1. Closed invariant curves and their bifurcations in two-dimensional maps
Tutorial Workshop DDSA
Discrete Dynamical Systems and Applicatons
Urbino, June 30-July 3 March 2010
Closed invariant curves and their bifurcations in two-dimensional maps
Anna Agliari
Catholic University
Via Emilia Parmense, 84.
29100 Piacenza (Italy)

2. SADDLE CONNECTIONS and
Aim of the Lecture
To show some global bifurcations related to
the appearance/disappearance of closed invariant curves;
the interaction between coexisting cycles and closed invariant curves;
SADDLE CONNECTIONS and
HOMOCLINIC TANGLES

3. CONTENTS INVARIANT SETS HOMOCLINIC AND HETEROCLINIC ORBITS
INVARIANT CLOSED CURVES
GLOBAL BIFURCATIONS OF INVARIANT CURVES
A Propedeutic Example
A Financial Market Model
A Business Cycle Model

4. S is an invariant set for the map G if
Let G be a map, defined in IR2
S is an invariant set for the map G if
Examples:
{p*} where G(p*) = p*
{p1, p2, …,pk} where G(pi)=pi+1, i=1,…,k-1 and G(pk)=p1
Attracting sets and attractors
Stable and unstable manifolds of a saddle point (or cycle), if G is invertible

5. Attracting Set
A closed invariant set A is an attracting set if there exists some neighborhood U of A such that
The set
is the basin of attraction of A
An attractor is an attracting set which contains a dense orbit.

6. Repelling Set
A closed invariant set A is a repelling set if however close to A there are points whose trajectories go away from A, that is, if there exists some neighborhood U of A such that
A repellor is a repelling set which contains a dense orbit.

7. w-Limit and a-Limit Sets
A point p is a w-limit point of x if there are points
on the forward orbit of x such that
A point p is a a-limit point of x if there are points
on the forward orbit of x such that
The a- (resp. w-) limit sets a(x), w(x) are the sets of a and w limit points of x, i.e. the sets of accumulation points of G-n(x) and Gn(x), as n goes to infinity.

8. Local Stable and Unstable Manifolds for saddle points
Let p* a saddle point for the map G, U a neighborhood of p*, ES the eigenspace associated with l1, with |l1|<1, EU the eigenspace associated with l2, with |l2|>1
The local stable manifold of p* in U is the set
The local unstable manifold of p* in U is the set
The local stable manifold is tangent to ES at p*
The local unstable manifold is tangent to EU at p*
They are as smooth as the map G
Stable Manifold Theorem

9. Stable and Unstable Sets
The (global) stable set of p* is the set
If G is a diffeomorphism,
it is an invariant set
it is a manifold
If G is an endomorphism,
it may be strictly mapped into itself, and backward invariant
it may be not connected U
The (global) unstable set of p* is the set
If G is a diffeomorphism,
it is an invariant set
it is a manifold
If G is an endomorphism,
it is invariant, but it may be not backward invariant
self intersections are allowed

10. Homoclinic Orbit
Let G be a diffeomorphism in IR2 and S a saddle point
The point q is said homoclinic to the saddle point S iff s
if a homoclinic point exists, then there must be infinitely many of such points
in the neighborhood of S, the stable and unstable sets have a tangle
The homoclinic orbit associated with q is the set
where and

11. Transversal intersection
Homoclinic Tangle
Generally, homoclinic bifurcations for maps occur via homoclinic tangles, in a certain parameter range. S
Quadratic tangencies of the manifolds at discrete points
homoclinic orbits appear S
Transversal intersection S
Quadratic tangencies of the manifolds at discrete points
homoclinic orbits disappear

12. During the homoclinic tangles...
In the parameter range in which the manifolds intersect transversally
A chaotic repellor exists, made up of infinetely many (countable) repelling cycles and uncountable aperiodic trajectories
Such a chaotic repellor appears at the first homoclinic tangency and disappears at the second one
Smale-Birkhoff Theorem

13. Heteroclinic Orbit The heteroclinic orbit connecting S2
Let S1 and S2 be two saddle points for the map G
A point q belonging to a neighborhood U(S2) is said heteroclinic from S2 to S1 if as n increases
and q belongs to the local unstable set of S2 in U(S2), i.e.
The heteroclinic orbit connecting S2
to S1 associated with q is the set
where
and

