VISVESVARAYA NATIONAL INSTITUTE OFTECHNOLOGY

Project ReportBifurcation analysis and applications

May 5,2016

Project Supervisor:Dr.G.Naga Raju

Department of Mathematics

Prashant PatelMS14MTH013

M.Sc. Mathematics

CONTENTS

1 Preliminary Remarks

2 Nonlinear Systems: Local Theory

2.1 Some Preliminary Concpts and Definitions

2.2 Stability, Liapunov Stability, Asymptotically Stable

2.3 Linearization

2.4 Hartman-Grobman Theorem

2.5 Routh-Hurwitz Criterian

3 Bifurcation of equilibrium points

3.1 Saddle Node Bifurcation

3.2 Transcritical Bifurcatiion

3.3 Pitchfork Bifurcation

3.4 Hopf Bifurcation

3.5 Selkov Model of Hopf Bifurcation

3.6 Sotomayor Theorems

3.7 Hopf Bifurcation Theorem

4 Population models

4.1 Two-species Competition Models

4.2 Prey-Predator Models

5 Conclusion

6 References

1

Preliminary Remarks

Differential equations are used in physics, chemistry,

biology, economics, engineering to model the real world problems. Few of the

applications of differential equations are as follows:

1 Population models

2 Prediction of weather

3 Prediction of stock prices

4 Atmospheric pollution

5 Diffusion of materials

6 Epidemic models

7 Pattern formation (e.g. stripes in Zebra, spots in leopards)

8 In general nonlinear systems cannot be solved explicitly therefore qualitative

behaviour of the system can change as parameters are varied.

9 The dynamics of the one dimensional ODE is very limited: all solutions either

settle down to equilibrium or head out to ±∞10 In higher dimensional phase space a wider range of dynamical behaviour is

possible.

11 What is interesting about nonlinear systems?

Answer: dependence on parameters.

12 modeling in ecology is to predict the behaviour of system for different

parametric conditions.

2

Non Linear Systems:Local theory

Equillibrium Solution

Consider a general autonomous vector field

dx

dt= f(x, ν), x ∈ Rn, ν ∈ R, (1)

an equillibrium solution is a point x ∈ Rn such that f(x) = 0, that is a solutionwhich does not change in time.

Stability

let x(t) be any solution of dx/dt = f(x) then x(t) is stable if solutions startingclose to x(t) at a given time remain close to x(t) for all later time.It is asymptotically stable if nearby solutions actually converge to x(t) as t tendsto ∞.

Liapunov Stabilityx(t) is said to be stable if given ε > 0 there exist a δ > 0 such that for any othersolution y(t) of dx/dt = F (x) satisfying, then for t > t0, t0 ∈ R

|x(t)− y(t)| < ε,

|x(t0)− y(t0)| < δ.

Asymptotic Stabilityx(t) is said to be asymptotically stable if it is Liapunov stable and if there exist aconstant b > 0 such that if

|x(t0)− y(t0)| < b,

then limt→∞|x(t)− y(t)| = 0.

Theorem 1. (For Asymptotically Stable) Suppose all of the eigen-values off ′(x) have negative real parts. Then the equillibrium solution x = x of the nonlinear vector field

dx/dt = f(x), x ∈ Rn

is asymptotically stable.

Hyperbolic Fixed PointsLet x = x be a fixed point of dx/dt = f(x), x ∈ Rn. Then x is called a hyperbolicfixed point if none of the eigen values of f ′(x) = J have zero real part.

Saddle, Stable and unstable node and CenterA hyperbolic fixed point of a vector field is called saddle if some, but not all of theeigenvalues of the associated linearization have real parts greater than zero. If allthe eigenvalues have negative real parts then the hyperbolic fixed point are calledstable node. and if all the eigenvalues have positive real parts then the hyperbolicfixed points are called unstable node. If the eigenvalues are purely imaginary andnon zero the non hyperbolic fixed point is called a center.

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Stable and unstable node

4

Examples of Saddlele, stable node, stable spiral and center

5

Linearization

Let dxdt

= f(x) be a vector field. Let x∗ be a fixed point and let η(t) = x(t) − x∗be a small perturbation away from x∗. To see whether the perturbation grows ordecays we derive a differential equation for η.

