Preface
This section gives an introduction to the most important feature of nonlinear ordinary differential equations: critical or equilibrium points and their stabilities. These points are analyzed without actual solving the differential equation. Stability theory deals with the effect of disturbances on timeprocesses in real systems. It is also important to know how solutions behave near equilibrium points, including basin of attraction.
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Introduction to Linear Algebra with Mathematica
Glossary
Stability
We turn our attention to a very important part of the qualitative analysis of differential equationsstability, which can be considered as the study of the effects caused by disturbances. The stability problems originated from mechanics and engineering where possible perturbations should be taken into account. The requirement of keeping some parameters (as the frequency and the voltage) constant under acting disturbances clearly leads to a stability problem. Thus, equilibrium solutions that correspond to configurations in which the physical system does not move, can only be observed in everyday situations when they are stable. An unstable equilibrium can be discovered within a short period of time because slight perturbations in the system or its physical surroundings will immediately dislodge the system far away from equilibrium. It is hard to catch up details for a moving car (unstable), but one can easily observe a parking car (which is in a stable position).
A physical system in perfect balance may be very difficult to achieve (a pencil standing on its point is unstable) or easy to maintain (oscillating pendulum near its downward position). If initial conditions are nudged a little bit, but the resulting solution of the differential equation undergoes a substantial movement away from the equilibrium, we describe this situation as unstable. Without some basic theoretical understanding of the nature of solutions, equilibrium points, and stability properties, one would not be able to understand when numerical solutions (even those provided by standard packages) are to be trusted. Moreover, when testing a numerical scheme, it helps to have already assembled a repertoire of nonlinear problems in which one already knows one or more explicit analytic solutions. Further tests and theoretical results can be based on first integrals (also known as conservation laws) or, more generally, Lyapunov functions, periodic solutions, and chaotic regions. To facilitate understanding, we mostly concentrate our attention on two dimensional case.
The solution set of each equation f_{k}(p_{1}, p_{2}, …, p_{n}) = 0, k = 1, 2, …, n, is referred to as the kth nullcline.
In the following, we will also assume that the equilibrium is an isolated critical point. This means that the point p ∈ ℝ^{n} is an isolated stationary point, if there is a positive number r such that the rneighborhood \( U_r = \ {\bf x}  {\bf b} \ < r \) of b contains no critical points other than b. Here
Example 1: The equilibrium points of the nonlinear planar system
The xnullcline is a parabola 4 x = y²  2 x², and ynullcline is the straight line y = −2 x. Their intersection consists of two stationary points: the origin (0,0) and (2,−4).
n1 = ContourPlot[4*x  y^2 + 2*x^2 == 0, {x, 1, 3}, {y, 5, 1},
ClippingStyle > Automatic, ColorFunction > Hue,
ContourStyle > Thick];
n2 = ContourPlot[2*x + y == 0, {x, 1, 3}, {y, 5, 1}, ClippingStyle > Automatic, ColorFunction > Black, ContourStyle > Thick]; point = Graphics[{PointSize[0.04], Point[{{0, 0}, {2, 4}}, VertexColors > {Red, Purple}]}]; Show[n1, n2, point] 

Figure 1: Intersections of two nullclines.  Mathematica code 
The theory of stability was introduced and further developed by the celebrated Russian mathematician Alexandr Lyapunov (also Liapunov) in 1892. His monograph, entitled: The General Problem of the Stability of Motion, includes so many fruitful ideas and results of primary significance that the whole theory of stability can be divided into two periods, viz. PreLyapunov period and PostLyapunov periods.
First of all Lyapunov provided a rigorous definition of motion stability. The absence of such a definition had often caused misunderstandings since otherwise a motion which is stable in one sense can be unstable in another. Then he suggested two main methods for analyzing the stability problems of motions. Of these, the second method, also called The Direct Lyapunov Method is widely popular due to its simplicity and efficiency
Example 2: Consider the nonlinear system of differential equations \[ \dot{x} = y + x \left( x^2 + y^2 \right) , \qquad \dot{y} = x + y \left( x^2 + y^2 \right) . \]
Upon plotting the phase portrait, we see that the origin is a stable critical point.
sp = StreamPlot[{y + x*(x^2 + y^2), x + y*(x^2 + y^2)}, {x, 1,
1}, {y, 1, 1}, PerformanceGoal > "Quality", StreamPoints > 35,
PlotTheme > "Classic"]


