Notes on Nonlinear Dynamics and Chaos by S.

Strogatz

Stuart Truax

February 7, 2021

These notes primarily use Nonlinear Dynamics and Chaos by S. Strogatzas their basis, and the descriptive language is often borrowed from, or a para-phrasing of, that text. The section numbers in this document do not followthe section numbers in Strogatz’s text, as I have condensed and reorganized thematerial to emphasize the areas of particular interest to me during my study.Moreover, I have included related material from other sources as appropriate,for the subject matter has many deep connections to other fields.

The graphics are generated from my own code, which can be found in theassociated github repository.

–S. Truax

1 Dynamical Systems

Def : General dynamical system:

x1 = f1(x1, ..., xn)

...

xn = fn(x1, ..., xn)

1.1 Higher-Order Time Derivatives

System with higher-order time derivatives are described by creating extra vari-ables in the state vector that represent the higher-order derivatives. For exam-ple, a one-dimensional damped harmonic oscillator is described by:

x1 = x2

1

x2 = − b

mx2 −

k

mx1

2 Flows on The Line

”Flows on The Line” refer to 1-D systems. For 1-D systems, oscillations areimpossible, and phase portraits consist of x as the independent variable, andx = f(x) as the dependent variable.

Trajectories are plotted as arrows on the x-axis.

2.1 Fixed Points and Stability

Def : A fixed point is a point x for which:

x = f(x) = 0

Stability can be determined graphically by sketching the values of f(x) inthe neighborhood of a fixed point x∗. Flows emanate outward from unstablefixed points (sources) and flow towards stable fixed points (sinks). A fixedpoint that attracts some trajectories and repels others is called half-stable.

2.2 Linear Stability Analysis

Proof : Let x∗ be a fixed point. Define:

η(t) = x(t)− x∗

As a small perturbation about the fixed point. It follows that:

η(t) =d

dt(x− x∗) = x

Hence:η = x = f(x) = f(x∗ + η)

Applying a Taylor’s expansion yields:

f(x∗ + η) = f(x∗) + ηf ′(x∗) +O(η2)

Since f(x∗) = 0, it follows that:

η = ηf ′(x∗) +O(η2)

2

Two conditions arise from this equation:

• f ′(x∗) 6= 0, which yields two cases:

– f ′(x∗) < 0 =⇒ η(t) decays exponentially.

– f ′(x∗) > 0 =⇒ η(t) grows exponentially.

• f ′(x∗) = 0 : the O(η2) term is not negligible, and a nonlinear analysis isrequired to determine stability.

2.3 Characteristic Time Scale of a Fixed Point

Def : A characteristic time scale of a fixed point x∗ can be defined as :

τ =1

|f ′(x∗)|

2.4 Existence and Uniqueness

Thm.: If f(...) is uniformly Lipschitz continuous on some interval of t, thenthere exists a unique solution on the interval.

3 Bifurcations

Bifurcations diagrams are really required to do the graphical analysis neces-sary for this section. Such diagrams have the bifurcation parameter r as theindependent variable and the state variable x as the dependent variable.

3.1 Normal Forms

Bifurcations of a dynamical system can be generalized and classified accordingto their normal forms. Consider the dynamical system:

x = f(x, r)

where r is the bifurcation parameter. Further consider a neighborhood abouta fixed point x = x∗ and a critical value of the bifurcation parameter r = rc.One can now perform a Taylor series expansion of x:

x = f(x, r) = f(x∗, rc)+(x−x∗)∂f∂x

∣∣∣(x∗,rc)

+(r−rc)∂f

∂r

∣∣∣(x∗,rc)

+1

2(x−x∗)2 ∂

2f

∂x2

∣∣∣(x∗,rc)

+. . .

3

In the assumed neighborhood of (x∗, rc), the term f(x∗, rc) = 0 will vanishdue to the fixed point condition f(x∗, r) = 0.

What is left is a polynomial form of f(x, r) whose low-order terms define thebifurcation.

3.2 Saddle-Node Bifurcations

Saddle-node bifurcations correspond to the creation or annihilation oftwo fixed points from a single fixed point.