η(t) = x(t)− x∗ (2)

dη

dt=d(x− x∗)

dt=dx

dt,

since x∗ is constant,Thus

dη

dt=dx

dt= f(x) = f(x∗ + η),

Now using Taylor’s expansion we obtain

dη

dt= f(x∗ + η) = f(x∗) + η ∗ f ′(x∗) +O(η2), (3)

where o(η2) denotes quadratically small terms in η. Finally note that f(x∗) = 0since x∗ is a fixed point. Hence

dη

dt= η ∗ f ′(x∗) +O(η2),

Now if f ′(x∗) 6= 0 the O(η2) terms are negligible and we write the approximation

dη

dt= η ∗ f ′(x∗). (4)

This is a linear equation in η and is called the linearization about x∗. It showsthat the perturbation grows exponentially if f ′(x∗) > 0 and decays if f ′(x∗) < 0.If f ′(x∗) = 0 the O(η2) terms are not negligble and a non linear analysis is neededto determine stability.

Example Using linear stability analysis determine the stability of the fixed pointsfor f(x) = sin(x).

The fixed points occur where f(x) = sin(x) = 0. Thus x∗ = kΠ is the fixed point,where k is an integer.

f ′(x∗) = cos(kΠ) = {1, k is even,−1, k is odd},

Hence x∗ is unstable if k is even and stable if k is odd.

Here in figure (a) −2Π, 2Π, 4Π are unstable fixed points and −Π,Π, 3Π are stablefixed points.

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Routh-Hurwitz Stability Criterion

If A is an m × m matrix, then the characteristic equation of A is given by:λm + a1λ

m−1 + a2λm−2 + ....... + am = 0, where, a′is, i = 1, 2, · · · · ·m are real

numbers.Define

D1 = a1,

D2 = det

(a1 a31 a2

)and

Dk = det

a1 a3 a5 . . . a2k−11 a2 a4 . . . a2k−20 a1 a3 . . . a2k−30 1 a2 . . . a2k−4. . . . . . .. . . . . . .0 0 0 . . . ak

, k = 1, 2, · · · · ·m.

where,aj = 0 for j > m. Then, the roots of characteristic equation have negativereal parts if and only if Dk > 0 for all k = 1, 2, · · · ·m.

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Theorem 2. Hartman Grobman theorem

Consider a vector fielddx

dt= f(x), x ∈ Rn, (5)

Where f is defined on a sufficiently large open set of rn. Suppose that equation(5) has a hyperbolic fixed point at x = x0 i.e.

f(x0) = 0,

and when f ′(x0) has no eigen values on the imaginary axis. Consider the associ-ated linear vector field

dξ

dt= f(x0) ∗ ξ, ξ ∈ Rn. (6)

Then we have following theorem:The flow generated by equation (5) is c0 conjugate to the flow generated by equation(6) in a neighbourhood of the fixed point x = x0 or to say that the two autonomoussystem of differential equations such as (5) and (6) are said to be topologicallyequivalent in a neighbourhood of the origin or to have the same qualitative struc-ture near the origin.

Bifurcation of Equilibrium Points

The change in qualitative behaviour (e.g. equilibrium points or periodic solutionsor their stability properties) as a parameter passes through a critical point isknown as bifurcation.

Saddle-Node Bifurcation

It is Basic mechanism by which equilibrium points are created or destroyed . Asthe bifurcation parameter passes through the bifurcation point, two equilibriadisappear, so there are no equilibria afterward. One of the two equilibria is stableanother is unstable, before they disappear.Example Consider the vector field

dy

dt= f(y, a) = a− y2, y ∈ R′, a ∈ R′, (7)

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It is easy to verify that

f(0, 0) = 0,∂f

∂y(0, 0) = 0 ,

The set of all fixed points is given by a − y2 = 0 ⇒ a = y2. This represents aparabola in the a− y plane.For

y =√a⇒ f ′ = −2

√a (stable),

Fory = −

√a⇒ f ′ = 2

√a (unstable).

In the figure arrows along the vertical lines represents the flow generated by (7)along the x- direction. Thus for a < 0 (7) has no fixed points and the vector fieldis decreasing in x. For a > 0 (7) has two fixed points. A simple linear stabilityanalysis shows that one of the fixed points is stable and the other fixed point isunstable. THis is an example of Saddle-Node Bifurcation. (x, a) = (0, 0) is abifurcation point and a = 0 is bifurcation value.

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Transcritical Bifurcation

In a Transcritical Bifurcation, there are two equilibria one stable and other unsta-ble. When the Bifurcation point is passed, the unstable becomes stable and stableone becomes unstable.Example consider the vector field

dy

dt= f(y, a) = ay − y2, y ∈ R′, a ∈ R′, (8)

It is easy to verify that

f(0, 0) = 0,∂f

∂y(0, 0) = 0,

The set of all fixed points is given by ay − y2 = 0 ⇒ y(a− y) = 0⇒ y = 0, a = 0are fixed points.

case(1) For a < 0, y = 0 (stable), y = a (unstable)

case(2) For a > 0, y = 0 (unstable), y = a (stable)

Thus an exchange of stability has occured at a = 0.

