Basic theory of Ordinary Differential Equations
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Abstract
Differential Equations are somewhat pervasive in the description of natural phenomena and the theory of Ordinary Differential Equations is a basic framework where concepts, tools and results allow a systematic approach to knowledge. This same book aims to give a concrete proof of how the modeling of Nature is based on this theory and beyond. This appendix is intended to provide some concepts and results that are used in the text, referring to the student background and to textbooks for a full acquaintance of the material. We actually mention [2,3,5,7,10] as basic references on the subject.
Keywords
Periodic Orbit Hopf Bifurcation Stable Manifold Negative Real Part Floquet MultiplierDifferential Equations are somewhat pervasive in the description of natural phenomena and the theory of Ordinary Differential Equations is a basic framework where concepts, tools and results allow a systematic approach to knowledge. This same book aims to give a concrete proof of how the modeling of Nature is based on this theory and beyond. This appendix is intended to provide some concepts and results that are used in the text, referring to the student background and to textbooks for a full acquaintance of the material. We actually mention [2,3,5,7,10] as basic references on the subject.
1 A.1 The Cauchy problem
For simplicity, we assume that the function F (t, x) is defined everywhere in ℝ^{n} ^{+} ^{1}
A first basic result on the Cauchy problem (A.1), concerns existence and uniqueness of a solution
Theorem A.1 (Existence and uniqueness). Let the function F(t, x) be continuously differentiable inℝ^{ n } then, for any Y ^{0}, there exists an interval (t _{0} − δ, t _{0} + δ) and a unique continuously differentiable function Y(t), defined for t ∈ (t _{0} − δ, t _{0} + δ)and satisfying (A.1).
We see that the existence stated by Theorem A.1 is local, and in general the solution cannot be extended beyond a maximal finite interval. Thus (A3) defines a function of the variables (t,t _{0},Y _{0}) on a region Ω that, in general, is strictly contained in ℝ^{ n+2}, because for each t _{0} and Y _{0} the solution is not in general globally defined. However, we have
Since we deal with models requiring not only existence and uniqueness of the solution but also global existence, we need conditions ensuring that the solution is indeed defined for all t ≥ 0. In fact we have
Actually boundedness of the solution is a requirement occurring in many significant results. If estimate (A.4) is satisfied only for t ≥ t _{0}, then the solution is global in the future.
A second condition concerns the structure of the system.
Then the solution exists globally.
We are interested in describing the behavior of solutions that exists globally in time. The analysis leads to concept and results of qualitative analysis.
2 A.2 Equilibria and their stability
Equilibria may be isolated points or even form a continuum in the phase space. A given equilibrium may be stable, asymptotically stable or unstable, according with the following
 stable, if for any δ > 0 it exists δ such that$$ \begin{array}{*{20}c} {\left {{\mathbf{Y}}_0  {\mathbf{Y}}^* } \right < \delta } \hfill & \Rightarrow \hfill & {\left {{\mathbf{Y}}(t,0,{\mathbf{Y}}_0 )  {\mathbf{Y}}^* } \right < \varepsilon } \hfill & {{\text{for}}} \hfill & {t \geqslant 0;} \hfill \\ \end{array} $$
 asymptotically stable, if it is stable and there exists δ such that$$ \begin{array}{*{20}c} {\left {{\mathbf{Y}}_0  {\mathbf{Y}}^* } \right < \delta } & \Rightarrow & {\mathop {\lim }\limits_{t \to + \infty } {\mathbf{Y}}(t,0,{\mathbf{Y}}_0 ) = {\mathbf{Y}}^* } \\ \end{array} ; $$

unstable if it is not stable.

if F′(y*) x003C; 0 the equilibrium y* is asymptotically stable;

ifF′(y*) x003C; 0 it is unstable.
We note that the previous statement gives only sufficient conditions and if we have F′(y*) = 0 we cannot conclude either for stability nor for instability.
3 A.3 Linear systems
The analysis of the linear case is especially important not only because it is possible to give a full description of the solutions, but also because, in the the nonlinear cases, local analysis in a neighborhood of an equilibrium point resorts on linearization (as in the one dimensional case) to draw conditions for stability.