The canonical form of a system with a saddle node bifurcation is:

x = r + x2

where r is the bifurcation parameter. Variation of r yields the following setsof equilibria:

• r < 0, two fixed points, one stable and one unstable

• r = 0, a single half-stable fixed point at x∗ = 0

• r > 0, no fixed points

4

3.2.1 Normal Form of the Saddle-Node Bifurcation

x = a(r − rc) + b(x− x∗)2

where a = ∂f∂x

∣∣∣(x∗,rc)

and b = 12∂2f∂x2

∣∣∣(x∗,rc)

3.3 Pitchfork Bifurcations

Pitchfork bifurcations correspond to systems with an inherent notion of sym-metry, and correspondingly symmetry breaking. Two types of pitchfork bifur-cations, subcritical and supercritical, correspond to first- and second-orderphase transitions under the Ehrenfest classification of phase transitions.

Remember that the order of the phase transition corresponds to the lowestderivative of the free energy that is discontinuous in the phase transition. Thebehavior of the Landau theory is captured by supercritical pitchfork bifurca-tions.

The supercritical/second-order pitchfork bifurcation is born by small am-plitude changes in the state variable x.

The subcritical/first-order pitchfork bifurcation corresponds to large am-plitude changes or jumps in the state variable x.

3.3.1 Supercritical Pitchfork Bifurcations

5

The normal form for the supercritical pitchfork bifurcation is:

x = rx− x3

Symmetry in this system is expressed through the invariance of reflectionsabout x = x. That is, substituting x→ −x does not change the equation:

−f(x) = f(−x)

which is the defining property of an odd function. System dynamicsdescribed by odd functions will generally capture symmetry. Corre-spondingly, the symmetry is broken by an additive constant (see the imperfectionparameter).

The bifurcation has the following sets of equilibria:

• r < 0, one stable fixed point at x∗ = 0

• r = 0, one weakly stable fixed point at x∗ = 0, with linear terms vanishing.

• r > 0, three fixed points, two stable at x∗ = ±√r, and one unstable at

x∗ = 0

Application Note: Systems of the form x = −x + βtanh(x) exhibit thispitchfork bifurcation as β is varied. Note that the Taylor series expansion oftanh(x) is indeed an odd function of x. Since x = −x+βtanh(x) is an activationfunction for neural networks, neural networks can model this kind of bifurcation,and thus model the corresponding phase transitions. Graphical analysis showsthat any sigmoid function can capture such behavior. (see Strogatz example3.4.1)

3.3.2 Lifschitz’s Buttocks

A potential function is defined as:

x = f(x) = −dVdx

For f(x) = rx− x3, V (x) becomes:

V (x) = −1

2rx2 +

1

4x4

Assuming our potential function to represent a thermodynamic potentiallike the free energy G, and plotting a family of V ’s as a function of x, each

6

member corresponding to a value of r, one can observe the ”Lifschitz’s Buttocks”behavior of second-order phase transitions (see Figure 26.4 of [2]).

3.3.3 Subcritical Pitchfork Bifurcations

The normal form for the subcritical pitchfork bifurcation is (note only a signchange from the supercritical case):

x = rx+ x3

• r < 0, three fixed points, two unstable at x∗ = ±√r, one stable fixed point

at x∗ = 0

• r = 0, one weakly stable fixed point at x∗ = 0, with linear terms vanishing.

• r > 0, one unstable at x∗ = 0

Effectively it is the inverted behavior of the supercritical case. In real sys-tems, a −x5 term exists in f(x), meaning that the two unstable equilibria arestabilized by the −x5 term for rs < r, rs some stability threshold of r. Thisalters the bifurcation diagram by adding two stable branches above and belowx = 0 for r > rs.

This has the implication that these stable branches of the system can beaccessed with a sufficiently large amplitude of x (see Figure 3.4.7 of Strogatz).

7

Such branches are called large amplitude branches, and represent equilibria towhich the system can ”jump” to if excited sufficiently strongly. Once in theselarge amplitude equilibria, r must be varied to less than rs to reset the system,thus creating hysteretic behavior.