14. Invariant Closed Curve
In a discrete map, a closed curve is an invariant set (i.e., a set mapped into itself) of the phase-space corresponding to quasi-periodic or periodic trajectories.
P* becomes unstable, merging with a repelling closed curve GU, existing when P* is still stable subcritical type
P* becomes unstable and an attracting closed curve GS appears, surrounding it supercritical type
Neimark-Sacker bifurcation
When the eigenvalues of the Jacobian matrix evaluated at the fixed point P* are complex and cross the unit circle then:

15. Dynamics on an Invariant Closed Curve
On an invariant closed curve G:
dense quasiperiodic orbits
a finite number of periodic orbits, saddles and nodes.
Rotation number r:
the average number of turns of a trajectory around the fixed point
Rational number q/p
p period of the cycle,
q number of turns
Irrational number
each point on G gives rise to a trajectory which never comes on the same point, and the closure of the trajectory is exactly G
Arnold Tongues

16. Arnold Tongues
inside the Arnold tongues the rotation number is rational
m1(a)
schematic
SN bifurcations
bifurcation point (a=ao)
Infinitely many tongues, of thickness  d (q-2)/2
(d is the distance from the unit circle)

17. Saddle-Node Connection
Frequency locking:
Two cycles appear via SN bifurcation
The invariant closed curve is given
by a saddle-node connection
The cycles disappear via SN bifurcation.

18. Saddle-Focus Connection
The unstable set of a saddle cycle connects the periodic points of a focus cycle
It is not homeomorphic to a circle
Repelling Invariant Closed Curves
The stable set of a saddle cycle issues from the periodic points of a repelling cycle

19. Saddle Connections Structurally unstable situations
The mechanisms associated with the appearance and disappearance of closed invariant curves we shall see involve saddle-connections: a branch of the stable set of a saddle point (o cycle) merges with a branch of the unstable one, forming an invariant closed curve.
Homoclinic loop
Heteroclinic loop
Double homoclinic loop
Structurally unstable situations
Global bifurcations
Dealing with maps, they are replaced by homoclinic tangles

20. A Propedeutic example
where -1 < a < 1
k > 0
b, c, d real numbers
T is a family of two-dimensional nonlinear endomorphisms
T depends on 5 parameters: we shall consider the sub-family
E* = (0,0) is a fixed point of T

21. Local Stability of E* When and a Neimark-Sacker bifurcation occurs.
The Neimark-Sacker bifurcation is of subcritical type if
Subcritical
Neimark bifurcation
a = : 3 Resonant case
a = : 4 Resonant case

22. Just after the NS bifurcation…
The attractor is a closed curve, but far from the fixed point E*
Two invariant closed curves, one attracting and one repelling appear when E* is stable. The repelling one decreases in size, merging with E* at the bifurcation value, leaving the attracting one as the unique attractor.

23. Corridor Stability
The system converges to its dynamic equilibrium for small perturbations, but shows no such tendency for larger shocks. Indeed, due to the existence of a repelling curve which bounds the basin of attraction of the stable fixed point, small shocks of the system have no effects on its dynamical behaviour, but large enough shocks may lead to an aperiodic (or periodic with large period) fluctuations or to an unfeasible trajectory.

24. Hysteresis Effect
Consider the system close to its equilibrium, for k < kN. As k passes the critical value kN, the trajectories of the system converge to a large closed invariant curve. The loss of stability in such a bifurcation thus recall a scenario of catastrophe. Moreover, if the parameter k is decreased again, the system does not return to its previous equilibrium but rests in steady oscillation.
6.185
6.205
6.185

25. Two Parameter Bifurcation Diagram7 7 4 5
Bifurcation
curve
Bifurcation
curve -1 -1 1 1
Subcritical NS bifurcation in (− , )
Periodicity regions exist even if the Neimark bifurcation is of subcritical type.
But they open below the bifurcation curve, i.e. when E* is stable
COEXISTENCE OF ATTRACTORS

