1. Stability of Differential Equations
Prof.(Dr.) Nita H. Shah
Department of Mathematics, Gujarat University,
Ahmedabad

2. Dynamical System
A dynamical system is a concept in Mathematics where a fixed rule describes the time dependence of a point in a geometric space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in the pipe, the number of fish each springtime in lake etc.
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3. Once if the system can be solved, given an initial point, it is possible to determine all future positions, a collection of points known as trajectory. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. Therefore, notion of stability has been introduced in the study of dynamical systems.
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4. The main questions of interest in this discussion are:
How to compute stability boundaries of equilibria in the parameter space?
How to predict qualitative changes in system's behaviour occurring at these equilibrium points?
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5. What is bifurcation?
In a dynamical system, a parameter is allowed to vary, then the differential system may change. An equilibrium can become unstable and a periodic solution may appear or a new stable equilibrium may appear making previous equilibrium unstable.
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6. A bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden “qualitative” change in its behaviour. The value of parameter at which these changes occur is known as “bifurcation value” and the parameter that is varied is known as the “bifurcation parameter”.
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7. So, in a little finer language we can say that- the study of changes in the qualitative structure of the flow of a differential equation as parameters are varied is called bifurcation theory. At a given parameter value, a differential equation is said to have stable orbit structure if the qualitative structure of the flow does not change for sufficiently small variations of the parameter.
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8. A parameter value for which the flow does not have stable orbit structure is called the bifurcation value, and the equation is said to be at a bifurcation point.
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9. Equilibrium Points
In dynamical systems, only the solution of linear systems may be found explicitly. The problem is that in general, real life problems may only be modelled by non-linear systems. The main idea is to approximate a non-linear system by a linear one.
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10. Let us first establish our notations for differential equations
Let us first establish our notations for differential equations. Let I be an open interval of the real line R and let ; be a real-valued differentiable function of a real variable t. We will use the notation to denote the derivative , and refer to t as time or the independent variable.
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11. Also let be a given real valued function. Now consider ……
Also let be a given real valued function. Now consider …….(1) where x is an unknown function of t and f is a given function of x. Equation (1) is called scalar autonomous differential equation; scalar because x is one dimensional (real valued) and autonomous because the function f does not depend on t.
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12. We say that a function x is a solution of (1) on the interval I if
We will often be interested in a specific solution of (1) which at some initial time has the value x0. Thus we will study x satisfying
…….(2)
Eq (2) is referred to as an initial value problem and any of its solutions is called a solution through x0 at t0.
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13. A point is called an equilibrium point (also critical point, steady state solution etc.) ofif
When is an equilibrium point, the constant function
for all t is a solution.
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14. Example:
Consider the logistic growth equation where x(t) denotes population density at time t, a and B are positive constants, B is the carrying capacity. Then by setting right hand side function equal to zero, i.e. we obtain two equilibrium points .
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15. Stability of Equilibrium
Now we introduce the concept of stability of an equilibrium point . Roughly speaking, an equilibrium point is stable if all solutions starting near stay nearby. If, in addition, nearby solutions tend to as , then is asymptotically stable. Precise definitions are given below:
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16. Definition: An equilibrium point of eq
Definition: An equilibrium point of eq. (1) is said to be stable if for any given , there is a , depending on such that, for every for which , the solution x(t) of (1) through at t = 0 satisfies the inequality The equilibrium point is said to be unstable if it is not stable.
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17. Definition: An equilibrium point is said to be asymptotically stable if it is stable and in addition there is an r > 0 such that for all satisfying
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18. Important results:
1. An equilibrium point of is stable if there is a such that Similarly, is asymptotically stable if and only if there is a such that An equilibrium point of is unstable if there is a such that
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19. 2. Suppose that f is a differentiable function with continuous first derivative and is an equilibrium point of i.e. . Suppose also that . Then the equilibrium point is asymptotically stable if and unstable if .
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20. On the basis of above result, we can define that: Definition: An equilibrium point of is called a hyperbolic equilibrium if . If then is called a non-hyperbolic or degenerate equilibrium point.
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21. Example:
Consider a differential equation . Then equilibrium points are Now This implies and
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22. Phase Portrait
First we define the orbit of the solution x(t) to be the set An orbit can be a point, a simple closed curve, or the homeomorphic image of an interval. A geometric picture of all the orbits of an autonomous differential equation is called its phase portrait or phase diagram.
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23. Examples:
1. Consider the differential equation . The equilibrium points of this equation are i.e. The function is positive on the interval , negative on , positive on and negative on . Therefore, its phase portrait can easily be drawn as follow:
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24. STTP @Parul University, Jan 1, 2016