3.4 Imperfect Bifurcations: Breaking the Symmetry

The pitchfork bifurcation exhibits symmetry via the odd polynomial in its nor-mal form. The symmetry of such dynamics can be broken by the inclusionof a 0-th order term h (i.e. the imperfection parameter). The imperfectionparameter thus captures the asymmetry of the system. For example:

x = h+ rx− x3

When h is non-zero the symmetry of the system (a supercritical pitchforksystem) is broken.

The fixed points of this system can be found by solving for the intersectionsbetween the curves x = h and x = rx − x3 for several values of r and h, as inthe below plot:

As can be seen, the intersections/fixed points vary in number from one forh > hcr, to two for h < hcr and r ≥ 0.

The maximum and minimum of the cubic curves (which occur for r ≥ 0)correspond to saddle node bifurcations (two equlibria emerging from none) and

8

occur at:

hcr = ±2r

2

√r

3

Since the number and type of equilibria vary according to two parameters, hand r, a surface plot of the equilibria can be obtained by horizontally stackingbifurcation diagrams of r for a continuum of h values. Such plots are calledstability diagrams. A stability diagram for the supercritical pitchfork is shownbelow:

There is a region of instability in the fold of the surface, with the remainder ofthe surface representing stable upper and lower branches in any given bifurcationdiagram evaluated at an h-slice. This is called a cusp catastrophe, so calledbecause a system that travels along a trajectory on the upper surface would ”falloff” the cusp onto the lower stable portion of the surface. The discontinuouschange in state constitutes the catastrophe. Notice the hysteretic nature of thejump. These are the same dynamics as seen in the overdriven Duffing oscillator.

The stability diagram for the subcritical pitchfork is shown below. Noticethe stable and unstable regions have switched places relative to the supercriticalpitchfork (a stable region resides in the cusp).

9

4 Linear Systems

4.1 Linear Systems in 2 Dimensions

Def : A 2-d linear system has the form:

x = ax+ by

y = cx+ dy

where a, b, c, d are constant parameters. The parameters b and c define thecoupling between the state variables x and y, whereas the parameters a and ddefine the intrinsic growth rates of the respective state variables.

The system is compactly written in matrix form:

x = Ax

where:

A =

(a bc d

)

10

and

x =

(xy

)

Properties of the linear system:

• For solutions x1 and x2, c1x1 + c2x2 is also a solution.

• x∗ = 0 is always a fixed point.

4.2 Solution Procedure for 2-D Linear Systems

Form of the Solutions: Solutions to 2-D linear systems are of the form:

x(t) = eλtv

v some direction in the phase plane, and λ some characteristic growth ratefor that direction. et being the eigenfunction of the linear differential operator,the solutions to the eigenvalue problem for the system are used in conjunctionwith this eigenfunction.

The solutions to the eigenvalue problem:

Av = λv

are found by solving the equation:

(A− λI)v = 0

The appropriate values of λ must first be found. If non-trivial eigenvaluesexist, the above equation must have non-trivial solutions (i.e. v = 0 cannotbe the only solution). This is equivalent to saying that A − λI must be non-invertible, is singular, is not one-to-one/injective, and det(A− λI) = 0.

Solution Procedure:

1. Solve the quadratic equation in λ from:

det(A− λI) = 0

which is:

11

λ2 − τλ+ ∆ = 0

where:

τ = trace(A) = a+ d

∆ = det(A) = ad− bc

which gives:

λ1 =τ +√τ2 − 4∆

2

λ2 =τ −√τ2 − 4∆

2

2. For each λi, solve for vi:

(A− λiI)vi = 0

3. Form the solution:

x(t) = c1eλ1tv1 + c2e

λ2tv2

4.3 Stability Classes

There are several notions of stability for a given fixed point of a system.

Def.: A fixed point x∗ is attracting if there is some δ > 0 such thatlimt→∞ x(t) = x∗ whenever ||x(0)− x∗|| < δ.

That is, the trajectory eventually returns to an arbitrarily small neighbor-hood of x∗, with no constraint on where it may wander to in the interim.

