Discrete and Continuous Dynamical Systems: Applications and Examples

Yonah Borns-Weil and Junho Won Mentored by Dr. Aaron Welters

Fourth Annual PRIMES Conference

May 18, 2014

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 1 / 32 Overview of dynamical systems

What is a dynamical system?

Two flavors: Discrete (Iterative Maps)

Continuous (Differential Equations)

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 2 / 32 Fixed points Periodic points (can be reduced to fixed points) Stability of fixed points By approximating f with a linear function, we get that a fixed point x∗ is stable whenever |f 0(x∗)| < 1.

Iterative maps

Definition (Iterative map)

A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R → R. Basic Ideas:

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 3 / 32 Periodic points (can be reduced to fixed points) Stability of fixed points By approximating f with a linear function, we get that a fixed point x∗ is stable whenever |f 0(x∗)| < 1.

Iterative maps

Definition (Iterative map)

A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R → R. Basic Ideas: Fixed points

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 3 / 32 Stability of fixed points By approximating f with a linear function, we get that a fixed point x∗ is stable whenever |f 0(x∗)| < 1.

Iterative maps

Definition (Iterative map)

A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R → R. Basic Ideas: Fixed points Periodic points (can be reduced to fixed points)

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 3 / 32 By approximating f with a linear function, we get that a fixed point x∗ is stable whenever |f 0(x∗)| < 1.

Iterative maps

Definition (Iterative map)

A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R → R. Basic Ideas: Fixed points Periodic points (can be reduced to fixed points) Stability of fixed points

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 3 / 32 Iterative maps

Definition (Iterative map)

A (one-dimensional) iterative map is a sequence {xn} with xn+1 = f (xn) for some function f : R → R. Basic Ideas: Fixed points Periodic points (can be reduced to fixed points) Stability of fixed points By approximating f with a linear function, we get that a fixed point x∗ is stable whenever |f 0(x∗)| < 1.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 3 / 32 Getting a picture: “cobwebbing”

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 4 / 32 If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it’s stable. If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r √ √ r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on.

A famous example: the logistic map

We consider xn+1 = rxn(1 − xn) on the interval [0, 1]. Properties vary based on r:

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r √ √ r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on.

A famous example: the logistic map

We consider xn+1 = rxn(1 − xn) on the interval [0, 1]. Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it’s stable.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 √ √ r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on.

A famous example: the logistic map

We consider xn+1 = rxn(1 − xn) on the interval [0, 1]. Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it’s stable. 1 If 1 < r ≤ 3, 0 is unstable, 1 − r is stable.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 2-cycle becomes 4-cycle, then 8-cycle, and so on.

A famous example: the logistic map

We consider xn+1 = rxn(1 − xn) on the interval [0, 1]. Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it’s stable. If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r √ √ r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 A famous example: the logistic map

We consider xn+1 = rxn(1 − xn) on the interval [0, 1]. Properties vary based on r: If 0 ≤ r ≤ 1, only fixed point is x∗ = 0, and it’s stable. If 1 < r ≤ 3, 0 is unstable, 1 − 1 is stable. r √ √ r+1± (r−3)(r+1) If 3 < r ≤ 1 + 6, no stable fixed points, but 2r is stable 2-cycle. 2-cycle becomes 4-cycle, then 8-cycle, and so on.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 5 / 32 The case of a stable 2-cycle

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 6 / 32 The orbit diagram

We can plot the points in stable cycles with respect to r:

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 7 / 32 The first Feigenbaum constant

n Let rn be where stable 2 cycle begins. The distance between rn’s converges roughly geometrically, up to r∞. Definition (δ) The first Feigenbaum constant is defined as r − r δ = lim n n−1 ≈ 4.669 ... n→∞ rn+1 − rn

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 8 / 32 The first Feigenbaum constant: not just for one map?

Yeah, but why do we care about δ?

Consider the sine map xn+1 = r sin πxn. Guess what its orbit diagram looks like?

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 9 / 32 Sine map orbit diagram

No, I didn’t accidentally repeat the previous image...

...it looks exactly the same! Not only that, if you try to calculate δ here, you’ll get the same number!

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 10 / 32 Univerality of δ

Theorem 1 (Universality of δ) If f 00 0 1 f 00(x)2 D f (x) = (x) − < 0 sch f 0 2 f 0(x) in the bounded interval and f experiences period-doubling, then letting {rn} be defined for this new map, r − r lim n n−1 = δ. n→∞ rn+1 − rn

Essentially, δ is a “universal constant!”

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 11 / 32 Now, the continuous case ... Continuous dynamical systems involve analyzing differential equations. They describe systems that change over time.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 12 / 32 Belousov’s discovery in 1950’s exhibits a periodical behavior.

