Bifurcation

“ The world, in short, is just as CONTINUOUS WITH ITSELF for the believers in free will as for the rigorous determinists, only the latter are unable to believe in points of bifurcation as spots of really indifferent equilibrium or as containing shunts which there—and there only, NOT BEFORE—direct existing motions without altering their amount. ”

“ An increase in the difference of the two external inputs tilts the potential in favor of the population with the greatest input, an effect also seen in linear, connectionist models, e.g. [11] . Modulations of the input common to both populations affect the curvature of the potential. For common inputs below the bifurcation, the potential exhibits a dimple, reflecting an attracting spontaneous state, while above the bifurcation the spontaneous state is repelling. This modulation of the potential via changes in the common input is a consequence of the nonlinearity of the system. ”

“ In nature it is of course only the stable types of motion which can persist for more than a short time. Thus the task of the physical evolutionist is to determine the forms of bifurcation, at which he must, as it were, change carriages in the evolutionary journey so as always to follow the stable route. ”

“ However, we must admit that the causality theory of nature has its strong suit. The reason why the bifurcation of nature is always creeping back into scientific philosophy is the extreme difficulty of exhibiting the perceived redness and warmth of the fire in one system of relations with the agitated molecules of carbon and oxygen, with the radiant energy from them, and with the various functionings of the material body. Unless we produce the all-embracing relations, we are faced with a bifurcated nature ”

“ The degree of stability at any epoch will depend on the fitness of some leading feature of the government to suit the slowly altering circumstances, and that feature corresponds to the characteristic denoted by a in the physical problem. A time at length arrives when the stability vanishes, and the slightest shock will overturn the government. At this stage we have reached the crisis of a point of bifurcation, and there will then be some circumstance, apparently quite insignificant and almost unnoticed, which is such as to prevent the occurrence of anarchy. ”

“ Strong recurrent excitation therefore naturally leads to a subcritical bifurcation and hence hysteresis. Network models describing the activity of neuronal populations in two-choice decision-making tasks have made use of this working-memory property which allows for a decision to be made and then ‘held in mind’, e.g. ”

“ Decline and reversal.—We have seen that under the continuous action of light, response begins to decline. Sometimes this process is very rapid, and in any case, under continued light, the deflection falls.
(1) The decline may nearly reach zero. If now the light be cut off there is a rebound towards recovery downwards, which carries it below zero, followed by an after-oscillation (fig. 108, a) .
(2) If the light be continued for a longer time, the decline goes on even below zero; that is to say, the response now becomes apparently negative. ”

“ Hence, if the surfaces of discontinuity are constrained to pass through corresponding fixed lines in the surfaces of discontinuity belonging to the varied state just mentioned, there must be a state of stable equilibrium for the system thus constrained which will differ infinitely little from the given state of the system, the stability of which is in question, and will have the same entropy, quantities of components, and exterior, but less energy. ”

“ Machines lay broken, tilted, flickering, whining, wheezing, like the bodies of the wounded. Delicate instruments were smashed to bits. The involuted field that Pud had flung through the vortex had evidently burst, as he had feared—for the vortex had vanished. So, probably, had the universe the field had burst in. The two fields that had interlocked were ruined, each having contaminated the other beyond use. Other energy-fields, having absorbed an excess of energy from the tharn, were bloated monstrosities or burned-out husks. ”

“ As soon as this equilibrium is lost, the balance is disturbed, and one of these forces gets stronger than the other, motion sets in, and creation begins. This state of things goes on cyclically, periodically. That is to say, there is a period of disturbance of the balance, when forces begin to combine and recombine, and things project outwards. At the same time, everything has a tendency to go back to the primal state of equilibrium, and the time comes when that total annihilation of all manifestation is reached. ”

“ For if the accelerative force, acting in parallel directions and equally, causes no perturbation in the motions of the parts of the system, it must of course, when it acts unequally, cause a perturbation somewhere, which will be greater or less as the inequality is greater or less. The excess of the greater impulses acting upon some bodies, and not acting upon others, must necessarily change their situation among themselves. And this perturbation, added to the perturbation arising from the inequality and inclination of the lines, makes the whole perturbation greater. ”

“ Now it is a general law of frictionless oscillation that, if a slowly varying periodic force acts on a system which would oscillate quickly if left to itself, the maximum excursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction of the excursion; but, if a quickly varying periodic force acts on a system which would oscillate slowly if left to itself, the maximum excursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction opposite to that of the excursion. ”

“ An enormous avalanche begins by the displacement of one atom, and the conflagration of a town by the fall of a match. Almost everything comes from almost nothing, one might think. It is only the first crystallization which is the affair of mind; the ultimate aggregation is the affair of mass, of attraction, of acquired momentum, of mechanical acceleration. History, like nature, illustrates for us the application of the law of inertia and agglomeration which is put lightly in the proverb, “Nothing succeeds like success.” ”

“ If a more considerable derangement were produced by any accidental disturbance, the consequence would be that the supplementary mechanism would cause the whole system to become locked, so that not a wheel would be capable of moving; the impelling power would necessarily lose all its energy, and the machine would stop. ”

“ This pink phosphorescence (Saunders assumed his lecture pose) is a new world forming, a twin world to Earth, speeding, crashing through space with mighty velocity, erratically circling this globe in continually narrowing rings to sure disaster. The stellar twins cannot escape collision, then—either we absorb or it absorbs. The Infinite forms a mighty barrier against which all matter swirls grinding void. Should Earth be the under world it forms the barrier impeding the mad rush of that splendid pale mystery which, as it approaches, absorbs all life, pulverizing this globe. ”