Def.: A fixed point x∗ is Lyapunov stable if for each ε > 0 there exists aδ > 0 such that ||x(t)− x∗|| < ε whenever ||x(0)− x∗|| < δ.

That is, the trajectory is confined to a neighborhood of x∗ for all t > 0.

Def.: A fixed point x∗ is asymptotically stable if is both Lyapunov stableand attracting.

Def.: A fixed point x∗ is unstable if is neither Lyapunov stable norattracting.

12

4.4 Phase Diagrams

Vector fields defined on the plane of points (x, y) (i.e the phase plane) arereferred to as phase diagrams.

4.4.1 Behaviors in the Phase Plane

Fast vs. Slow Eigendirections:

Case 1: Stable Nodes As a solution approaches a stable node, it willtypically become increasingly parallel to the slower eigendirection. That is theeigenvector v with smallest |λ|. As t→ −∞, solutions are parallel to the fastereigendirections.

Case 1: Unstable Nodes As a solution exits an unstable node, it willbe initially parallel to the faster eigendirection. That is the eigenvector v withlargest |λ|. As t→∞, the solutions becomes more parallel to the slower eigendi-rection.

13

4.5 Classification of Phase Diagrams and Fixed Points ofa System with only Real Eigenvalues

Example Matrix Matrix andEigenvectorproperties

Equilibria Phase Diagram

A =

(a 00 −1

)a < −1

• λ1, λ2 < 0

• |λ1| > |λ2|

• Non-singular

Stable node at(0,0)

A =

(a 00 −1

)a = −1

• λ1, λ2 < 0

• |λi| = |λj |

• Non-singular

Stable symmetri-cal node or star at(0,0)

A =

(a 00 −1

)−1 < a < 0

• λ1, λ2 < 0

• |λ1| < |λ2|

• Non-singular

Stable node at(0,0)

14

Example Matrix Matrix andEigenvectorproperties

Equilibria Phase Diagram

A =

(a 00 −1

)a = 0

• λ1 = 0

• λ2 < 0

• Singular

• At leastone zeroeigenvalue

Line of fixedpoints along x-axis(i.e. the eigendi-rection of the 0eigenvalue)

A =

(a 00 −1

)a > 0

• λ1 > 0

• λ2 < 0

• |λ1| = |λ2|

• Non-singular

Saddle Pointat (0,0) , Stablemanifold of thesaddle point isthe axis of thedirection of thenegative eigenvalue(i.e. solutionsdecay toward (0,0)along this direc-tion). Unstablemanifold is theeigendirectionof the positiveeigenvalue.

4.6 Oscillatory Behavior in Terms of Eigenvalues

Recall that the eigenvales of the system have the general form:

λ1,2 =1

2(τ ±

√τ2 − 4∆)

Complex eigenvalues, which cause oscillatory behavior, are generated for:

τ2 − 4∆ < 0

15

The eigenvalues can be expressed in terms of the oscillatory parameters ofthe damping α and the radial oscillation frequency ω as:

λ1,2 = α± iω

yielding these handy equations for the oscillation parameters in terms of thetrace and determinant of the matrix A:

α = τ/2

ω =1

2

√4∆− τ2

16

4.7 Classification of Phase Diagrams and Fixed Points fora System with Complex Eigenvalues

Example Matrix Matrix andEigenvectorproperties

Equilibria Phase Diagram

A =

(0 a−a 0

)a > 0

• Im(λ1) > 0

• Re(λ1) = 0

• Im(λ2) < 0

• Re(λ2) = 0

• Non-singular

Center at (0,0)

A =

(a a−a a

)a > 0

• Im(λ1) > 0

• Re(λ1) 6= 0

• Im(λ2) < 0

• Re(λ2) 6= 0

• Non-singular

Spiral at (0,0)

4.8 Linear Systems of Note

4.8.1 Damped Harmonic Oscillators

Physical systems that store potential and kinetic energy will exhibit resonancemodes, which are characteristic states of oscillation representing a periodictransformation of kinetic into potential energy, and vice versa. These resonancephenomena occur in all energy domains, and examples can be readily seen in

17

electrical (e.g. RLC circuits), mechanical (e.g. mass-spring-damper systems),and even chemical (e.g. Belusov-Zhabotinsky reactions) energy domains.