Oscillating chemical reactions

Chemical reactions: governed by differential equations involving concentrations of the reactants and products. Multi-step reactions can exhibit complicated dynamical behaviors.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 13 / 32 Oscillating chemical reactions

Chemical reactions: governed by differential equations involving concentrations of the reactants and products. Multi-step reactions can exhibit complicated dynamical behaviors. Belousov’s discovery in 1950’s exhibits a periodical behavior.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 13 / 32 Continuous dynamical systems and oscillating chemical reactions

Figure: Periodic behavior of an oscillating chemical reaction.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 14 / 32 Continuous dynamical systems: one–dimensional case

x˙ = f (x) The continuous time dynamics˙x of a system is governed by its current statex.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 15 / 32 Flow and vector fields Stable and unstable fixed points (x ˙ = 0)

Continuous dynamical systems: one–dimensional case

Example:x ˙ = r + x2, where r is a parameter.

Figure: The phase portrait of the systemx ˙ = r + x 2.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 16 / 32 Continuous dynamical systems: one–dimensional case

Example:x ˙ = r + x2, where r is a parameter.

Figure: The phase portrait of the systemx ˙ = r + x 2.

Flow and vector fields Stable and unstable fixed points (x ˙ = 0)

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 16 / 32 Continuous dynamical systems: bifurcations

Example:x ˙ = r + x2, where r is a parameter.

Figure: The phase portrait of the systemx ˙ = r + x 2.

Bifurcation: a qualitative change in the vector field.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 17 / 32 Vector fields: represented as arrows on the plane (phase portrait).

Continuous dynamical systems: two–dimensional case

x˙ = f (x, y) y˙ = g(x, y).

Figure: A two-dimensional vector field.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 18 / 32 Continuous dynamical systems: two–dimensional case

x˙ = f (x, y) y˙ = g(x, y).

Figure: A two-dimensional vector field.

Vector fields: represented as arrows on the plane (phase portrait).

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 18 / 32 Linearization near a fixed point

Linearized systems: near a fixed point (x∗, y ∗),

u = x − x∗, v = y − y ∗,

  ∂f ∂f !   u ∂x ∂y x = ∂g ∂g . v ∂x ∂y y The matrix ∂f ∂f ! ∂x ∂y A = ∂g ∂g ∂x ∂y (x∗,y ∗) is called the Jacobian matrix at the fixed point (x∗, y ∗).

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 19 / 32 Linearization near a fixed point

Figure: A two-dimensional vector field.

The eigenvectors and eigenvalues (λ) of A determine the eigendirections near (x∗, y ∗). Behavior of the flow near a fixed point is governed by the stable manifolds (λ < 0) and unstable manifolds (λ > 0).

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 20 / 32 Back to oscillating chemical reactions...

Figure: Periodic behavior of an oscillating chemical reaction.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 21 / 32 The BZ (Belousov-Zhabotinsky) reaction

Main reaction steps:

− + d[I2] k1a[MA][I2] MA + I2 → IMA + I + H ; = − (1) t k1b + [I2] 1 d[ClO ] [ClO ] ClO + I − → ClO− + I ; 2 = −k 2 (2) 2 2 2 2 t 2 [I −]

− − + − ClO2 + 4I + 4H → Cl + 2I2 + 2H2O;

d[ClO−] [I −] 2 = −k [ClO−][I −][H+] − k [ClO−][I ] (3) dt 3a 2 3b 2 2 u + [I −]2

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 22 / 32 The BZ (Belousov-Zhabotinsky) reaction

Main reaction steps:

− + d[I2] k1a[MA][I2] MA + I2 → IMA + I + H ; = − (4) t k1b + [I2] 1 d[ClO ] [ClO ] ClO + I − → ClO− + I ; 2 = −k 2 (5) 2 2 2 2 t 2 [I −]

− − + − ClO2 + 4I + 4H → Cl + 2I2 + 2H2O;

d[ClO−] [I −] 2 = −k [ClO−][I −][H+] − k [ClO−][I ] (6) dt 3a 2 3b 2 2 u + [I −]2 =⇒Very complicated.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 23 / 32 Simplified model of the BZ reaction

4xy x˙ = a − x − 1+x2 ,  y  y˙ = bx 1 − 1+x2 . − − Here, x and y are dimensionless concentrations of I and ClO2 .

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 24 / 32 Simplified model of the BZ reaction

Figure: The phase portrait of the simplied model of the BZ reaction.

Fixed point where the nullclines intersect tangentially

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 25 / 32 Analysis of the dynamical system

Behavior of vector fields near a fixed point in a linearized system is determined by the determinant ∆ and the trace τ of the Jacobian matrix.