Figure 1: Phase portrait.  Mathematica code 
■
Example: Consider the nonlinear autonomous system of equations
sp=StreamPlot[{2*x*y, y^2  x^2}, {x, 1, 1}, {y, 1, 1},
PerformanceGoal > "Quality", PlotTheme > "Classic"];
Introducing polar coordinates
ar = Graphics[{Red, Thickness[0.008], Arrowheads[0.07], Arrow[{{0, 1}, {0, 0}}]}]; line = Graphics[{Red, Thickness[0.008], Arrowheads[0.07], Arrow[{{0, 0}, {0, 1}}]}]; Show[line, ar, sp]
\[
x= t\,\cos\theta , \qquad y = r\,\sin\theta ,
\]
we obtain
\[
\dot{r} = r^2 \sin\theta , \qquad \dot{\theta} = r\,\cos\theta .
\]
We derive the separable equation
\[
\frac{{\text d}r}{{\text d}\theta} = \frac{\dot{r}}{\dot{\theta}} = r\,\tan\theta \qquad \left( \theta \ne \frac{\pi}{2} , \quad \theta \ne \frac{3\pi}{2} \right) .
\]


Figure 1: Phase portrait.  Mathematica code 
The solution of the latter becomes
Example: ■
An important generalization of the previous definition is uniform asymptotic stability when both of the initial moment t_{0} and of the initial position x_{0} of the the perturbed motion do not affect the longterm behavior of the solution.
There are known other definitions of stability (for instance, in Poincaré sense); however, the definitions above (in Lyapunov sense) are mostly used. Before the end of the nineteen century, most of research and applications of differential equations was centered around the analytic expression of solutions of differential equations. Upon the influence of the famous French mathematician Henri Poincaré (18541912), the topic of differential equations shifted to the study of the global properties of the solutions without solving the differential equations themselves. The important contributions of Poincaré came at the eve of twentieth century slightly later the Lyapunov discovery. Stability applications have seeped into industry such as in engineering applications, where the common practice is to run a process in “steady state”. For the system that the engineer is interested in, it is frequently of much greater importance to know that the system is approaching a stable equilibrium and will remain there for long time periods, than to have an exact computation of short term transient behavior. Computers make it possible to find approximately the solutions of differential equations on a finite interval of time, but they do not answer the qualitative aspects of the global behavior of phase curves.
A fundamental problem in the theory of differential equations is to study the motion of the system u sing the vector field that induces the differential equations. Qualitative analysis involves questions of the type: Do the solutions go to infinity, or do they remain bounded within a certain region? What conditions must a vector field satisfy in order for the solutions to remain within a given region? Do nearby solutions act similarly to a particular solution of interest? These are questions of qualitative type, in contrast with analytic methods that tend to search for a formula to express each solution of a differential equation. Some authors separate asymptotic stability property when t → ∞ from stability and consider asymptotic stability without requirement for critical point to be stable.
x*
A typical quasasymptotically stable critical point may have the following phase portrait.
c1 = Graphics[{Blue, Thickness[0.01], Circle[{0, 0}, 0.4]}];
c2 = Graphics[{Blue, Thickness[0.01], Circle[{1.6, 0}, {2, 1}]}; c3 = Graphics[{Blue, Thickness[0.01], Circle[{1.3, 0}, {1.7, 0.85}]}]; c4 = Graphics[{Blue, Thickness[0.01], Circle[{1.0, 0}, {1.4, 0.7}]}]; c5 = Graphics[{Blue, Thickness[0.01], Circle[{0.7, 0}, {1.1, 0.55}]}]; Show[c1, c2, c3, c4, c5] 

