajectory cannot escape, and we can use Poincaré-Bendixson theorem.
  • Assuming for a contrary that the previous situation is impossible, i.e. that always \(y_G > y_A,\) and taking \(y_A>1,\) we come to conclusion that between point \(E\) and \(G\) there is also a point \(F(x_F, 1).\)

  • The trajectory between points \(F\) and \(G\) allows estimates

    \[\begin{multline*}y_G = y_F + \int_{x_F}^{x_G=1}\frac{dy}{dx}dx = 1 + \int_{x_F}^{x_G=1}\frac{\frac{dy}{dt}}{\frac{dx}{dt}}dx = 1 + \int_{x_F}^{x_G=1}\left(\underbrace{1-x^2}_{\leq 0} \underbrace{- \frac{x}{y}}_{\leq x} \right) dx \\ \leq 1 + \frac12(x_F^2 - 1). \end{multline*}\]

    The point \(F(x_F, 1)\) should lie in the region \(x + (x^2 - 1) * y \leq 0,\) hence \(x^*\leq x_F \leq -1,\) where \(x_1\) is the negative solution of the equation \(x^2 + x - 1 = 0,\) i.e. \(x^* = \frac{1 + \sqrt{5}}{-2}.\) Continuing estimates for \(y_G,\) we obtain

    \[y_G \leq 1 + \frac{1}{2}((x^*)^2 - 1) = 1 - \frac12x^* = \frac{5+\sqrt{5}}{4} < 2.\]
  • (Deyvid Penkov:) alternatively, in the last step, to prove that for some trajectory \(y_G < y_A,\) take a circle \(x^2 + y^2 = R^2,\) which is tangential to the left-most branch of the curve \(y = x / (1 - x^2).\) The solution in the region \(x<-1, y>0\) will have to enter the circle, and thus \((-1,y_G)\) lies inside that circle, and thus \(y_G\) is bounded by \(\sqrt{R^2 - 1}.\)
• Damped oscillator / van der Pol oscillator in the electric circuit consisting of
  • electromotive force \(\mathcal{E}(t),\)
  • resistor \(R,\) / semiconductor,
  • capacitor \(C,\)
  • inductance coil \(L.\) The rules that connect the current \(I\) and the voltage drop \(V\) is as follows: \(V_R = R I,\quad V_{\mathrm{sc}} = I_{\mathrm{sc}}(I_{\mathrm{sc}}^2 - a^2), \quad \frac{d V_C}{dt} = \frac{1}{C} I_C,\quad \frac{d I_L}{d t} = \frac{1}{L} V_L.\)

• the previous derivation leads to van der Pol oscillator with a parameter \(\mu>0,\)

  \[x'' + \mu(x^2-1)x'+x=0.\]

  • for \(\mu\to+\infty\) the model exhibits relaxation oscillations: slow modes are followed by fast modes. The period can be heuristically estimated as \(T=\mu(3-4\ln2)+\mathcal{O}(1)\) (not rigorous reasoning).

  • for \(\mu\to 0,\) the model is near the exactly solvable linear oscillations, thus, heuristically, \(x(t) = A \cos(t) + \mathcal{O}(\mu)\). Computing the energy change over one cycle on the periodic orbit, we obtain, heuristically, \(|A|=2.\)

Please finish yourself with the tasks from the previous lesson, Problems Week 3.

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Week 5 (9.11.2023)

Lecture:

• finished Caratheodory theory: generalized Banach theorem and generalized Pickard theorem, file “Kapitola16. Carathéodoryho teorie.pdf”
• started Optimal Control theory, motivational exmaples: parking problem, Moon landing problem, rare-visited-house heating problem.

Practice:

• Problems 1(a, b, c, d), 2a from page 5, chapter Optimal Control, online practice book.

Comments/issues:

• in Problem 2a: is the function Caratheodory?

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Week 6 (16.11.2023)

Lecture:

file Kapitola 18: Optimální regulace:

• Theorem 18.1 (structure of \(\mathcal{R}(t)\) for linear problem \(x'=Ax+Bu\))
• Theorem 18.2 (connection between observability of \(x'=Ax\) via \(y=Bx\) and controllability of \(x'=A^Tx+B^Tu\))
• Theorem 18.6 (properties of \(\mathcal{R}(t)\) for constrained controls)

Practice:

• online practice book, chapter Optimal Control, problems on page 19-20: 16, 17, 15.

Please take a look at other problems on pages 19-20.

Comments/issues/clarifications:

• difference between weak and weak star convergence, see László Erdős notes, Bob Gardner notes.