26. Square Separate Map
-0.02
0.03
0.015
0.02
-0.05
-0.05
0.05

27. Saddle-Node Bifurcation for closed curves
At the bifurcation
“Saddle-node” bifurcation for closed invariant curves

28. Cycles of Period 5
0.27
0.31
5.4 6
0.03
Bifurcation
path
-0.05
-0.05
0.05
A saddle-node bifurcation has occurred: two cycles of period 5 exist

29. Appearance of Closed Curves
0.015
Appearance of Closed Curves
-0.02
-0.02
0.03

30. Enlargement in

31. Saddle Connection attracting cycle
A focus cycle and a saddle cycle exist
At the bifurcation:
The merging of the a1 and w1 branches of two distinct periodic points gives a saddle connection
A repelling closed curve and a saddle-focus connection exist

32. Saddle Connection repelling cycle
A focus cycle and a saddle cycle exist
At the bifurcation:
The merging of the a1 and w1 branches of two distinct periodic points gives a saddle connection
An attracting closed curve and a
saddle-focus connection exist

33. Summing up … Repelling node cycle (a) (b) (c)
If the cycle C, involved in the global bifurcation with the saddle S, is
attracting then the closed curve appearing after the first step is repelling,
together with an attracting saddle-connection.
If the cycle C, involved in the global bifurcation with the saddle S, is
repelling then the closed curve appearing after the first step is attracting,
together with a repelling saddle-connection.
Repelling node cycle
(a)
(b)
(c)

34. Transversal intersection
At the Bifurcation
Quadratic tangencies of the manifolds
homoclinic orbits appear
Homoclinic tangle
Transversal intersection
Quadratic tangencies of the manifolds
homoclinic orbits disappear

35. 0.02
Period 4 Cycles
-0.02
0.02
0.015
-0.02
-0.02
0.015
0.015
At the bifurcation, the two invariant curves no longer merge
-0.02
-0.02
0.02

36. Saddle-Node Bifurcation
Only the stable set of a saddle cycle is involved in the bifurcation

37. A financial market with imitators
The model describes a single asset stock market where
n is the number of traders
(i=1,...,n) is the investment attitude of agent i s.t.
if agent i wants to buy an asset’s unit at t
(optimistic investor)
if agent i wants to sell an asset’s unit at t
(pessimistic investor)
The investors’ average opinion is, then,
Let nb and ns be the totality of buyers and sellers with
nb + ns= n
Thus, we get

38. Transition Rates
In a unit time period, the variations
in the buyers’ group are nsgbs - nbgsb
in the sellers’ group are nbgsb-nsgbs
Transition rate from buyers to sellers
Transition rate from sellers to buyers is
b is the strenght of the fundamentalist reaction (difference between the market price and the fundamental value k)
g is the imitation effect
n is the speed of opinion change
m measures the remaining factors affecting the asset price in traders opinion

39. The Model
The evolution over time of the average investment attitude can be expressed as
The adjustment price mechanism is
where
is the speed of adjustment
is the volume of trading per agent.

40. The Map
The dynamics of both the average investement attitude xt and the asset
price pt are described by the following two dimensional nonlinear map
By means of an axis translation we can express the price pt as the deviation from the fundamental value k , obtaining

41. Local Stability Region
Parameter plane (g – m – 1, n)
Neimark - Sacker
bifurcation curve
Flip bifurcation curve n
g -m-1
Bifurcation
path
Fixed point
P*=(0,0)
The Neimark-Sacker bifurcation may be either of super- or subcritical type

42. Noninvertible Map Riemann foliation LC-1a LCb P* LCa LCa Z3 LC-1b Z1x
Riemann foliation

43. l = 0.5 n = 0.23 m = 1.25 b = 1 q = 6
Appearance of the repelling closed curve involved in the subcritical N-S bifurcation
g = 2.22 F5 F6 F4 P* F3 F1 Gu F5 F2 F6 F4 P* F1
B() F3 F2
An attracting focus cycle of period 6 exists as well as a saddle cycle of the same period
g =

44. Saddle connection After the bifurcation Unstable set Stable setP* Gu S2 S3 S4 S6 S1 S5
g = 2.22
Unstable set
g = F2 F3 F4 F5 F6 F1 P* S2 S3 S4 S6 S1 S5 Gu
Stable set
The unstable set of the saddle cycle reaches the two attractors, the stable one comes from the boundary of the set bounded trajectories.

