dynamics of happiness in love

STUDY OF DYNAMICS OF HAPPINESS IN LOVE

PRESENTED BY

PRADYUMN PATHAK

MSC(MATHEMATICS)

Dayal Bagh Educational

Institute Agra

Email id: pathakpradyumn70@gmail.com

ABSTRACT

This Research paper deals with the dynamics of happiness in love. An obvious difficulty in any model of love and happiness is defining what is meant by love and happiness & quantifying it in some meaningful way.

There are many type of love, including intimacy, passion, commitment and each type consists of a complex mixture of feelings. Although love and happiness are not identical, but love is a one source of happiness,

We define a Mathematical model using the Autonomous system of two linear differential equations with constant coefficients to study the dynamics of happiness in love, by considering two lovers following two type of strategies with positive rate of happiness (happy minded) or negative rate of happiness (not happy minded) & then analyze this system by using the characteristic roots of characteristic equation of the Coefficient matrix of the Matrix representation of this Autonomous system for different cases & gives phase plane trajectory for each cases. On the basis of this analysis we give conclusion.

PreliminariesConsider a two dimensional system of linear differential equations = .

Phase Portrait

Representative set of solutions of the system of linear differential equations ,

plotted as parametric curves (With t as the parameter) on the Cartesian plane

tracing the path of each Particular solution

(x, y) = (x1(t), x2(t)) , −∞ < t <∞ .

Phase Plane

Cartesian plane where the phase portrait resides.

Trajectories

The parametric curves traced by the solutions.

Equilibrium solution(critical point)

An equilibrium solution of the system = is a point (x1, x2) where = 0,

that is, where x1’= 0 = x2

’. An equilibrium solution is a constant solution of the

system, and is usually called a critical point.

Stability of critical point

Asymptotically stable All trajectories of its solutions converge to the critical point as t→∞. A critical point is asymptotically stable if all of B’s eigenvalues are

negative, or have negative real part for complex eigenvalues.

Unstable All trajectories (or all but a few, in the case of a saddle point) start out at the

critical point at t→-∞, and then move away to infinitely distant out as t→∞. A critical point is unstable if at least one of B’s eigenvalues is positive, or

has positive real part for complex eigenvalues.

Stable (or neutrally stable) Each trajectory move about the critical point within a finite range of

distance. It never moves out to infinitely distant, nor (unlike in the case of asymptotically stable) does it ever go to the critical point.

A critical point is stable if B’s eigenvalues are purely imaginary for complex eigenvalues .

INTRODUCTION

Previous work on this topic Some dynamical models of love, which is certainly one source of happiness,

have been inspired by the seminal work of Strogatz (1988, 1994) was originally intended more to motivate students than as a serious description of love affairs.essentially the same model described earlier by Rapoport(1960) , & it has also been studied by Radzicki(1993).

Related discrete dynamical models of verbal interaction of married couples have been advanced by Gottman(2002).

Also dynamical model of love & dynamical model of happiness presented by J.C. Sprott (2004, 2005) which deals with the dynamics of happiness & dynamics of love.

Happiness For many people pursuit of happiness is a dominant goal of life. Happiness is a feeling which is dynamic & it would be a dull & lifeless person

indeed who did not experience ups & downs According to Buddhist philosophy true happiness is a mood not related to

circumstances.

The model presented here is not sensitive to the precise definition of happiness. It is essential to understand that the happiness described here is the subjective feeling of the individual in love with other individual rather than the judgment of someone else.

PROPOSED MODEL

It is always difficult to define a love & happiness . As they depend on so many factors. Many things can influence happiness besides love. love and happiness are not identical.

We are dealing with the happiness of individuals when they are in love

with following strategies.

STRATEGIES OF LOVERS TRUE LOVER:

When happiness of one lover is dependent on the happiness of other lovers who loves him i.e. one lover is happy due to the happiness of all other lovers who loves him & also unhappy due to the unhappiness of some other lovers who loves him.

PRACTICAL LOVER:

Lover is only happy when he get success in love otherwise he is not happy i.e. happiness in this case depend upon the success which is defined by lover.

MATHEMATICAL FORMULATION We define this model for two lovers; say X & Y both love each other.

 Hx(t) represent the happiness of X at time t & Hy(t) represent the happiness of

Y at time t simply denoted by Hx & Hy. Hx > 0 means X is happy in love.

Hx < 0 means X is unhappy in love.

Hx = 0 means X is neither happy nor unhappy i.e. X has no feelings, emotions for the other lover Y.

If Hx & Hy both are zero then there is the end of love story.

GENERAL MODEL FOR TWO LOVERS X & Y

This System of linear differential equation is called an Autonomous System.

1

MATRIX REPRESNTATION OF THE MODEL

Matrix is called COMMUNITY MATRIX of the above system.