The total amount on energy involved in the potential-kinetic energy transfercan be incrementally diminished by energy dissipation processes (i.e. entropyproduction), which are represented by a damping parameter ζ.

Consider the general second-order differential equation describing such aresonant system for a degree-of-freedom x:

x+ 2ζωnatx+ ω2natx = 0

where ζ is the damping parameter and ωnat is the natural resonance fre-quency of the system in radians per second. Such an equation can be derivedusing, for example, Newton’s Second Law, Kirchoff’s Voltage Law, or, mostgenerally, the Euler-Lagrange equation of classical field theory.

Recasting the equation into matrix form using x = v yields the following:

A =

(0 1

−ω2nat −2ζωnat

)

and

x =

(xv

)so:

(xv

)=

(0 1

−ω2nat −2ζωnat

)(xv

)

The non-zero damping and complex eigenvalues of the system yield a stablespiral at (0,0), as shown below:

18

As t → ∞, the system eventually comes to rest at (0,0) due to the energydissipation represented by the damping.

Removing the damping, the ideal harmonic oscillator is obtained:

x+ ω2natx = 0

(xv

)=

(0 1

−ω2nat 0

)(xv

)

which has a center about (0,0), and oscillates forever in the absence of dis-sipation, as shown below:

19

Note that the trajectories about the center are described by ellipses of theform:

ω2natx

2 + v2 = C

which is the same thing as saying that the system Hamiltonian ( the sumof potential and kinetic energies) is constant. In other words, it expresses con-servation of energy.

4.8.2 Relationship Dynamics: Strogatz’s Romeo and Juliet

Steve Strogatz published a whimsical and insightful model of male-female courtshipusing a 2-D linear system of differential equations [1]. Consider the followingquantities:

• R(t) - Romeo’s love for Juliet at time t

20

• J(t) - Juliet’s love for Romeo at time t

Now consider the system parameterized by a, b, c, and d, which variouslydescribe the coupling and growth/decay rates of their respective interests in thecourtship.

(R

J

)=

(a bc d

)(RJ

)

Consider the case of equal caution between the lovers:

A =

(a bb a

)

where a < 0 is a measure of cautiousness and b > 0 a measure of respon-siveness. Consider the general situations that arise, depending on the relativemagnitudes of a and b, as outlined by the first two cases of the below table (anextra, separate situation of Love-Hate is added for completeness):

21

Situation Parameters Equilibria Phase DiagramMutual Indiffer-ence: The rela-tionship eventuallyfades due a mu-tual indifference ef-fect. Caution leadsto apathy.

A =

(a bb a

)• a2 > b2

• a < 0

• b > 0

Stable node at (0,0)

Explosive: De-pending on theirinitial feelings,the relationshipbecomes deeplyaffectionate (1stquadrant) or deeplyhateful (3rd quad-rant). In bothcases, the feelingsare eventuallyextremely mutual(the solutions ap-proach the limits ofline R = J).

A =

(a bb a

)• a2 < b2

• a < 0

• b > 0

Saddle node at(0,0)

Love-Hate: Anoscillation betweenlove and hate, withone growing to hatethe other, the otherreciprocating, thenthe former growingto love the other,the other recipro-cating, and so on.

The matrix

A =

(0 a−b 0

) Center at (0,0)

22

Situation Parameters Equilibria Phase DiagramStale Romeo:Romeo doesn’tchange at all.Case 1: Depend-ing on where Julietbegins relative toRomeo (i.e. aboveor below the lineR = J) , she growsinfinitely affection-ate or infinitelyhateful.Case 2: Julieteventually adoptsthe same feeling asRomeo (i.e. goes tothe line R = J)

A =

(0 0a b

)Case 1:

• a < 0,

• b > 0

Case 2:

• a > 0

• b < 0

Line of unstable(case 1) or sta-ble (case 2) fixedpoints on R = J

Case 1 Phase Diagram

Opposites At-tract: If theirinitial feelings arethe same, theyremain in thatstate forever. Forunequal initialfeelings, two casesemerge, dependingon the coefficients.The partner withdecaying intrinsiclove (cautious-ness) and growingresponsiveness(Romeo in case1) will require theother partner tohave mutual or bet-ter feelings for therelationship to gowell (1st quadrant).Otherwise it willdescend into hate(3rd quadrant).