λ _{ i } are the eigenvalues of A (i = 1,⋯,p);

m _{i} is the multiplicity of the eigenvalue λ _{ i } (i = 1,⋯,p);

v _{ ij } are the projection of the initial vector Y _{0} onto the generalized eigenvectors of A (i =1,⋯ ,p j = 0, ⋯ ,m _{i} − 1).
Thus we can state
 1.
Asymptotically stable, if and only if ℜλ_{ i } < 0 for all i=1,⋯ ,p.
 2.
Unstable, if there exists k such that ℜλ_{ k } > 0.
 3.
Stable, if ℜλ_{ i } ≤ 0 for all i = 1,⋯, p, and all eigenvalues with null real part are simple.

E ^{s} (stable space), the direct sum of all the E _{i} with ℜλ_{i} < 0;

E ^{c} (central space), the direct sum of all the E _{i} with ℜλ_{i} = 0;

E _{u} (unstable space), the direct sum of all the E _{i} with ℜλ_{i}> 0.

if Y _{0} ∈ E ^{s} then \(\mathop {\lim }\limits_{t \to + \infty } \left {Y(t,Y_0 )} \right = 0\) and \( \mathop {\lim }\limits_{t \to  \infty } \left {Y(t,Y_0 )} \right = + \infty ; \)

if Y _{0} ∈ E ^{u} then \(\mathop {\lim }\limits_{t \to  \infty } \left {Y(t,Y_0 )} \right = 0\) and \(\mathop {\lim }\limits_{t \to + \infty } \left {Y(t,Y_0 )} \right = + \infty ;\)

if Y _{0} ∈ E ^{c} then the solution Y(t,Y _{0}) may be constant, run on a circle, or \(\mathop {\lim }\limits_{t \to \pm \infty } \left {Y(t,Y_0 )} \right = + \infty \)

the origin is asymptotically stable if and only if trace (A) < 0 and det(A) > 0;

if det(A) < 0, then the origin is a saddle point;

if det(A) > 0 and trace(A) > 0, then the origin is completely unstable.
Theorem A.6. All the roots of the polynomial (A.10), with a _{0} > 0, have negative realpart ifandonly if the principal minors of ((H _{ ij })) are all positive.
Assume that the equation satisfies the conditions (A.11) for α < α* [or α > α*], while they are violated at α = α*. Then, either α_{3}(α_{*}) = 0 in which case 0 is a root of (A.12) for α α* or α_{1}(α*)α_{2}(α*) − α_{3} (α^{*}) = 0, in which case (A.12) has two purely imaginary roots at α α*.
4 A.4 The nonlinear case
Then the origin O is an equilibrium for (A.14) and we have

if ℜλ _{i} < 0⋯for all i= 1,⋯,p, then the origin O is asymptotically stable;

if there exist k such that ℜ λ _{ k } > 0, then the origin O is unstable.

if ℜλ_{i} < 0 for all i= 1,…,p, then Y* is asymptotically stable;

if there exist k such that ℜλ _{k} > 0, then Y* is unstable.
We stress the local nature of the previous result and also note that the critical cases, when one or more eigenvalues have null real part, are not decidable on the basis of the linearization.
A few additional results are in order. First we have
Then we consider the case of a saddle point, i.e. the case when the Jacobian JF(Y*) has both eigenvalues with positive real part and eigenvalues with negative real part. We have

\( \mathcal{M}^s (Y^* ) \) and \( {\mathcal{M}^u}({Y^*}) \) are respectively tangent to E ^{s} andE ^{u} at Y*
 there exist a neighborhood \( \mathcal{U} \) of Y* such that$$ \begin{array}{*{20}c} {\mathcal{M}^s ({\mathbf{Y}}^* ) = \left\{ {\begin{array}{*{20}c} {\left. {{\mathbf{x}} \in U} \right\mathop {\lim }\limits_{t \to + \infty } {\mathbf{Y}}(t,{\mathbf{x}}) = {\mathbf{Y}}^* } & {and} & {{\mathbf{Y}}(t,{\mathbf{x}}) \in \mathcal{U}} & {for\,t \geqslant 0} \\ \end{array} } \right\},} \\ {\mathcal{M}^u ({\mathbf{Y}}^* ) = \left\{ {\begin{array}{*{20}c} {\left. {{\mathbf{x}} \in U} \right\mathop {\lim }\limits_{t \to + \infty } {\mathbf{Y}}(t,{\mathbf{x}}) = {\mathbf{Y}}^* } & {and} & {{\mathbf{Y}}(t,{\mathbf{x}}) \in \mathcal{U}} & {for\,t \leqslant 0} \\ \end{array} } \right\}.} \\ \end{array} $$
5 A.5 Limit sets

ω(x) ≢∅;

ω(x) is closed and connected;

ω (x) is invariant;

\(\mathop {\lim }\limits_{t \to \pm \infty } d\left( {Y(t,x),\omega (x)} \right) = 0;\)
The last statement of the Theorem gives a precise account of how the solution tends to ω(x).