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Pitchfork Bifurcation

In bifurcation theory, a field within mathematics, a pitchfork bifurcation is aparticular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcationshave two types - supercritical or subcritical.

Example Consider the vector field

dx

dt= f(x, µ) = µ ∗ x− x3, x ∈ R′, µ ∈ R′, (9)

It is easy to verify that

f(0, 0) = 0,∂f

∂x(0, 0) = 0 ,

The set of all fixed points is given by µ∗x−x3 = 0⇒ x(µ−x2) = 0⇒ x = 0, x2 = µare fixed points.

case(1) For µ < 0, x = 0 (stable)

case(2) For µ > 0, x = 0 is still a fixed point but two new fixed points have beencreated at µ = 0 and are given by x2 = µ. In this process x = 0 become unstablefor µ > 0, with the other two fixed points are stable.

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Hopf-Bifurcation

The Hopf-Bifurcation refers to the local appearance or disappearance of a periodicsolution from an equilibrium as a parameter crosses a critical value. The Hopf-Bifurcation typically occurs when a complex conjugate pair of eigenvalues of thejacobian matrix at an equilibrium point become purely imaginary. This impliesthat a Hopf-Bifurcation can only occur in systems of dimension tw or higher. TheHopf-Bifurcation ensures the local existence of a periodic solution.

The Hopf bifurcation in the Selkov system

The Selkov model exhibits a Hopf-Bifurcation. As the b parameter increases from0.2 to 0.975 the model switches from a stable equilibrium point to a limit cyclenear b = 1.0 and back to a stable equilibrium point near b = 0.2.

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Theorem 3. (Sotomayor Theorems):Let us consider a system

dx

dt= f(x, ν), x ∈ Rn, ν ∈ R, (10)

Suppose the f(x∗, ν∗) =0 and that the n× n matrix A ≡ Df(x∗, ν∗) has a

simple eigenvalue λ = 0 with eigenvector v and that AT has an eigenvector w

corresponding to the eigenvalue λ = 0. Furthermore, suppose that A has k

eigenvalue with negative real part and (n-k-1) eigenvalues with positive real parts

and that the following conditions are satisfied

wTfν(x∗, ν∗) 6= 0 and wT [D2f(x∗, ν∗)(v, v)] 6= 0, (11)

Then the system (10) experiences a Saddle-node bifurcation at the equilibrium

point x∗ as the parameter ν passes through the bifurcation value ν = ν∗

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If the conditions (11) are changed to

wTfν(x∗, ν∗) = 0andwT [Dfν(x∗, ν∗)v] 6= 0, wT [D2f(x∗, ν∗)(v, v)] 6= 0. (12)

Then above system (10) experiences a transcritical bifurcation at the equilibriumpoint x∗ as the parameterν varies through the bifurcation value ν = ν∗

Theorem 4. (Hopf-Bifurcation Theorem(1942))Suppose(10) has an equilibrium at (x∗, ν∗) satisfying following:

(H1) : Dxf(x∗, ν∗) ,

has a simple pair of purely imaginary eigenvalues and no other eigenvalues withzero real parts.Then (H1) implies that there is a smooth curve of equilibrium points (x(ν), ν)

with x(ν∗)=x. The eigenvalues λ(ν), λ(ν) of Dxf(x(ν)ν∗) which are imaginary at

ν = ν∗ vary smoothly with ν. If, moreover,

(H2) :d

dν(Reλ(ν))|ν=ν∗ 6= 0 .

is satisfied,then there exists a unique branch of periodic solution of the system(10)

near (x∗, ν∗).

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Population Models

Two-Species Competition Models

A general model for two interacting species that compete for a common foodsupply with population sizes x1(t) and x2(t) is given by

dx1dt

= x1F (x1, x2),dx2dt

= x2G(x1, x2), (13)

Subject to the positive initial condition

x1(0) > 0, x2(0) > 0 ,

Where F (x1, x2) and G(x1, x2) are the growth rates of both the species respec-tively. The growth rates F (x1, x2) and G(x1, x2) satisfy the following assumptions:

(1) An increase in population of one species will result in a decrease of the growthrate of the other as the two species compete for the same resources. Hence

∂F

∂x2(x1, x2) < 0,

∂G

∂x1(x1, x2) < 0 ,

(2) If either of the population becomes very large, both population tend to de-crease. Hence there exist k > 0 such that

F (x1, x2) < 0, G(x1, x2) < 0,

if x1 ≥ k or x2 ≥ k

(3) In the absence of the other species, both species have a positive growth rateup to a certain population and then a negative growth rate beyond it. Thereforethere are constants k1 > 0, k2 > 0 such that

F (x1, 0) > 0 for x1 < k1 andF (x1, 0) < 0 for x1 > k1 ,

G(0, x2) > 0 for x2 < k2 andG(0, x2) < 0 for x2 > k2 .

