All Questions
Tagged with nonlinear-dynamics bifurcation
21
questions
0
votes
0
answers
25
views
Logistic map: bifurcation and domain of attraction
Let $f(x) = \mu x(1-x)$ be the logistic map, the question is divided into 3 parts:
Part (1): what can you say about the domain of attraction of the 2-cycle in $3<\mu<1+\sqrt 6$?
My attempt: let $...
- 305
0
votes
0
answers
51
views
nonlinear odes: stabilizing terms in a subcritical pitchfork bifurcation
I am reading through Strogatz's book on nonlinear odes and dynamical systems. One thing that is a little confusing is his description of stabilizing higher order terms to control the dynamics of a ...
- 2,531
0
votes
1
answer
101
views
Dynamic system, bifurcation and equilibrium points
I'm a bit stuck in solving the following exercise: consider the dynamical system
$$\dot{x} = 2 + 3\mu x - x^3$$
I have to find the equilibrium points and the stability type, and the the bifurcations ...
- 3,482
0
votes
1
answer
111
views
Deriving a Hopf Bifurcation – Perturbation Method vs. Jacobian Matrix
When deriving a Hopf bifurcation of a dynamical system, the usual process is:
Find a fixed point $(x_0, y_0)$
Perturb the system about the fixed point $(x_0+\tilde{x}, y_0+\tilde{y})$
Linearize, ...
1
vote
0
answers
52
views
Calculating Bifurcation points of trig equations
I am going through Nonlinear Dynamics and Chaos by Strogatz and I am getting stuck on a few questions:
To calculate the stability of a fixed point:
Work on where it intersects the x axis ( fixed ...
- 79
1
vote
0
answers
118
views
long term behavior of solution to a first order nonlinear differential equation
Consider the following system of first order nonlinear autonomous ODEs (derivatives are taken with respect to $t$):
$$
\begin{cases}
\dot{x} = -2xy^2+1 \\
\dot{y} = -2x^2y+1 \\
x(0) = x_0,\;y(0)=y_0
\...
- 997
4
votes
1
answer
209
views
2D bifurcation problem
I come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you ...
- 1,575
1
vote
0
answers
133
views
Analysis of Time-Varying Differential Equation
I'm trying to analyze the a differential equation of the form:
$$
\dot{x} = -K_1\sin(\omega t + x) - K_2\sin(x)
$$
where $K_1 > 0$, $K_2 > 0$, and $\omega >0$ are positive real constants and $...
- 83
2
votes
0
answers
39
views
Show $h$ and $g$ are commutative in the canonical form of ODE.
I come a cross this problem in my nonlinear analysis course. I know how to find the normal forms of any order. However, the commutative isometry! And in the third point the professor put two ...
- 1,575
2
votes
0
answers
86
views
I want to understand the characteristics of a particular non-linear hierarchical dynamical system
I would like to study the characteristics of the non-linear dynamical system detailed below; in particular, I would like to find its set of fixed points; and to compute (i) the maximum values of ...
- 1,013
1
vote
0
answers
69
views
Limit cycle continuation folding over itself
I was doing some computations in MATCONT. First I continued equlibrium point. Hopf bifurcation occured for $a = 6$ and branching point/pitchfork for $a = -2$. Continuing forward and backward this ...
- 41
0
votes
1
answer
284
views
How to know when a transcritical bifurcation occurs (Example 3.2.1 Strogaz Nonlinear Dynamics and chaos)
Picture of Question + solution
Hi,
In this question After we expand $\dot{x}$ around $x^*$ = 0 , we get $\dot{x}$ = (1-ab)x + ($\frac{1}{2}$a$b^2$)$x^2$ + O($x^3$).
How do we jump from this to knowing ...
3
votes
1
answer
180
views
Dynamical System that exhibits a fold bifurcation of Limit Tori?
Fold Bifurcations of a fixed points (i.e. saddle node bifurcations) and Fold bifurcations of limit cycles (i.e. when a stable limit and unstable limit cycle annihilate) are observed in plenty of ...
- 213
1
vote
1
answer
171
views
How to calculate fixed points and plot bifurcation diagram for non-linear ODE system
I am trying to understand how to analyse a system of coupled, non-linear ODEs taken from this paper. I want to perform a fixed point analysis and plot a bifurcation diagram to show how fixed points ...
- 29
2
votes
1
answer
268
views
Forced Duffing equation $\ddot x +x+\varepsilon(bx^3+k \dot x+ax−F\cos(t))=0$ bifurcation analysis
For the forced Duffing oscillator in the limit where the forcing, detuning, damping, and nonlinearity are all weak:
$$\ddot x +x+\varepsilon(bx^3+k \dot x+ax−F\cos(t))=0$$
where $0<\varepsilon<&...
- 409