A =

(a b−b −a

)Case 1:

• a < 0,

• b > 0

Case 2:

• a > 0

• b < 0

Line of stable nodeson R = J

Case 1 Phase Diagram

23

Situation Parameters Equilibria Phase DiagramSame Styles: Ifinitial feelings areboth positive orboth negative, amutual result ofthe same sign willeventually obtain.If otherwise, itwill depend on therelative magnitudeof the respectivefeelings. All solu-tions converge toR = J

The matrix

A =

(a bb a

)• a < 0,

• b > 0

Line of stable nodeson R = J

Out of Touch:The partners haveno coupling to theirown respectivestates (i.e. areneither cautiousnor impetuous),but are responsiveto the state ofthe other partner.Two cases emerge:Case 1: Oppositecoupling (a la thecyclical sex gamesof Dawkins 1976,see [3] p.130 andAppendix J, pp.200-202), whichcauses oscillations.Case 2: Couplingsof the same sign,which lead to eithermutual indifference(at (0,0)), extremehate (along R = Jto 3rd quadrant),or extreme affec-tion (along R = Jto 1st quadrant)

The matrix

A =

(0 ab 0

)Case 1:

• a > 0,

• b < 0

Case 2:

• a > 0

• b > 0

Case 1: Center at(0,0)Case 2: Saddlepoint at (0,0), sta-ble manifold alongR = −J

Case 1 Phase Diagram

Case 2 Phase Diagram

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5 Nonlinear Systems on the Phase Plane

5.1 Nonlinear Systems

Consider 2-D systems:

x1 = f1(x1, x2)

x2 = f2(x1, x2)

where fi(x1, x2) can be nonlinear.

Written in vector notation:

x(t) = f(x)

5.2 Existence, Uniqueness, and Topology

Theorem: Consider the initial value problem x = f(x), x(0) = x0. If f is con-tinuous and all of its partial derivatives δfi/δxj for i, j = 1, .., n are continuousfor x on some open connected set D ⊂ Rn, then for x0 ∈ D, the initial valueproblem has a solution x(t) on some interval (−τ, τ) about t = 0. Furthermore,the solution is unique.

Corollary: Within D, trajectories never intersect.

The effect of the corollary is that if a trajectory is an orbit, it will neverescape it, and orbits contained within an orbit will never veer into the outerorbit.

5.3 Linearization and the Jacobian

Linearization about fixed points in a 2-D nonlinear system allows a local phasediagram to be constructed using methods identical to those of 2-D linear sys-tems.

Consider the system:

x = f(x, y)

y = g(x, y)

with (x∗, y∗) a fixed point. Consider now the quantities:

25

u = x− x∗, v = y − y∗

which represent a perturbation about the fixed point. Forming differentialequations for these quantities proceeds as follows in this example for u:

u = x

since x∗ is constant. Using substitution:

u = f(x∗ + u, y∗ + v)

A Taylor series expansion about (u = 0, v = 0) yields:

u = f(x∗, y∗) + u∂f∂x |(x∗,y∗) + v ∂f∂y |(x∗,y∗) +O(u2, v2, uv)

and since f(x∗, y∗) = 0 becomes the following:

u = u∂f∂x |(x∗,y∗) + v ∂f∂y |(x∗,y∗) +O(u2, v2, uv)

the same expression for v is :

v = u ∂g∂x |(x∗,y∗) + v ∂g∂y |(x∗,y∗) +O(u2, v2, uv)

The system for u and v can thus be expressed as:

(uv

)=

(∂f∂x

∂f∂y

∂g∂x

∂g∂y

)(x∗,y∗)

(uv

)+ quadratic term vectors

The matrix:

A =

(∂f∂x

∂f∂y

∂g∂x

∂g∂y

)(x∗,y∗)

is the Jacobian and represents the local linear dynamics about the fixedpoint (x∗, y∗). It must be numerically evaluated for each fixed point one wishesto analyze.