Quasasymptotically stable critical point  Mathematica code 
An equilibrium x* is unstable if there exists a positive number ε such that for any δ ≤ ε such that there is at least one solution x = φ(t) that starts within δneighborhood of the critical point and be at distance more that ε from the equilibrium solution.
The following (pathological) examples demonstrate that an equilibrium solution can be unstable, but asymptotically stable. This means that there exists a positive number ε such that the magnitude of every nearby solution exceeds it, but eventually all solutions approach the critical point as t → ∞.
Example: Consider the following system \eqref{EqPlanar.1} due to Thomas Brown (Rassias, J.M., CounterExamples in Differential Equations, page 95 )
circle1 = Graphics[{Thick, Blue, Circle[{0.5, 0.0}, 0.5]}];
circle2 = Graphics[{Thick, Blue, Circle[{1.0, 0.0}, 1.0]}]; arrow1 = Graphics[{Arrowheads[0.04], Blue, Arrow[{{0.98, 0.15}, {1.0, 0}}]}]; arrow2 = Graphics[{Arrowheads[0.045], Blue, Arrow[{{1.98, 0.15}, {2.0, 0}}]}]; circle3 = Graphics[{Thick, Blue, Circle[{1.5, 0.0}, 1.5]}]; arrow3 = Graphics[{Arrowheads[0.045], Blue, Arrow[{{2.98, 0.15}, {3.0, 0}}]}]; c1a = Graphics[{Thick, Blue, Circle[{0.5, 0.0}, 0.5]}]; a1a = Graphics[{Arrowheads[0.04], Blue, Arrow[{{0.98, 0.15}, {1.0, 0}}]}]; c2a = Graphics[{Thick, Blue, Circle[{1.0, 0.0}, 1.0]}]; a2a = Graphics[{Arrowheads[0.045], Blue, Arrow[{{1.98, 0.15}, {2.0, 0}}]}]; c3a = Graphics[{Thick, Blue, Circle[{1.5, 0.0}, 1.5]}]; a3a = Graphics[{Arrowheads[0.045], Blue, Arrow[{{2.98, 0.15}, {3.0, 0}}]}]; c4a = Graphics[{Thick, Blue, Circle[{2.2, 0.0}, 2.2, {0, 4.5}]}]; c4 = Graphics[{Thick, Blue, Circle[{2.2, 0.0}, 2.2, {Pi, 1.6}]}]; a4 = Graphics[{Arrowheads[0.05], Blue, Arrow[{{0.32, 1.19}, {0.24, 1.1}}]}]; a4a = Graphics[{Arrowheads[0.05], Blue, Arrow[{{0.32, 1.19}, {0.24, 1.1}}]}]; exp = Plot[2.08 + Exp[x], {x, 0.4, 2.1}, PlotStyle > {Thick, Blue}]; expa = Plot[2.08 + Exp[x], {x, 2.6, 0.4}, PlotStyle > {Thick, Blue}]; xa = Graphics[{Arrowheads[0.05], Black, Arrow[{{4.0, 0}, {5.0, 0}}]}]; xt = Graphics[ Text[StyleForm["x", FontSize > 18, FontColor > Black], {4.8, 0.4}]]; ya = Graphics[{Arrowheads[0.05], Black, Arrow[{{0.0, 2.0}, {0.0, 3.7}}]}]; yt = Graphics[ Text[StyleForm["y", FontSize > 18, FontColor > Black], {0.4, 3.6}]]; Show[circle1, arrow1, circle2, arrow2, circle3, arrow3, c1a, a1a, c2a, a2a, c3a, a3a, c4, c4a, a4, a4a, exp, expa, ya, yt, xa, xt] 

Some solution trajectories.  Mathematica code 
■
Example: Consider the planar system
StreamPlot[{x  y + (x*y  x^3  x*y^2)/Sqrt[x^2 + y^2],
x + y  (x^2 + y^3 + y*x^2)/Sqrt[x^2 + y^2]}, {x, 2, 2}, {y, 2,
2}, Mesh > 11, MeshShading > {Red, None}]