ω(x) = Y*, when Y* is an equilibrium and lim Y(t x)=Y*

ω(x) = O+(x) when Y(t,x) is aperiodic solution (Y(t,x) =Y(t + T,x)).
A useful result concerning limit sets is the following (see [9, Sect. 8.2]), known as ButlerMcGehee Lemma, that we state in a very simple form, originally due to Freedman and Waltman
Proposition A.3. Let Y* be an isolated equilibrium of (A.20), and let Y* ∈ ω(x) but ω(x) ≠ Y*. Then ω(x) includes a point x _{1} ∈ \( \mathcal{M}^s (Y^* ) \), x _{1} ≠ Y*, (hence the whole orbit through x _{1}) and a point x _{2} ∈ \( \mathcal{M}^s (Y^* ) \), x _{2}≠Y* (hence the whole orbit through x _{2}).
6 A.6 Planar case: PoincaréBendixson theory
Theorem A.13. Consider the autonomous system (A.18) and suppose that the orbit O _{+}(Y _{0}) is bounded. If the ωlimit set ω(Y _{0}) does not contain any equilibrium point, then it is a periodic orbit.
This result has several consequences for planar systems. In fact we have

ω(Y _{0}) is an equilibrium;

ω(Y _{0}) is a periodic orbit;

ω(Y _{0}) is a singular cycle, i.e. it is the union of a finite number of orbits joining such points as t → ω∞ or as t → − ∞ (either heteroclinic or homoclinic orbits).
The previous theorem restricts the possible outcomes and any information to exclude existence of periodic orbits is very useful. Indeed we have
Finally, a simple statement that may help is the following
Theorem A.16. For system (A.18), the region enclosed by a periodic orbit must contain at least one equilibrium.
7 A.7 Planar competitive and cooperative systems
A class of differential equations for which a large body of theory has been developed is that of cooperative or competitive systems. An extensive account of the theory, not limited to differential equations, including many results of interest for Population Dynamics can be found in [8].
In the general case of problem (A.1) with (A.2) we have the following
The class of competitive systems fits exactly the class of models designed to describe competition, as those we have considered in Chap. 7.
We limit ourselves to consider planar systems, for which a simple result holds, allowing to determine the global asymptotic behavior of the solutions. In fact we have
Theorem A.17. For a planar competitive system, all the solutions are eventually monotone. Then any bounded solutions goes to an equilibrium point as t →+∞.
8 A.8 Lyapunov functions
Then we have

V(Y*)= 0;

V(x) > 0 for x∈\( u \) and x ≠ Y*;

V?(x) ≤ 0 on \( u \)
then Y* is stable. If moreover

V (x) < 0 for x≠Y*,
then Y* is asymptotically stable.
Finally, information about the ωlimit set come from the following result:
9 A.9 Persistence
We take from [9] two results simplified to suit system (A.19). Before stating the theorems, some definitions are needed.
Note that, if M is a hyperbolic equilibrium, it is weakly repelling if its stable manifold has no intersection with the interior of \(\mathbb{R}_ + ^n\).
Finally, a collection {M _{1},...,M _{n}} of subsets of X _{0} is cyclic if, after possibly renumbering the sets, there exist y _{12}(t), y _{23}(t), ..., y _{ nȡ21},n(t), y _{n,1}(t) solutions of (A.19) belonging to X _{0} such that lim_{ t→−∞} d(y _{ ij },M _{i}) = 0, lim_{ t→−∞} d(y _{ ij },M _{ j }) = 0. If {M _{1},...,M _{ n }} is not cyclic, it is acyclic.
Theorem A.20 (Theorem 8.17 in [9]). Let \( \cup _{x \in X_0 } \omega (x) = \cup _{i = 1}^n M_i \) where each Mi is isolated, compact and weakly repelling. If {M _{1},...,M _{ n }} is acyclic, then (A.19) is weakly uniformly persistent.
A second very useful result is the following
Then if (A.19) is weakly uniformly persistent, it is persistent.
The two Theorems can be combined in the following result
Let \( \cup _{x \in X_0 } \omega (x) = \cup _{i = 1}^n M_i \) where each M _{ i } is isolated, compact and weakly repelling. if {M _{ 1 },...,M _{ n }} is acyclic, then (A.19) is persistent.
10 A. 10 Elementary bifurcations
The theory examines the possible changes in the qualitative structure of the system, as the parameter α varies. We refer to [6] (more elementary and computational) or [10] for a comprehensive introduction to the topic. Here we present simply the recipes for the simplest (and common in mathematical biology) equilibrium bifurcations. Equilibrium bifurcation means qualitative changes that relate to changes in the properties of an equilibrium of system (A.20).
Correspondingly to the family of differential equations (A.20), we will consider family of equilibria Y _{α} depending on α; as seen above, its stability properties, are determined by the real sign of eigenvalues of the Jacobian matrix J ( α ) = JF ( α, Y _{α} ). Neglecting cases where system (A.20) has eigenvalues with 0 real part for all α in an interval (bifurcation theory considers only generic properties, which intuitively means for almost all systems), changes in stability may occur only when one eigenvalue crosses the imaginary axis at some value α*.