The Jacobian by itself is sufficient for the local analysis in most cases. Thequadratic terms become important in the local analysis for borderline casesincluding centers, stars, degenerate nodes, and non-isolated fixed points.

5.4 Stability of Fixed Points with High-Order NonlinearTerms

The stability of fixed points in a nonlinear dynamical system can be altered bythe presence of the nonlinear terms outside the Jacobian. Whether or not thesenonlinear terms have an effect on stability of the fixed point can be determined

26

by looking at the eigenvalues of the fixed point. Robust cases, in which thestability of the fixed point is influenced by nonlinear terms, can be differentiatedfrom marginal cases, in which the nonlinear terms must be considered.

Robust (Hyperbolic) Cases Marginal Cases

• Repellers (Sources):Re(λi) > 0 for all i

• Attractors (Sinks):Re(λi) < 0 for all i

• Saddles: Re(λi) < 0,Re(λj) > 0 , i 6= j

• Centers: Re(λi) = 0 forall i

• Higher-order and non-isolated fixed points: Atleast one λi such that λi =0

The robust cases for which Re(λi) 6= 0 are referred to as hyperbolic fixedpoints. The eigenvalue condition for hyperbolic fixed points is equivalent to thecondition f ′(x∗) 6= 0, which allows the following theorem:

Theorem in brief : (Hartman-Grobman Theorem) There exists a homoe-morphism (topological equivalence) between the phase portrait near a hyper-bolic fixed point and its linearization.

This motivates a further (informal) definition:

Def.: A fixed point is said to be structurally stable if its topology cannotbe changed by an arbitrarily small perturbation to its local vector field.

5.5 Lotka-Volterra Systems: Predator-Prey Dynamics

The Lotka-Volterra system has been used extensively to describe predator-preydynamics. It is a system of the form:

x = ax− bxy

y = −cy + dbxy

with the following definitions:

• x(t) - Population of prey at time t

• y(t) - Population of predators at time t

• a - the intrinsic population growth rate of the prey

• b - the population decay rate of prey due to predation

• c - the intrinsic population decay rate of the predators

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• d - the ratio of prey to predators required for a stable predator population(i.e. how many prey it takes to sustain a single predator)

The fixed points for the system are:

x∗ =

(00

),

(c/(db)a/b

)

with the the non-zero fixed point being a center. The phase diagram withseveral trajectories is shown below:

The oblong shape of the center trajectories is due to the d parameter be-ing greater than 1. The major axis of those trajectories is in the direction ofthe x- axis, indicating that the population of prey required to support a givenpopulation of predators is greater than that given predator population.

The direction of the center trajectories is counter-clockwise, indicating anovershoot of prey population will create a swell in the predator population. Theswell in predator population will subsequently prove unsustainable due to adwindling prey population, bringing the state closer to the fixed point at (0,0).However, due to a strong intrinsic growth rate of the prey population (i.e. thea parameter), the state swings back in the direction on the positive x-axis, andthe cycle continues.

The fixed point at (0,0) is a saddle point, with the y-axis as its stable man-ifold and the x-axis as its unstable manifold.

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5.6 Conservative Systems

Consider a force F (x) which is a nonlinear function of x. F (x) is assumed tohave no dependence on time t or x. The lack of a dependence on x implies thatthere is no dissipative component of F (x).

The motion of a particle in this system is given by Newton’s 2nd Law:

mx = F (x)

Using the previous definition of the potential:

dV

dx= −F (x)

the 2nd Law can be rewritten as:

mx+dV

dx= 0

Multiplying both sides by x gives:

mxx+dV

dxx = 0

which is the same as (via two reverse applications of the chain rule):

d

dt[1

2mx2 + V (x)] = 0

Integrating this equation gives:

E =1

2mx2 + V (x)

which is the Hamiltonian for the system. The form of the above equationfor a conservative system defines an ellipse in the phase space for a given E.