25. 2. Consider the differential equation
2. Consider the differential equation . The equilibrium points of this equation are i.e. The function is positive on the interval and . Therefore, its phase portrait will look as follow:
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26. STTP @Parul University, Jan 1, 2016

27. Bifurcation diagram
In case our system depends on parameters, the collection of the phase portraits corresponding to each choice of the parameter is called a bifurcation diagram.
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28. Bifurcation Types
We divide bifurcations into two classes: 1. Local bifurcations 2. Global bifurcations
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29. Local Bifurcations
A local bifurcation occurs when a parameter change causes the stability of an equilibrium point to change. Here, the topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point.
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30. Some of local bifurcations are: Saddle-node bifurcation
Trans-critical bifurcation
Pitchfork bifurcation
Period doubling bifurcation
Hopf bifurcation
etc.
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31. Global Bifurcations
Global bifurcations occur when ‘larger’ invariant sets, such that periodic orbits collide with equilibria. This causes changes in the topology of the trajectories in the phase space which can not be confined to a small neighbourhood.
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32. Some of global bifurcations include: Homoclinic bifurcation
Heteroclinic bifurcation
Infinite period bifurcation
Blue-sky catastrophe
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33. Now let us discuss some of these bifurcations with examples.
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34. Saddle-node bifurcation
A saddle-node bifurcation or tangent bifurcation is a collision and disappearance of two equilibria in dynamical systems. Consider the differential equation , for a is real. ⇒ equilibrium points are .
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35. Therefore, if a < 0, then we have no real solution, If a > 0, then we have two real solution. We now consider each of the two solutions for a > 0, and examine their linear stability. First, we add a small perturbation: Substituting this in the given differential equation, gives
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36. and since the term in brackets on the RHS is trivially zero, therefore Solving this, we get From this, we see that for and for
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37. The bifurcation diagram below, therefore the saddle-node bifurcation at a = 0 corresponds to the creation of two new solution branches. One of these is linearly stable, and the other is linearly unstable. a
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38. Trans-critical bifurcation
In a trans-critical bifurcation, two families of fixed points collide and exchange their stability properties. The family that was stable before the bifurcation is unstable after it. The other fixed point goes from being unstable to being stable.
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39. Now consider the dynamical system , for x, a, b real
Now consider the dynamical system , for x, a, b real. Again, a and b are control parameters. We can find two steady states to this system We now examine the stability of each of these states in turn, following the usual procedure.
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40. For the state , we add a small perturbation which yields with the linearized form has the solution
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41. Therefore, perturbations grow for a > 0 and decay for a < 0
Therefore, perturbations grow for a > 0 and decay for a < 0. So the state is unstable if a > 0, the state is stable if a < 0. Now for the state , we add small perturbation which yields the linearized form
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42. has the solution Therefore, perturbations grow for a > 0 and decay for a < 0. So the state is stable if a > 0, the state is unstable if a < 0.
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43. It can be easily seen that the bifurcation point a = 0 corresponds to an exchange of stabilities between the two solution branches.
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44. The pitchfork bifurcation
In pitchfork bifurcation one family of fixed points transfers its stabilities properties to two families after or before the bifurcation point. If this occurs after the bifurcation point then pitchfork bifurcation is called super-critical. Similarly a pitchfork bifurcation is called sub-critical if the non-trivial fixed points occur for values of the parameter lower than the bifurcation value.
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45. Consider the dynamical system for a, b real Again, a and b are control parameters. We can find two steady states to this system
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46. As usual, we now examine the linear stability of each of these states in turn. First we write the perturbation for that gives with the solution So we see that the state is unstable if a > 0, the state is stable if a < 0.
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47. For the states and , setting gives with the solution Thus it is obvious that the state and is stable if a > 0, the state and is unstable if a < 0.
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48. Supercritical pitchfork bifurcation in
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49. Hopf bifurcation
It comes in two dimensional system. Also, GLOBAL BIFURCATIONS take place in higher dimensions.
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50. We need to introduce a small piece of notations: We will denote the set of all continuous functions with continuous derivatives by . Analogously, we will use to indicate the functions with continuous derivatives up through order n. if the domain of functions is a subset U of ℝ, then we will use notation etc.
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51. To emphasize the dependence of a solution x(t) of eq(2) through x0 at t0=0 on the initial condition, we will often use the notation for this solution. In other words
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52. At each point on the (t,x)-plane where f(x) is defined, the right hand side of (1) gives the value of derivative which can be thought of as the slope of a line segment passing through that point. The collection of all such line segments is called the direction field of the differential equation (1).
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53. The graph of a solution of (2) x0, i. e
The graph of a solution of (2) x0, i.e. the subset of the (t, x)-plane defined by is called the trajectory. A trajectory is tangent to the line segments of the direction field at each point on the plane it passes through. Since f(x) is independent of t, on any line parallel to the t-axis, the line segments of the direction field all have the same slope.
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54. To each point x on the x-axis, we can associate the directed line segment from x to x + f(x). We can view this directed line segment as a vector based at x. The collection of all such vectors is called the vector field generated by (1).
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55. Projections of trajectories through x0 onto the x-axis are called orbits. Precise definition can be given as Definition: The positive orbit , negative orbit , and orbit of x0 are defined, respectively as the following subset of the x-axis:
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56. STTP @Parul University, Jan 1, 2016