Here a, b є [-1, 1]-{0}. a-rate of happiness of X . b-rate of happiness of Y . a & b can’t be zero as happiness can’t be constant in love, as it is always vary with

time. a>0 means X is happy minded & a<0 means X is not happy minded. b>0 means Y is happy minded & b<0 means Y is not happy minded. Value of A depend on lovers For true lover A = 0 For practical lover A = 1(when there is a success),A = -1(when there is no

success).

Since A works for only practical lover so A is termed as “Practical Happiness Factor”

DEFINING MODEL FOR DIFFERENT TYPES OF LOVER1. X is true lover & Y is practical lover

(a) When there is a success in love (b) When there is no success in love

2. X & Y are both true lovers

3. X & Y are both practical lovers

(a) When there is a success in love (b) When there is no success in love

(c) When there is success for X only

MATHEMATICAL ANALYSIS Equation (1) which define a general model have a single equilibrium at

Hx=Hy=0, Corresponds to the end of love story.

1. When a>0 & b>0

Take a=b=1 in all cases.

Case 1: X is true lover & Y is practical lover

For equation 1(a) the characteristic equation is having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y get success in love

Saddle point (0,0)

For equation 1(b) the characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y don't get success in love

Saddle point (0,0)

Case 2: Both X and Y are true lovers:-

The characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are true lover

Saddle point (0,0)

Case 3: X and Y are both practical lovers:-

For equation 3(a) the characteristic equation is , having one root is 0 & other is 2>0.Hence Hx=Hy=0 is a proper node. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the proper node (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers and they get success in love

unstable proper node(0,0)

For equation 3(b) the characteristic equation is , having one root is 0 & other is -2<0.Hence Hx=Hy=0 is a proper node. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the proper node (0,0) is stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers & they don't get success in love

stable proper node(0,0)

For equation 3(c)the characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lover, only X get success in love

Saddle point (0,0)

2. When a>0 & b<0

 Take a = 1 & b = -1 in all cases. 

Case 1: X is true lover & Y is practical lover:-

For equation 1(a) the characteristic equation is , having complex conjugate as roots with positive real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the focus (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y get success in love

unstable focus(0,0)

For equation 1(b) the characteristic equation is , having complex conjugate as roots with negative real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the focus (0,0) is stable & it is asymptotically stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

hx

hy

X is true lover & Y is practical lover & Y don't get success in love

stable focus (0,0)

Case 2: Both X and Y are true lovers:-

The characteristic equation is , having complex conjugate as roots with real part is zero. Hence Hx=Hy=0 is a center. In Figure arrow indicates the center (0,0) is stable but not asymptotically stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

hx

hy

Both X & Y are true lover

center (0,0)

Case 3: X and Y are both practical lovers:-

For equation 3(a) the characteristic equation is , having complex conjugate as roots with positive real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the focus (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

hx

hy

Both X & Y are practical lovers and they get success in love

unstable focus (0,0)

For equation 3(b) the characteristic equation is , having complex conjugate as roots with negative real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicates the focus (0,0) is stable & it is asymptotically stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers & they don't get success in love

stable focus (0,0)

For equation 3(c) the characteristic equation is , having both roots equal to zero. Hence Hx=Hy=0 is a node. In Figure arrow indicates the proper node (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lover, only X get success in love

node (0,0)

3. When a<0 & b>0

Take a = -1 & b=1 in all cases. 

Case 1: X is true lover & Y is practical lover:-

For equation 1(a) the characteristic equation is , having complex conjugate as roots with positive real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the focus (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y get success in love

unstable focus (0,0)

For equation 1(b) the characteristic equation is , having complex conjugate as roots with negative real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the focus (0,0) is stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y don't get success in love

stable focus (0,0)

Case 2: Both X and Y are true lovers:-

The characteristic equation is , having complex conjugate as roots with real part is zero. Hence Hx=Hy=0 is a center. In Figure arrow indicates the center (0,0) is stable but not asymptotically stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are true lover

center (0,0)

Case 3: X and Y are both practical lovers:-

For equation 3(a) the characteristic equation is , having complex conjugate as roots with positive real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the focus (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers and they get success in love

unstable focus (0,0)

For equation 3(b) the characteristic equation is , having complex conjugate as roots with negative real part. Hence Hx=Hy=0 is a focus. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicates the focus (0,0) is stable & it is asymptotically stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers & they don't get success in love

stable focus (0,0)

For equation 3(c) the characteristic equation is , having both roots equal to zero. Hence Hx=Hy=0 is a node. In Figure arrow indicates the proper node (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lover, only X get success in love

node (0,0)

4. When a<0 & b<0  

Take a = -1 & b = -1 in all cases.