Def.: A conserved quantity E(x) for a system x = f(x) is a real-valuedfunction that is:

• Constant on trajectories (i.e. dEdt = 0)

• Non-constant on every open set.

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A intuitive consequence of this definition is that a conservative systemcannot have any attracting fixed points.

An example of such a nonlinear conservative system is the following:

Given the potential (here mass m = 1)

V (x) = − 12x

2 + 14x

4

the force can be found via:

−dVdx = x− x3

which gives the dynamics:

x = x− x3

This can be rewritten in system form as:

x = y

y = x− x3

The fixed points for the system are:

x∗ =

(00

),

(10

),

(−10

)

with the first fixed point being a sadle node, followed by two centers.

The phase diagram is shown below:

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Trajectories that begin and end at the same fixed point are called homoclinicorbits, which are often found in conservative systems. heterooclinic orbits con-nect two different fixed points. In the case of the above system, the homoclinicorbits are the ones starting from the fixed point (0, 0). Homoclinic orbits lackperiodicity, for they only return to their starting point as t→ ±∞.

5.6.1 Energy Surfaces

The Hamiltonian (total energy) of each point in the phase space can be com-puted as:

E =1

2mx2 + V (x)

Plotting the energy E along with the phase space variables x and x, anenergy surface can be produced, as shown below:

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The centers of the system represent local minima of the energy surface. Thesaddle point of the phase space is exactly that for the energy surface.

5.6.2 Centers in Conservative Systems

The following theorem effectively states that in second-order conservative sys-tems (linear or nonlinear), centers occur at the minima of the energy surface.

Theorem: Consider the system x = f(x), where x = (x, y) ∈ R2, andf continuous differentiable. Suppose there exists a conserved quantity E(x)and suppose that x∗ is an isolated fixed point. If x∗ is a local of E, then alltrajectories sufficiently close to x∗ are closed.

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5.7 Reversible Systems

Reversible systems express the property of time reversal symmetry. Theycan be defined as follows:

Def : A reversible system is any second-order system that is invariantunder t→ −t and y → −y.

Graphically, reversible systems will be symmetric about the x-axis in phasespace, with the time direction of the trajectories reversed after the reflectiontransformation about the x-axis to complete the symmetry. This leads to amore general definition of reversible systems for higher-order systems:

Def : A reversible system is a system x = f(x) that is invariant undert→ −t and x→ R(x), where R(x) is map such that R2(x) = x.

As with conservative systems (which are not the same as reversible systems),centers have special properties in reversible systems:

Theorem: Suppose the origin x∗ = 0 is a linear center for the continuouslydifferentiable and reversible system

x = f(x, y)

y = g(x, y)

Then all trajectories sufficiently close to the origin are closed trajectories.

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p.1176, Section 12.1

“In general, if a map or flow contracts volumes in phase space, it is calleddissipative. Dissipative systems commonly arise as models of physical situa-tions involving friction, viscosity, or some other process that dissipates energy.In contrast, area-preserving maps are associated with conservative systems, par-ticularly with the Hamiltonian systems of classical mechanics.

A Hopf Decompositions

Rigorously defining conservative and dissipative dynamical systems requires un-derstanding the machinery behind Hopf decompositions and its associated re-currence theorem. Strogatz’s statement about the preservation of phase spacevolumes by conservative systems is based on the notion of incompressibility.

More information can be found at [4]

References

[1] S. Strogatz, Nonlinear Dynamics and Chaos, 2nd. Ed., Westview Press,Philadelphia, PA, 2015.

[2] T. Lancaster and S.J. Blundell, Quantum Field Theory for the Gifted Ama-teur, Oxford University Press, Cambridge, UK, 2014.

[3] J.M Smith, Evolution and the Theory of Games, Cambridge UniversityPress, Oxford, UK, 1982.

[4] Hopf Decomposition,https://en.wikipedia.org/wiki/Hopf decomposition

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