Figure 2: Quasistable critical point (1,0).  Mathematica code 
Another unprecedentedly ugly example of quisistable system: All solutions nearby 0 goes back to 0, since nearby solutions in the left half plane naturally converge to 0, and nearby solutions in the right half plane first converge to the xaxis, then wind back to the left half plane.
StreamPlot[{x^2 + ((x^2 + 1/16) (y  1/2)^3) Boole[y >= 1/2], y + (y  1)^2 y Boole[y >= 1]  500 (y  2)^4 Boole[y >= 2] + (x + 1)^3 Boole[x <= 1] + (x  1)^3 Boole[x >= 1]}, {x, 4, 4}, {y, 4, 4}, PlotTheme > "Web"]


Figure 3: Quasistable critical point (1,0).  Mathematica code 
■
Example: Consider an electric field E given by
Module[{tocartesian, cartesianlist, field, cartesianfield}, tocartesian = {Overscript[r, "^"] > x/r Overscript[x, "^"] + y/r Overscript[y, "^"], Overscript[\[Theta], "^"] > (y/Sqrt[x^2 + y^2]) Overscript[x, "^"] + x/Sqrt[x^2 + y^2] Overscript[y, "^"], r > Sqrt[x^2 + y^2], \[Theta] > ArcTan[x, y]};
cartesianlist = (a_ Overscript[x, "^"] + b_ Overscript[y, "^"]) > {a, b};
field = rfield Overscript[r, "^"] + thetafield Overscript[\[Theta], "^"];
cartesianfield = FullSimplify[field //. tocartesian] /. cartesianlist;
StreamPlot[cartesianfield, opts]]
Now we can feed it our original rθ field definition directly
PolarStreamPlot[{3/(4 r^4) (3 Cos[\[Theta]]^2  1),
3/(4 r^4) Sin[2 \[Theta]]}, {x, 3, 3}, {y, 3, 3},
PlotTheme > "Detailed"]


Figure 4: Quasistable electric field.  Mathematica code 
We plot the phase portrait for the given system of nonlinear differential equations
f1[x_, y_] = x^2 *(y  x)*y^5/(x^2 + y^2)/(1 + (x^2 + y^2)^2);
f2[x_, y_] = y^2 *(y  2*x)/(x^2 + y^2)/(1 + (x^2 + y^2)^2); sp = StreamPlot[{f1[x, y], f2[x, y]}, {x, 1, 1}, {y, 1, 1}, PerformanceGoal > "Quality", StreamPoints > 35, PlotTheme > "Classic"]; line = Graphics[{Red, Thickness[0.01], Line[{{1, 0}, {1, 0}}]}]; Show[line, sp] 

Figure 1: Phase portrait.  Mathematica code 
■
If every solution of the autonomous system \eqref{EqStability.1} approaches some solution y(t), not necessarily critical point, as t → +∞, then it is called globally asymptotically stable.
The stability of equilibrium solutions of constant coefficient linear systems of differential equations depends completely on the roots of the characteristic polynomial det(λI − A) = 0. The following theorem, credited to the German mathematician Adolf Hurwitz (1859–1919).
Planar Systems
The concepts of asymptotic stability, stability, and instability can be easily visualized by considering twodimensional models. Let φ(t) = [x(t), y(t)] be the trajectory and x^{∗} = (x^{∗}, (y^{∗}) be an isolated critical point to the autonomous system. We say that a point P((x, y) approaches he critical point x^{∗} if \( \lim_{t\to\infty} x(t) = x^{\ast} \) and \( \lim_{t\to\infty} y(t) = y^{\ast} \) ✴
We consider systems of first order ordinary differential equations in normal form on the plane
 Alligood, K.T., Sauer, T.D., and Yorke, J.A., Chaos. An Introduction to Dynamical Systems, 1996, SpringerVerlag, New York, QA614.8.A44 1996
 Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, SpringerVerlag, Berlin, Heidelberg, 2014. ISBN13 : 9783662015308
 Hahn, W. Stability of Motion, SpringerVerlag, Berlin, Heidelberg, 1967.
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