Case 1: X is true lover & Y is practical lover:-

For equation 1(a) the characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y get success in love

saddle point (0,0)

For equation 1(b) the characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

X is true lover & Y is practical lover & Y don't get success in love

saddle point (0,0)

Case 2: Both X and Y are true lovers:-

The characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Fig 4.2 arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are true lover

saddle point (0,0)

For equation 3(a) the characteristic equation is , having one root is 0 & other is 2>0.Hence Hx=Hy=0 is a proper node. In Fig 4.3(a) arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the proper node (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers and they get success in love

unstable proper node (0,0)

For equation 3(b) the characteristic equation is , having one root is 0 & other is -2<0.Hence Hx=Hy=0 is a proper node. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicates the proper node (0,0) is stable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lovers & they don't get success in love

stable proper node (0,0)

For equation 3(c) the characteristic equation is , having two real roots with opposite sign. Hence Hx=Hy=0 is a saddle point. In Figure arrow indicates the dynamics of happiness of X & Y as time increasing. As arrow indicate the saddle point (0,0) is unstable.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

Hx

Hy

Both X & Y are practical lover, only X get success in love

saddle point (0,0)

INTERPRETATION

When X is true lover and Y is practical lover and Y get success in love

When X & Y both are happy minded then they are mutually happy and their happiness increases in love as time increases so love story never ends.

When X is happy mind but Y is not happy minded then both are happy in love most of the time but as time increases happiness of Y decreases but this situation is not stable so love story never ends.

When Y is happy minded but X is not happy minded then both are happy in love most of the time but as time increases happiness of X decreases but this situation is not stable so love story never ends.

When X & Y both are not happy minded then Y is happy & X is not happy as time increases but this situation is not stable so love story never ends.

When X is true lover and Y is practical lover and Y don’t get success in love

When X & Y both are happy minded then they are mutually happy for some time & mutually unhappy for most of the time in love as time increases but this situation is not stable love story never ends.

When X is happy minded but Y is not happy minded then both are unhappy in love most of the time & this situation is stable so there is end of love story.

When Y is happy minded but X is not happy minded then both are unhappy in love but as time increases happiness of X & Y increases to zero (but they remain unhappy) but finally there is end of love story.

When X & Y both are not happy minded then they are unhappy or happy mutually & this situation is not stable so love story never ends.

When X & Y both are true lovers

When X & Y both are happy minded then they are mutually happy or unhappy in love as time increases but this situation is not stable so love story never ends.

When X is happy minded but Y is not happy minded then there is endless cycle of happiness & unhappiness in love as time increases, love story never ends.

When Y is happy minded but X is not happy minded then also there is endless cycle of happiness & unhappiness in love as time increases, love story never ends.

When X & Y both are not happy minded then they are mutually happy or unhappy in love as time increases& this situation is not stable so love story never ends.

When X & Y both are practical lovers and both get success in love

When X & Y both are happy minded then they are mutually happy or unhappy in love as time increases but this situation is not stable so love story never ends.

When X is happy minded but Y is not happy minded then both are happy in love for most of the time but as time increases happiness of Y decreases for some time but this situation is not stable so love story never ends.

When Y is happy minded but X is not happy minded then both are happy in love for most of the time but as time increases happiness of X decreases for some time but this situation is not stable so love story never ends.

When X & Y both are not happy minded then both are happy in love for most of the time but as time increases happiness of X & Y decreases for some time but this situation is not stable so love story never ends.

When X & Y both are practical lovers and both don’t get success in love

When X & Y both are happy minded then both are unhappy in love & their happiness decreases as time increases but this situation is stable there is end of love story.

When X is happy mind but Y is not happy minded then both are unhappy in love for most of the time but as time increases happiness of X increases to zero, there is the end of love story.

When Y is happy minded but X is not happy minded then both are unhappy in love but as time increases happiness of Y increases to zero, there is the end of love story.

When X & Y both are not happy minded then both are unhappy in love & their happiness increases to zero as time increases there is end of love story.

When X & Y both are practical lovers and only X get success in love

When X & Y both are happy minded then both are mutually happy in love & their happiness increases as time increases, love story never ends.

When X is happy minded but Y is not happy minded then both are happy in love for most of the time but as time increases they are unhappy for some time, but this situation is not stable so love story never ends.

When Y is happy minded but X is not happy minded then both are happy in love for most of the time but as time increases they are unhappy for some time, but this situation is not stable so love story never ends.

When X & Y both are not happy minded then X is happy & Y is unhappy for most of the time in love but as time increases X is unhappy for some time, but this situation is not stable so love story never ends. 

CONCLUSIONS Love Story never end.(There is always some feeling between the two lovers). When both lovers are true lover When both lovers are practical & they get success When one lover is true lover & other is practical lover & practical lover get

success.

(Whether both lovers are happy minded or not happy minded, or one of them is happy minded & other is not happy minded.)

When one lover is true lover & other is practical lover & practical lover don’t get success in love

(When both lovers are happy minded or unhappy minded.)

Love story ends.(There is no feeling between the two lovers) When both lovers are practical lover & they don’t get success in love.

(Whether both lovers are happy minded or not happy minded, or one of them is happy minded & other is not happy minded.)

When one lover is true lover & other is practical lover & practical lover don’t get success.

(When only one of the lover is happy minded.)

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