45. Homoclinic tangle S1
At the closure of the homoclinic tangle the behavior of the two branches is modified S1
g =
g = S1
The boundaries are well defined, but some oscillations are clearly visible
g =
The crossing of the branches a2 and w1

46. Business Cycle Model Kt is the capital stock
Discrete version of the Kaldor model:
where Yt is the income (or output) level
Kt is the capital stock
variations over time in Yt depend on the firm reaction to the difference between investment demand and saving (the parameter a > 0 measures such a reaction)
the depreciated capital stock is increased by the gross investment It = It(Yt,Kt) ; the parameter d (0 < d < 1) is the depreciation rate)

47. Consumption where Y* denotes the equilibrium income
Ct is an increasing function of Yt
c0 - c1 < Ct < c0 + c1
For large value of income, Ct remains nearly constant
Around Y*, Ct quickly increases at a rate close to c2
Ct(Y*) = c0
c2 (0 < c2 < 1) is the maximum consumption propensity

48. Investment Gross investment: where is the desired stock of capital:
and k > 0 is the capital-output ratio (exogenous)
The net investment is proportional to the difference between the long run (or desired) capital stock and the current one. The parameter b > 0 is the stock adjustment parameter.

49. The Map where x = Kt - kY* y = Yt - Y*
Substitute the consumption and investment functions in the model
Change the variables in order to have (0,0) as a fixed point
where x = Kt - kY*
y = Yt - Y*
and the symbol ’ denotes the unit advancement operator.

50. Properties of T
It depends on 6 parameters: a, d, k, b, c1, c2 : in our analysis
It is symmetric with respect to the origin
It is invertible if
holds. Otherwise it is a Z1-Z3-Z1 map.
In our analysis we shall consider a parameter range in which T is invertible.

51. Fixed Points If the map T has a unique fixed point
If two further fixed points exist, P* and Q*, symmetric with respect to E*
They are such that x = ky and y can be obtained from the equation

52. Stability regions of E*
Bifurcation path
Stability regions of E*

53. Stable and unstable manifolds of E*
E* Saddle x y
P* and Q* attracting foci
Stable and unstable manifolds of E*

54. The stable manifold of E* has many and many convolutionsx y
The stable manifold of E* has many and many convolutions x y

55. Two invariant closed curves appear . . .x y y
one stable, with a wide basin of attraction
one unstable, from which the stable manifold of E* issues

56. x y
Trajectories t x

57. The repelling closed curve shrinks in the proximity of the saddle E*
In the proximity of E*, the invariant manifolds show some oscillations

58. Homoclinic Bifurcation of E*x y
Homoclinic Bifurcation of E* x y
A homoclinic tangle occurs

59. Transversal crossing in the rangex y
Transversal crossing in the range x y
The stable manifold is located at opposite side with respect to the unstable one

60. Homoclinic Bifurcation of a Saddle double homoclinic loop
Two disjoint closed invariant curves
constitute the basins of Q1 and Q2.
WS (O ) constitutes the boundary
that separates the basins
of Q1 and Q2.
Homoclinic tangle

61. Unstable Manifold of E*x y
Unstable Manifold of E* x y x y

62. Two repelling closed curves existy
Two repelling closed curves exist x
The two repelling closed curves shrink, merging with the stable fixed points at a subcritical Neimark-Hopf bifurcation t

63. Interaction between closed curve and coexisting cycles
The stable and unstable manifolds of the period 8 saddle cycle

64. y x
A new homoclinic tangle is coming, involving the inner branches of the invariant sets

65. Close to the first tangential contacty x
Close to the first tangential contact x y
the attracting closed curve exhibits many oscillations

66. The homoclinic tangle developsy x
The homoclinic tangle develops y x

67. a strange repellor existsy
In the parameter range of the homoclinic tangle
a strange repellor exists

68. x t t x

69. From an Attracting Closed Curve to a Heteroclinic Connection
Two coexisting attractors: a focus-cycle and a closed curve
At the bifurcation:
Only one attractor survives;
the invariant closed curve is replaced by the saddle-focus connection.
The merging of the inner branches of the manifolds gives a saddle connection