Figure: Classification of fixed points.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 26 / 32 Fixed point: x∗ = a/5, y ∗ = 1 + (x∗)2 = 1 + (a/5)2. Jacobian at the fixed point (x∗, y ∗) is

1  3(x∗)2 − 5 −4x∗  . 1 + (x∗)2 2b(x∗)2 −bx∗

The determinant and trace are given by

5bx∗ 3(x∗)2 − 5 − bx∗ ∆ = > 0, τ = . 1 + (x∗)2 1 + (x∗)2

Analysis of the dynamical system

The system is linearized near the fixed point.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 27 / 32 Jacobian at the fixed point (x∗, y ∗) is

1  3(x∗)2 − 5 −4x∗  . 1 + (x∗)2 2b(x∗)2 −bx∗

The determinant and trace are given by

5bx∗ 3(x∗)2 − 5 − bx∗ ∆ = > 0, τ = . 1 + (x∗)2 1 + (x∗)2

Analysis of the dynamical system

The system is linearized near the fixed point. Fixed point: x∗ = a/5, y ∗ = 1 + (x∗)2 = 1 + (a/5)2.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 27 / 32 The determinant and trace are given by

5bx∗ 3(x∗)2 − 5 − bx∗ ∆ = > 0, τ = . 1 + (x∗)2 1 + (x∗)2

Analysis of the dynamical system

The system is linearized near the fixed point. Fixed point: x∗ = a/5, y ∗ = 1 + (x∗)2 = 1 + (a/5)2. Jacobian at the fixed point (x∗, y ∗) is

1  3(x∗)2 − 5 −4x∗  . 1 + (x∗)2 2b(x∗)2 −bx∗

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 27 / 32 Analysis of the dynamical system

The system is linearized near the fixed point. Fixed point: x∗ = a/5, y ∗ = 1 + (x∗)2 = 1 + (a/5)2. Jacobian at the fixed point (x∗, y ∗) is

1  3(x∗)2 − 5 −4x∗  . 1 + (x∗)2 2b(x∗)2 −bx∗

The determinant and trace are given by

5bx∗ 3(x∗)2 − 5 − bx∗ ∆ = > 0, τ = . 1 + (x∗)2 1 + (x∗)2

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 27 / 32 τ > 0 if b < bc = 3a/5 − 25/a.

A bifurcation occurs at b = bc (the stability of the fixed point changes).

Analysis of the dynamical system

The fixed point is unstable if ∆ > 0 and τ > 0 (∆ > 0 is given to us).

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 28 / 32 Analysis of the dynamical system

The fixed point is unstable if ∆ > 0 and τ > 0 (∆ > 0 is given to us).

τ > 0 if b < bc = 3a/5 − 25/a.

A bifurcation occurs at b = bc (the stability of the fixed point changes).

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 28 / 32 Applying the Poincar´e-BendixsonTheorem

Theorem 2 (Poincar´e-BendixsonTheorem) Suppose that: 1 R is a closed, bounded subset of the plane; 2 x˙ = f (x) is a continuously differentiable vector field on an open set containing R; 3 R does not contain any fixed points; and 4 There exists a trajectory C that is “confined” in R, in the sense that it starts in R and stays in R for all future time. Then R contains a closed orbit.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 29 / 32 Applying the Poincar´e-BendixsonTheorem

Theorem 2 (Poincar´e-BendixsonTheorem) Suppose that: 1 R is a closed, bounded subset of the plane; 2 x˙ = f (x) is a continuously differentiable vector field on an open set containing R; 3 R does not contain any fixed points; and 4 There exists a trajectory C that is “confined” in R, in the sense that it starts in R and stays in R for all future time. Then R contains a closed orbit.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 29 / 32 Poincar´e-BendixsonTheorem tells us that there is a stable limit cycle (a closed orbit) around the fixed point. Chemically, this explains why the BZ reaction shows a periodic behavior.

Conclusion

Figure: A trapping box in the BZ reaction system.

Because distant vectors point to the unstable fixed point when b < bc , we can form a punctured trapping region for trajectories.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 30 / 32 Chemically, this explains why the BZ reaction shows a periodic behavior.

Conclusion

Figure: A trapping box in the BZ reaction system.

Because distant vectors point to the unstable fixed point when b < bc , we can form a punctured trapping region for trajectories. Poincar´e-BendixsonTheorem tells us that there is a stable limit cycle (a closed orbit) around the fixed point.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 30 / 32 Conclusion

Figure: A trapping box in the BZ reaction system.

Because distant vectors point to the unstable fixed point when b < bc , we can form a punctured trapping region for trajectories. Poincar´e-BendixsonTheorem tells us that there is a stable limit cycle (a closed orbit) around the fixed point. Chemically, this explains why the BZ reaction shows a periodic behavior.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 30 / 32 Conclusion

Changing the parameters a, b > 0, which depend on the rate constants and concentrations of slow reactants, results in a supercritical Hopf bifurcation. Change in stability. Formation of a stable limit cycle.

Figure: Supercritical Hopf bifurcation in the BZ reaction system.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 31 / 32 Bibliography

S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books Publishing, Cambridge, 1994.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 31 / 32 Acknowledgements

We thank our mentor Dr. Aaron Welters for mentoring us with studying nonlinear dynamical systems, as well as providing many useful advice in general. Also, our head mentor Dr. Tanya Khovanova and others have helped us with improving our presentation, and MIT—PRIMES provided us with this enjoyable opportunity to study mathematics. Finally, we wish to thank our parents for their continued support with our studies.

J. Won, Y. Borns-Weil (MIT) Discrete and Continuous Dynamical Systems May 18, 2014 32 / 32