Examples of Competition Model

The most popular example for two competing species is the classical Lotka-Volterracompetition model.The Lotka-Volterra equations, also known as the predator -preyequations, are a pair of first-order, non-linear, differential equations frequentlyused to describe the dynamics of biological systems in which two species interact,one as a predator and the other as prey. The populations change through timeaccording to the pair of equations:

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dx

dt= αx− βxy ,

dy

dt= δxy − γy .

Where, x is the number of prey, y is the number of some predator. dydt

and dxdt

represent the growth rates of the two populations over time, t represents time andα, β, γ, δ are positive real parameters describing the interaction of the two species.

Physical meaning of the equations

The Lotka-Volterra model makes a number of assumptions about the environmentand evolution of the predator and prey populations:(1) The prey population finds ample food at all times.(2) The food supply of the predator population depends entirely on the size of theprey population.(3) The rate of change of population is proportional to its size.(4) During the process, the environment does not change in favour of one speciesand genetic adaptation is inconsequential.(5) Predators have limitless appetite.As differential equations are used, the solution is deterministic and continuous.This, in turn, implies that the generations of both the predator and prey are con-tinually overlapping.Prey

When multiplied out, the prey equation becomes

dx

dt= αx− βxy ,

The prey are assumed to have an unlimited food supply, and to reproduce expo-nentially unless subject to predation; this exponential growth is represented in theequation above by the term αx. The rate of predation upon the prey is assumedto be proportional to the rate at which the predators and the prey meet; this isrepresented above by βxy. If either x or y is zero then there can be no predation.With these two terms the equation above can be interpreted as: the change in theprey’s numbers is given by its own growth minus the rate at which it is preyedupon.Predators

The predator equation becomes

dy

dt= δxy − γy ,

In this equation,δxy represents the growth of the predator population. (Note thesimilarity to the predation rate; however, a different constant is used as the rate at

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which the predator population grows is not necessarily equal to the rate at whichit consumes the prey).γy represents the loss rate of the predators due to eithernatural death or emigration; it leads to an exponential decay in the absence ofprey.

Hence the equation expresses the change in the predator population as growthfueled by the food supply, minus natural death.

Solutions to the equations

The equations have periodic solutions and do not have a simple expression interms of the usual trigonometric functions, although they are quite tractable.Ifnone of the non-negative parameters α, β, γ, δ vanishes, three can be absorbedinto the normalization of variables to leave but merely one behind: Since the firstequation is homogeneous in x, and the second one in y, the parameters β

αand

δγ, are absorbable in the normalizations of y and x, respectively, and γ into the

normalization of t, so that only αγ

remains arbitrary. It is the only parameteraffecting the nature of the solutions.A linearization of the equations yields a solution similar to simple harmonic mo-tion with the population of predators trailing that of prey by 900 in the cycle.

Frequency Plot

A simple example

Suppose there are two species of animals, a baboon (prey) and a cheetah (preda-tor). If the initial conditions are 80 baboons and 40 cheetahs, one can plot theprogression of the two species over time. The choice of time interval is arbitrary.

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Frequency Plot

One may also plot solutions parametrically as orbits in ”phase-space”, withoutrepresenting time, but with one axis representing the number of prey and theother axis representing the number of predators for all times. This is to say, elim-inating time from the two differential equations above results in only one such,

dy

dx=−y(δx− y)

x(βy − α),

whose solutions are closed curves; integrating d∗log(y)(α−βy)−d∗log(x)(γ−αx)yields an evident constant quantity V depending on the initial conditions, whichis conserved on each curve,

V = −δx+ γlog(x)− βy + αlog(y) .

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Phase Space Plot

An aside: These graphs illustrate a serious potential problem with this as a bi-ological model: For this specific choice of parameters, in each cycle, the baboonpopulation is reduced to extremely low numbers, yet recovers (while the cheetahpopulation remains sizeable at the lowest baboon density). In real-life situations,however, chance fluctuations of the discrete numbers of individuals, as well as thefamily structure and life-cycle of baboons, might cause the baboons to actually goextinct, and, by consequence, the cheetahs as well. This modelling problem hasbeen called the ”atto-fox problem an atto-fox being a notional 10−18 of a fox, inthe context of rabies modelling in the UK.

