By definition a bifurcation is a point where a system undergoes a qualitative change in its structure, and a bifurcation diagram is intended to help identify such points. Bifurcations can be classified into two types: continuous (also called subtle or supercritical) and discontinuous (also called catastrophic or subcritical). Discontinuous bifurcations usually exhibit hysteresis. Those that don't are called "explosive" (Smale 1967). It is also possible that your system has multiple attractors, and that a fixed initial condition will cross a basin boundary as the boundary moves in response to the change in parameters. In that case, the location of the jump will depend on your chosen initial conditions. This problem is usually avoided by changing the parameter slowly without returning to the original initial conditions, in which case hysteresis is observed by comparing the bifurcation diagram for increasing values of the parameter with the one for decreasing values of the parameter. In the region where hysteresis occurs, you should find (at least) two coexisting attractors.

Several phenomena can arise in bifurcation diagrams. I would suggest you show a figure, in which you define accurately what you plot and show what you call a jump. It is generally better to be too explicit than not enough.

Typically jump bifurcations arise when you have multiple (locally) stable solutions and one of these solutions loses stability at a parameter value causing the trajectories to "jump" to the nearest attracting solution.

In your case, at 2.6 Hz it looks as though a steady-state solution suddenly becomes unstable and then the trajectories jump to a stable two-cycle. I would conjecture that if you ran the bifurcation diagram generator again, but now start around 2.9 and run the parameter backwards (decreasing down to 1 or 0) you will find that this jump happens at a slightly different value. In particular, this jump would happen (probably) when the 2-cycle becomes unstable and the trajectories would then be attracted by the steady-state.

In the case of the second jump, between the values 2.6 and 2.9 it looks as though the lower branch undergoes a saddle-node bifurcation in which two steady-states collide and annihilate each other. This causes the trajectories to jump to another attracting steady-state which undergoes a series of period-doubling bifurcations into chaos. Again, try running things backwards and see if the parameter in which this jump happens at is a slightly different value, which would confirm my suspicions.

This phenomenon may be called "hysteresis". Jumpings depend on the direction of the parameter. See pictures. In general bifurcation diagram ranges of parameters, where two stable periodic solutions coexist, are visible.

Similar phenomena are described in the book:

Book Детерминированный хаос в нелинейных цепях с электрической ду...

By definition a bifurcation is a point where a system undergoes a qualitative change in its structure, and a bifurcation diagram is intended to help identify such points. Bifurcations can be classified into two types: continuous (also called subtle or supercritical) and discontinuous (also called catastrophic or subcritical). Discontinuous bifurcations usually exhibit hysteresis. Those that don't are called "explosive" (Smale 1967). It is also possible that your system has multiple attractors, and that a fixed initial condition will cross a basin boundary as the boundary moves in response to the change in parameters. In that case, the location of the jump will depend on your chosen initial conditions. This problem is usually avoided by changing the parameter slowly without returning to the original initial conditions, in which case hysteresis is observed by comparing the bifurcation diagram for increasing values of the parameter with the one for decreasing values of the parameter. In the region where hysteresis occurs, you should find (at least) two coexisting attractors.

dear Mohammadreza Atarodi

the bifurcation diagram that you are showing comes from a smooth system or a piecewise smooth one ?  "jumps" as observed there may be due to "border collision bifurcations" occurring in piecewise smooth systems

Hi Mohammadreza

I only across this question now so perhaps you have an answer already. I agree with Julien Sprott about changing the parameter slowly which is changing the shape of  the bifurcation from a pitchfork shape to a more abrupt shape. However, there is another phenomenon that I came across last year at a conference and its called stability loss delay (SLD). Here is an extract from a paper I submitted in 2015:

A Hopf bifurcation produces oscillatory behaviour from a steady state where the eigenvalues have zero value real parts but complex conjugate imaginary parts. The pitchfork shape at each bifurcation point had "tangential distortion" , or a jump, as you call it, for parameter variation using a ramp generator which was aggravated by the slope of the ramp generator signal. However, this was due to stability loss delay , a phenomenon referred to in a paper by Anatoly Neyshtadt . Non-trivial SLD was first observed in the work of M.A.Shishkova under the supervision of Lev Pontryagin and is an important phenomenon still not fully investigated. The behaviour of systems depends on a range of parameter values close to a critical bifurcation

point; where the bifurcation parameter changes slowly with time, and when a parameter value passes through a value which would normally cause a bifurcation, then the system delays the bifurcation, hence the name stability loss delay. At the point where it bifurcates, it does so quite abruptly, heading off at a tangent rather than the more gentle pitchfork bifurcation shape normally observed. For example, in the logistic equation system, the bifurcation occurs for growth values R = 3, provided time is suspended during parameter change (e.g. as in a two loop iteration in Matlab, with the inner loop representing the growth parameter change). However, in a real system where you change parameter as a slope, then the bifurcation is delayed and occurs later than R= 3 and the bifurcation shape is changed.  You can overcome this using a step change in the parameter.  Hope this helps.

Paul Tobin

I am agree with Prof. Sprott. But for simple situation you investigate bifurcation in pendulum equation, in damped case you have first type of bifurcation, in zero damped case, limit cycle start with non-zero frequency. Example of first case is saddle-node and second case is Hopf bifurcation. Some properties of your diagram is somehow confusing. Look at works on Josephson junction for physical and dynamical meaning of jump, one of them is hysteresis. 

Can you identify dynamical dimension of your system for more accurate analysis?