So, if \(X\) is a Banach space, then

• \(x_j\in X\) converges weakly to \(x\in X\) (\(x_j \rightharpoonup x\)), if for any \(f\in X^*\) we have \(f(x_j)\to f(x).\)
• \(f_j\in X^*\) converges star weakly to \(f\in X^*\) (\(f_j\overset{*}{\rightharpoonup}f\)), if for any \(x\in X\) we have \(f_j(x)\to f(x).\)

Now, for a sequence \(f_j\in X^*\) we have both the notions of weak and star weak convergence:

• \(f_j\rightharpoonup f,\) if \(\forall \widehat x\in X^{* *},\) we have \(\widehat x(f_j)\to \widehat x(f).\)
• \(f_j\overset{*}{\rightharpoonup}f,\) if \(\forall x\in X,\) we have \(f(x)\to f(x_j).\)

If an element \(\widehat x\in X^{* *}\) is identified with \(x\in X,\) then \(\widehat x(f) = f(x)\) for any \(f\in X^*.\)

But in general \(X^{* *}\supset X,\) hence weak convergence is a stronger notion than weak star convergence.

In particular, \(L^{\infty} = (L^1)^{*},\) but \((L^{\infty})^*=\mbox{set of all measures}\supset L^1.\)

So, requiring weak convergence instead of weak star convergence in Banach-Alaouglu theorem would read:

for any sequence \(u_j(.)\in\mathcal{U}\) there is a subsequence \(\tilde u_j(.)\) and \(u_0(.)\in\mathcal{U}\) such that for all measures \(d\mu\) we have \(\int \tilde u_j(s)d\mu(s)\to\int u_0(s)d\mu(s)\).

instead of

for any sequence \(u_j(.)\in\mathcal{U}\) there is a subsequence \(\tilde u_j(.)\) and \(u_0(.)\in\mathcal{U}\) such that for all \(\phi\in L^1\) we have \(\int \tilde u_j(s)\phi(s)ds\to\int u_0(s)\phi(s)ds\).

Update 21.11.2023: for instance, \(\sin(j t)\) converges star weakly to \(0\) in \(L^{\infty}(0, \pi)\) (Riemann-Lebesgue theorem, \(\int f(s)\sin(js)ds\to0\) as \(j\to\infty\)), but it does not converge to \(0\) weakly (for the later, we would need \(\int\sin(js)d\mu(s)\to 0\) for all measures \(d\mu(s)\), but this is violated for \(\delta(s-\frac{\pi}{2})\)).

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Week 7 (23.11.2023)

Lecture:

file Kapitola 18: Optimální regulace:

• Theorem about local controllability for the linear system \(x'=Ax+Bu\) with \(u:(0, t)\to[-1, 1]^m.\)
• Theorem 18.7 about global controllability for the above system.
• Theorem 18.8: bang-bang principle. Uses Krein-Milman theorem (we need a simple corollary of that theorem: a nonempty convex compact subset of a locally convex topological vector space has a nonempty set of extremal points. We use this for \(L^{\infty}(0, t, \mathbb{R}^m),\) which is a normed space and thus locally convex topological vector space. Moreover, as for a metric space, compactness is equivalent to sequential compactness).
• Theorem 18.9 about the existence of the minimal time.
• Theorem 18.10: (Pontrjagin maximal principal for minimal time and linear problem): started.

Practice:

• online practice book, chapter Optimal Control, problems on page 29: 21, 23, 24.

Please finish problem 24 and do problems 25-27.

Comments/issues/clarifications:

• problem 22 is not of the form appicable for the usage of Pontrjagin principle, since the integrand of the functional is not autonomous (it depends on \(x, u\), but also on \(t\) via the function \(z(t)\)).

UPDATE(30.11.2023): Pontrjagin principle works also for functions \(f(s, x(s), u(s)),\) so it is applicable.

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Week 8 (30.11.2023)

Lecture + Practice:

file Kapitola 18: Optimální regulace:

• Theorem 18.10: (Pontrjagin maximal principal for minimal time and linear problem).
• Theorem 18.11: (general Pontrjagin with fixed time, not fixed end value). We did it for more general functions \(f(s, x, u), r(s, x, u),\) not just \(f(x, u), r(x, u).\)
• Theorem 18.3 (local controllability for nonlinear systems).
• Theorem 18.4 (using Lemma 18.2): controllability of a linear system.

Comments/issues/clarifications:

• it follows from the general Pontrjagin, that \(\begin{cases}x_j' = \frac{\partial H}{\partial p_j}, \\ p_j' = -\frac{\partial H}{\partial x_j},\end{cases}\) i.e. they take the form of the Hamiltonian equations.