“ It is sufficient that the forces of attraction and expansion are two general, real, and fixed effects, for us to receive them for causes of particular ones; and impulsion is one of these effects, which we must not look upon as a general cause, known and demonstrated by our senses, since we have proved that this force of impulsion cannot exist nor act, but by the means of attraction, which does not fall upon our senses. Nothing is more evident, nay, certain, than the communication of motion by impulsion; it is sufficient for one body to strike another to produce this effect. ”

“ It takes some time for a disturbance to travel from one point to another, and its intensity may undergo a diminution as it recedes further from its point of origin. Suppose a disturbance originated at C; if two points are taken near each other, as A and B, the disturbance will reach them almost at the same time, and with the same intensity. The electric disturbance will be the same in both. The effect produced at A and B will balance each other and there will be no resultant current. ”

“ The smaller we suppose the increment of time δt, the greater must be the impressed forces, but the time-integral, or impulse, of each force will remain finite. The limiting value of the impulse, when the time is diminished and ultimately vanishes, is defined as the instantaneous impulse, and the momentum p, corresponding to any variable q, is defined as the impulse corresponding to that variable, when the system is brought instantaneously from a state of rest into the given state of motion. ”

“ We may apply similar principles to the consideration of the earth-moon system. It is true that we have here a series of dynamical phenomena, while the illustrations I have given of stable and unstable equilibrium relate only to statical problems; but we can have dynamical stability and dynamical instability, just as we can have stable and unstable equilibrium. Dynamical instability corresponds with the maximum of energy, and dynamical stability to the minimum of energy. ”

“ A system is called practically unstable when a very small (not necessarily indefinitely small) disturbance or variation in its condition will produce a considerable change. In the former part of this paper, in which the influence of surfaces of discontinuity was neglected, a system was regarded as practically unstable when such a result would be produced by a disturbance of the same order of magnitude as the quantities relating to surfaces of discontinuity which were neglected. ”

“ The oscillation produced by a single break is quite sufficient for a single experiment, and it is mere waste to have a series of useless oscillations. But the most serious objection to the production of secondary sparks, unless absolutely wanted, is their deteriorating action on the spark balls. Anyone who has tried to obtain an oscillatory discharge knows how easily the discharge becomes irregular, and the most fruitful source of trouble is often traced to the disintegration of the sparking surface. ”

“ Boehme calls them harshness, attraction, and anguish, which in modern terms are contraction, expansion, and rotation. The first two are in deadly antagonism, and being forced into collision, form an endless whirl of movement. These two forces with their resultant effect are to be found all through manifested nature, within man and without, and are called by different names: good, evil and life, God, the devil and the world, homogeneity, heterogeneity, strain, or the three laws of motion, centripetal and centrifugal force, resulting in rotation. ”

“ Where indeed in this crowded, topsy-turvey world, with its headlong changes and helter-skelter readjustments, its new tolerances and indifferences and accommodations, was there room for a character fashioned by slower sterner processes and a life broken under their inexorable pressure? And then, in a flash, she viewed the chaos from a new angle, and order seemed to move upon the void. If the old processes were changed, her case was changed with them; she, too, was a part of the general readjustment, a tiny fragment of the new pattern worked out in bolder freer harmonies. ”

“ Like an elastic and mobile object which, thrown into the air, attains by degrees the summit of its course, then possesses a zero velocity and is for a moment in equilibrium, and then falls on touching the ground to rebound, so the world should be subjected to huge oscillations which first bring it to a maximum of entropy till the moment when there should be produced a slow evolution in the contrary direction bringing it back to the state from which it started. ”

“ We have to consider what features, connected with the evolution, may be attributed to the operation of these dynamical laws. We have, in fact, to deduce the consequences which seem to follow from the fact that, in consequence of collisions, and in consequence of friction, an isolated system in space must gradually part with its initial store of energy, but that, notwithstanding any collisions and any friction, the total moment of momentum of the system suffers no abatement. ”

“ And let no one say that philoneism and progress are found as proportional reaction to misoneic action, an oscillation of the pendulum excluding the law of inertia. The pendulum itself does not oscillate, but remains perpetually motionless until moved; and its oscillations, even the smallest, are produced the most often by external causes entirely accidental. And the law of inertia is here also so constant, that if it did not find in the friction of the atmosphere a cause of impediment the movement once begun would continue ad infinitum. So a ball flies and rebounds, when a force propels it ”

“ What we term inertia, or the passiveness of matter, is but the equilibrium of opposing forces; and when this equilibrium is destroyed, as for example in chemical changes, motion and activity become visible, and continue until the equilibrium be restored. ”

“ The delay thus gained might not be more than a second in duration—but that second may be critical; for in the constant rising and falling of considerations in the mind, where two associated systems of them are nearly in equilibrium it is often a matter of but a second more or less of attention at the outset, whether one system shall gain force to occupy the field and develop itself, and exclude the other, or be excluded itself by the other. ”

“ If any analogy whatever existed between the human mind, on one side, and the laws of motion, on the other, the mind had already entered a field of attraction so violent that it must immediately pass beyond, into new equilibrium, like the Comet of Newton, to suffer dissipation altogether, like meteoroids in the earth’s atmosphere. ”