70. At the Bifurcation:

71. x y
The invariant curve comes into resonance with the cycle, forming an attracting set of the saddles and foci, with the foci the attractor within it. y x
An invariant closed curve still exists, given by the saddle-focus connection

72. The outer branches of the saddle cycle approach each otherx
The outer branches of the saddle cycle approach each other

73. From a Heteroclinic Connection to a Larger Closed Curve
At the bifurcation:
An attracting focus-cycle and a saddle-focus connection.
An attracting closed curve appears, surrounding the focus cycle.
The merging of the outer branches of the manifolds gives a saddle connection

74. x y x y x y

75. At the Bifurcation:

76. x t t x y x

77. A new attractor appears, during the transversal crossingx t y x t y
A new attractor appears, during the transversal crossing

78. y x y x
At the closure of the homoclinic tangle, an attracting closed curve surrounds the periodic points of the cycles

79. Again the Market Model
g = F2 F3 F4 F5 F6 F1 P* Gu
After the appearance of the repelling closed curve and before the occurrence of the subcritical NS
a more complex situation can be detected…

80. A 6-curves cyclical attractorP*
The unstable set connects the attracting closed curves
Due to a supercritical NS bifurcation of the focus cycle P* Gu P*
The stable set separates the basins of attraction of the 6 invariant closed curves
g =

81. Appearance of an Attracting Closed CurveGu Gs
The attracting closed curve surrounds the fixed point and the 6-curves cyclical attractor

82. Saddle Connection 6-curve cyclical attractorG1 G2 G4 G5 G6 G3 S2 S1 Gu
At the bifurcation:
A cyclical attractor connected to a saddle cycle. Gu G1 G2 G4 G5 G6 G3 S2 S1 S3 S4 S5 S6 Gs Gu G2 G1 S2 S1
An attracting closed curve appears, surrounding the cyclical attractor.
The merging of the outer branches of the invariant sets gives a saddle connection

83. Homoclinic tangle Transversal crossing g = 2.2428 g = 2.2429

84. Disappearance of the 6-curves cyclical Attractor
g =
g = Gu G1 G2 G4 G5 G6 G3 P* Gs Gu P*
Intermingled basins
Basin of attraction of the cyclical attractor made up by the stable set of the saddle cycle

85. Conjecture
Homoclinic loop made up by the inner branches of the invariant sets
A 6-curves cyclical repellor appears G2 S2 S1 Gs Gu Gs Gu S1 Gs Gu
After their merging, both the cyclical attractors and the cyclical repellor disappear F1 F2 S1

86. Main References … and references therein
Global bifurcations of invariant closed curves
- AA, G. I. Bischi, R. Dieci, L. Gardini (2005) "Global bifurcations of closed invariant curves in two- dimensional maps: A computer assisted study" International Journal of Bifurcations and Chaos, vol. 15 (4), pp
AA, G.I. Bischi, R. Dieci, L. Gardini (2006) "Homoclinic tangles associated with closed invariant curves in families of 2D maps" Grazer Math. Ber. N. 350, pp. 1-14
Appearance/disappearance of invariant closed curves
- AA (2007), On the bifurcation mechanisms causing the appearance of invariant closed curves, Grazer Math. Ber., N.351, pp. 1-20
- AA, Gardini, Puu (2005), Some global bifurcations related to the appearance of closed invariant curves, Computers and Mathematics in Simulation 68, pp
AA (2006) Homoclinic connections and subcritical Neimark bifurcations in a duopoly model with adaptively adjusted productions, Chaos Solitons & Fractals 29, pp
I. Foroni, AA (2010) “Complex dynamics associated with the appearance/disappearance of invariant closed curves” Submitted
Interactions between coexisting closed curves and cycles
- AA, Dieci R. (2006), Coexistence of attractors and homoclinic loops in a Kaldor-like business cycle model, In: T. Puu and I Sushko (eds.): Business cycle dynamics: Models and tools. Springer-Verlag, pp
AA, Dieci R., Gardini L. (2007), Homoclinic tangle in Kaldor's like business cycle models, Journal of Economic Behavior and Organization 62, pp
… and references therein