• function \(\eta(s)\) in the proof of the (linear, minimal time)-Pontrjagin can be chosen to be measurable, since we can take \(\eta_j(s) = \mathrm{sgn}(v_j(s)),\) where \(v(s) = h^Te^{-sA}B.\)

• in the proof of Theorem 18.4, compared to Martin Křepela notes, pages 21-22/42, the following clarifications are needed:

  • the first step is done using the transformation \(x = Vy,\) where \(V = (b_1, \ldots, b_n).\)
  • to establish the controllability of the system in Step 2, it is easier to go to conjugate observability problem, rather then inspect the Kalman matrix (In the first Step, it is the opposite, and the Kalman matrix is just identity matrix there).
  • the fourth step is done as follows: after constructing the basis \((v_1, \ldots v_n) =:V\) and the \(m\times n\) matrix \(\tilde F\) and establishing that \((A+B\tilde F) V = V\Lambda\) for appropriate matrix \(\Lambda,\) one makes two subsequent reductions of the control \(u:\) first, take \(u = \underbrace{\tilde F}_{m\times n} \underbrace{x}_{n\times1} + \underbrace{e_1}_{m\times 1}\underbrace{v}_{1\times1}.\) The system is reduced then to the form \(Vy' = (A+B\tilde F)x + Be_1v,\) where \(v\) is a one-dimensional control. Since \(Be_1 = v_1 = Ve_1,\) the system then reduces to the form \(y' = \Lambda y + e_1v,\) which is already treated in Step 1.

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Week 9 (7.12.2023)

Lecture:

file Martin Křepela notes, pages 35-42/42, Central manifold:

• Equivalence of properties INVariance, REDuction, FixedPoint.
• Theorem on existence of function \(\phi(x):\) started.

Practice:

• online practice book, chapter Center Manifold, exercises on pages 1-9 and problems on pages 10-11: exercises 1, 3, 4, problems 5, 7.

Please take a look at the rest of exercises and problems.

Comments/issues/clarifications:

• exercise 4 in the online book works already with \(\phi(u) = 0:\) indeed, in that case we have \(\phi(u)=\mathcal{O}(u)^3,\) and we need to expand \(u + \phi(u) - \sin(u+\phi(u)).\) It is already \(\frac16(u+\phi(u))^3 + \mathcal{O}(u)^5 = \frac16u^3+\mathcal{O}(u^5).\) So, instead of first plugging \(\phi(u)\) and then expanding in Taylor series, it is better to first expand \(\sin\) in Taylor series and then substitute \(\phi(u).\)

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Week 10 (14.12.2023)

Lecture:

• Theorem of existence of function \(\phi\) satisfyimg property (INV).
• Exercise: prove differentiability of \(\phi.\) (Idea: differentiate equations formally in \(p_0\) and try to adopt approach of the proof of the Theorem)
• Consequence: \(\nabla\phi(0) = 0.\) Follows by expansion of the \((M\phi)(p_0)\) at \(p_0\to0.\) We obtain \(\nabla\phi(0)\cdot A = B\cdot\nabla\phi(0).\) A non-zero solution \(\nabla\phi(0)\) would exist only in the case of \(A, B\) having a joint spectrum (Gantmacher, Theory of matrix, chapter on the equation \(XA=BX.\))

Practice:

• Deyvid Penkov: for two Caratheodory functions that are equal almost everywhere, the set of solutions coincide.
• Central manifold exercises: 1, 12. In 1, the existence of multiple functions \(\phi\) does not violate the Theorem on existence of \(\phi,\) since the conditions of the Theorem are not satisfied: \(f\) is not bounded globally.
• Optimal control exercises, 22: started.

Next time:

• Lecture (anticipated): approximation of the central manifold.
• Practice: writing test (a problem on general Pontrjagin maximal principle with fixed time).

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Week 11 (21.12.2023)

Lecture:

• approximation of the center manifold: confusion with choosing of the parameters (not resolved yet).

Practice:

• Practice: writing test (a problem on general Pontrjagin maximal principle with fixed time).

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Week 12 (4.1.2024)

Lecture: files Martina Křepela’s notes (pages 30-34 / 42), Michal Kozák’s notes (36-43 / 43) bifurcation.

• bifurcations in 1d:
  • saddle-node (sedlo-uzel, appearance/annihilation of two stationary points),
  • transcritical (transkritická, a pair of stable and non-stable stationary points interchanges stability type),
  • pitchfork (vidlíčková, 1 stationary point changes to 3 stationary points).

Practice:

• Practice: writing test (a problem on general Pontrjagin maximal principle with fixed time).

Next time

• (anticipated): Hopf bifurcation in 2D (appearance/disappearance of a periodic orbit).

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Další materialy:

• Poincaré-Bendixsonova věta: Poincaré-Bendixson 1, Poincaré-Bendixson 2, Poincaré-Bendixson 3

• Periodic orbits

• Henon map

• Oliver Knill’s lectures on dynamical systems

• La Salle invariance principle

• • Van der Pol 1: relaxation, estimates of period in strongly nonlinear regime and the amplitude for the weakly nonlinear regime

  • Van der Pol 2: Manuela Girotti

  • Standard (damped) electric circuit oscillator

  • Van der Pol in electric circuit

• Chaos courses

• Lawrence C. Evans course on oprimal control

• Lemmon notes on Pontrjagin Maximum Principle