57. On the orbit we insert arrows to indicate the direction in which is changing as t increases. The flow of a differential equation is then drawn as the collection of all its orbits together with the direction arrows and the resulting picture is called the phase portrait of the differential equation.
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58. fig: (a) direction field along several trajectories, (b) vector field,
(c) Orbits, and (d) phase portrait of
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59. Dimension 2
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60. Let us first develop some basic notations in 2D which are parallel to 1D.
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61. Let I be an open interval of the real line ℝ and be two functions of a real variable t. Also, let be two given real valued functions in two variables.
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62. We will undertake a geometrical study of a pair of simultaneous differential equations of the form This system (3) is called the general planar system.
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63. In this discussion, we will use boldface letters to denote vector quantities. For instance, if we let , and , then eq(3) can be written as ……..(4) This equation looks the same as the scalar equation considered in dimension 1.
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64. But we must keep in mind that x is a two-vector and f is a vector-valued function. We will follow the convention of using subscript to denote the components of a vector and superscripts to label different vectors, e.g. In particular, an initial-value problem for eq(4)will be indicated by
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65. To begin our qualitative study, we now reconsider the system (4) and its flow from a geometric point of view. At each point of the -space where f(x) is defined, the RHS of eq(4) gives a value of the derivative which can be considered as the slope of a line segment at that point. The collection of all such line segments is called the direction field of the differential equation (4).
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66. The graph of the solution of eq(4) through , i. e
The graph of the solution of eq(4) through , i.e. the curve in the three-dimensional (t, x)-space defined by is called the trajectory through . Of course at each point through which it passes, a trajectory is tangent to a line segment of the direction field.
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67. Since f(x) is independent of t, on any line parallel to the t-axis, the line segments of the direction field all have the same slope. Therefore, it is natural to consider the projections of the direction field and the trajectories of eq (4) onto the -plane. More precisely, to each point x on the -plane, where f(x) is defined, we can associate the vector which should be thought of as being based at x.
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68. In other words, we can assign to the point x the directed line segment from x to x + f(x). The collection of all such vectors is called the vector field generated by (4), or simply the vector field f. Projections of trajectories onto the -plane are called orbits. More specifically, we make the following definition:
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69. Definition: The positive orbit , negative orbit , and orbit of are defined, respectively as the following subset of the x-axis:
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70. To compensate for the loss of time parameterization in orbits, on the orbit we insert arrows to indicate the direction in which is changing as t increases. The flow of a differential equation is then drawn as the collection of all its orbits together with the direction arrows; the resulting picture is called the phase portrait of the differential equation.
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71. Example: Consider the system Step 1: Equilibrium points
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72. Step 2: Stability of Equilibria As (due to solution of second ODE) Therefore, first ODE becomes: grows exponentially as Therefore, equilibrium point is unstable.
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73. Step 3: Phase Portrait: ……. (i) ……
Step 3: Phase Portrait: …….(i) …….(ii) The flow is vertical along the curve (i) and horizontal along the curve (ii).
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74. STTP @Parul University, Jan 1, 2016

75. More accurate picture can be seen as:
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76. Thank you!
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