Phase-space plot of a further example

A less extreme example covers:α = 2/3, β = 4/3, γ = 1 = δ. Assume x,y quantifythousands, each. Circles represent prey and predator initial conditions from x =y =0.9 to 1.8, in steps of 0.1. The fixed point is at (1,1/2).

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Phase-space plot

Dynamics of the system

In the model system, the predators thrive when there are plentiful prey but, ulti-mately, outstrip their food supply and decline. As the predator population is lowthe prey population will increase again. These dynamics continue in a cycle ofgrowth and decline.

Population equilibrium

Population equilibrium occurs in the model when neither of the population levelsis changing, i.e. when both of the derivatives are equal to 0.

x(α− βy) = 0,−y(γ − δx) = 0 ,

When solved for x and y the above system of equations yields y=0, x=0 and

y =α

β, x =

γ

δ.

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Hence, there are two equilibria,The first solution effectively represents the extinction of both species. If bothpopulations are at 0, then they will continue to be so indefinitely. The secondsolution represents a fixed point at which both populations sustain their current,non-zero numbers, and, in the simplified model, do so indefinitely. The levels ofpopulation at which this equilibrium is achieved depend on the chosen values ofthe parameters,α, β, γ, δ.

Stability of the fixed points

The stability of the fixed point at the origin can be determined by performing alinearization using partial derivatives, while the other fixed point requires a slightlymore sophisticated method. The Jacobian matrix of the predator-prey model is

J(x, y) =

(α− βy −βxδy δx− γ

).

First fixed point (extinction)

When evaluated at the steady state of (0, 0) the Jacobian matrix J becomes:

J(0, 0) =

(α 00 −γ

).

The eigenvalues of this matrix are λ1 = α, λ2 = −γ. In the model α and γ arealways greater than zero, and as such the sign of the eigenvalues above will alwaysdiffer. Hence the fixed point at the origin is a saddle point.The stability of this fixed point is of significance. If it were stable, non-zero pop-ulations might be attracted towards it, and as such the dynamics of the systemmight lead towards the extinction of both species for many cases of initial popula-tion levels. However, as the fixed point at the origin is a saddle point, and henceunstable, it follows that the extinction of both species is difficult in the model.(In fact, this could only occur if the prey were artificially completely eradicated,causing the predators to die of starvation. If the predators were eradicated, theprey population would grow without bound in this simple model): The popula-tions of prey and predator can get infinitesimally close to zero and still recover.

Second fixed point (oscillations)

Evaluating J at the second fixed point leads to:

J(γ

δ,α

β) =

(0 −βγ

δαδβ 0

),

The eigenvalues of this matrix are

λ1 = i√αγ, λ2 = −i√αγ .

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As the eigenvalues are both purely imaginary, this fixed point is not hyperbolic,so no conclusions can be drawn from the linear analysis. However, as illustratedabove, the system admits a constant of motion V, or, equivalently, exp(V),

K = yαexp(−βy)xγexp(−δx) ,

and the level curves, for each constant K, are closed orbits surrounding the fixedpoint: the levels of the predator and prey populations cycle, and oscillate aroundthis fixed point. Increasing K moves a closed orbit closer to the fixed point. Thelargest value of the constant K is obtained by solving the optimization problem.

K = yαexp(−βy)xγexp(−δx) =yαxγ

exp(δx+ βy)−→ max (x, y) > 0 ,

.The maximal value of K is thus attained at the stationary (fixed) point (γ

δ, αβ) and

amounts to

K∗ = (α

βe)α(

γ

δe)γ .

where e is Euler’s Number.

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Methodology

1 To analyze the various aspects of models we use the tools of nonlinear ODE.

2 The stability of various equilibrium points have been discussed with the helpof linearization techniques, Hartman-Grobman theorem, Routh-Hurwitz criterionand Bendixson-Dulac criterion.

3 Sotomayor’s theorem has been used to prove the existence of saddle node andtranscritical bifurcation.

4 The existence of periodic solution through Hopf- bifurcation has been proved byusing Hopf-bifurcation theorem.

5 Normal form theory is used to reduce the models into their corresponding canon-ical form of Bogdanov-Takens bifurcation.

6 Numerical simulations in MATLAB and MAPLE have been carried out to verifythe results obtained analytically.

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References:

1 L. Perko, Differential equation and dyanamical system, Springer(1996).

2 J. Guckenheimer and P. Holmes , Nonlinear Oscillations dynamical system

and Bifurcatioins of vector fields ( Springer, New York 1983).

3 Steven H.strogatz, Nonlinear dynamics and chaos, with applications to Physics,

Chemistry, Biology and engineering

4 Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and

Chaos

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