tangent bifurcation J(α:*) has eigenvalue 0 and F _{ α }(α*,Y _{α}* ) ≠ 0;

transcritical bifurcation J(α:*) has eigenvalue 0 but F _{ α }(α*,Y _{α}* ) = 0;

Hopf bifurcation J(α:*) has eigenvalues ±iα (necessarily n ≥ 2).
More generally, when some other technical conditions hold, (A.20) has two equilibria in a neighborhood ∪ of Y _{ α }* for α < α* and none for α > α**or vice versa. Furthermore, one equilibrium will havek eigenvalues with negative real part (and n − k with positive), and the other one k − 1, with 1 ≤k ≤ n;ifk = n, one equilibrium is asymptotically stable, and the other one is unstable.
Now, at α = 0 we have F _{ x }(0,0) = 0, but F a(0,0) = 0. For α < 0, the equilibrium \( y_0^* (\alpha ) \) is asymptotically stable because \( F_x (\alpha ,y_0^* (\alpha )) = \alpha \) and the equilibrium α is unstable because F _{ x }(α,α) = −α; for α > 0 vice versa.
The fact that F _{ α } = 0 can be considered to be nongeneric, but many classes of systems in mathematical population dynamics share the property that one state (that we call 0, thinking of population density) is an equilibrium for all parameter values. If the model has a biological interpretation, negative values are often not considered, so that the equilibrium α is neglected for α < 0. In this light, the transcritical bifurcation can be viewed not as an intersection of two equilibria that exchange their stability, but as the emergence of a new equilibrium at α = 0, as the equilibrium 0 loses its stability.
In this interpretation, one can distinguish a forward and a backward bifurcation; in the forward bifurcation (case above), the new equilibrium exists when 0 is unstable and inherits its stability; a backward bifurcation is exemplified by the equation y′(t) = (α +y(t))y(t), in which a positive equilibrium −α exists when the equilibrium 0 is asymptotically stable and is unstable. We insist that this distinction depends on the choice of restricting the system to one side of the 0 equilibrium.
Hopf bifurcation
The origin (0,0) is an equilibrium for all values of α, and its Jacobian has eigenvalues α ∓ i; hence they have negative real part for α < 0, positive real part for α > 0, while they are purely imaginary for α = 0.
Finally assume that the Jacobian at α* has eigenvalues ±i ω with ω> 0, while all other eigenvalues have nonzero real part. Necessarily it must be n ≥ k ≥ 2.
Under some technical conditions, one can compute a quantity α _{ 0}(see, for instance, (5.62) in [6]) such that if α _{ 0} ≠ 0, there exist a family of periodic orbits Γ(α) contracting to Y(α*) as α approaches α*. If α _{0} < 0 (supercritical Hopf bifurcation), Γ(α) exists for α > α* (and close to α*) and its stable manifold has dimension k (if k = n, i.e. x*(α) is asymptotically stable, for α < α*, the periodic orbit is attracting). If α _{0} > 0 (subcritical Hopf bifurcation), Γ(α) exists for α < α* (and close to α*) and its stable manifold has dimension k − 2 (thus, the periodic orbit is unstable).
We end this section by remarking that all these results are local: they depend only on the values of the function in a neighborhood of the equilibria, and they describe the behavior of the solutions only in the neighborhood. This is the reason why in the text we generally relied on methods that could yield the global structure of the solutions. However, local bifurcation theory is a very effective method to explore (especially through numerical bifurcation tools such as AUTO or MATCONT) the (local) behavior of a system, and also to get some insights into the global behavior.
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